Research ArticleImpact of Resource Blocks Allocation Strategies onDownlink Interference and SIR Distributions in LTE NetworksA Stochastic Geometry Approach
Anthony Busson 1 and Iyad Lahsen-Cherif2
1 INRIA CNRS UMR 5668 LIP University Lyon 1 France2CNRS UMR 8623 LRI University Paris Saclay France
Correspondence should be addressed to Anthony Busson anthonybussoninriafr
Received 26 January 2018 Revised 5 May 2018 Accepted 28 May 2018 Published 26 June 2018
Academic Editor Natalia Y Ermolova
Copyright copy 2018 Anthony Busson and Iyad Lahsen-CherifThis is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
We propose a model based on stochastic geometry to assess downlink interference and signal over interference ratio (SIR) in LTEnetworksThe originality of thiswork lies in the proposition and combination of resource blocks assignment strategies transmissionpower control and realistic traffic patterns into a stochastic geometry model For this model we compute the first two moments ofinterference They are used to parameterize its distribution from which we deduce the SIR distribution Outage and transmissionrates (modulation and coding rate) are then derived to evaluate the system performance Simulations that cover a large set ofscenarios show the accuracy of our proposal and allow us to compare these strategies withmore complex ones that aim tominimizeglobal interference Numerical evaluations highlight the behavior of the LTE network for different traffic patternsload eNodeBdensity and amount of resource blocks and offer insights about possible parameterization of LTE networks
1 Introduction
The amount of mobile data that cellular networks mustcarry is continuously increasing The capacity of the wirelesssystems must continuously increase in order to satisfy thegrowing demand of traffic from users and applicationsLong-Term Evolution and Long-Term Evolution-Advanced[1] (LTE-A) have been recently standardized to improve thenetwork capacity and support this traffic growth One ofthe solutions brought by LTE is the enhancement of theradio spectrum reuse The smallest radio resource that canbe allocated to a user is a resource block (RB) An RB is achannel (an OFDMA channel composed of a set of OFDMsubcarriers) for the duration of one time slot Consideringthe number of RB is finite they are reused in different cellsgenerating potential intercell interference The algorithmsthat assign RB to users located in different cells have thusan important role in the system performance A static RBassignment where disjoint resources are distributed to eachcell may lead to an inefficient resource usage as the unused RB
in a cell cannot be reused in another one Instead algorithmsthat assign RB can be centralized in a schedulercontrollerthat controls a certain number of neighborhood cells andadapts the RB assignments to the cells load Also it mayimprove the spatial reuse while ensuring a low level ofinterference
Several studies have proposed assignment strategies per-formed at the scheduler to minimize global interference[2 3] These strategies aim to minimize interference for agiven configuration and are evaluated exclusively throughsimulations However the assignment strategies have to beevaluated for more general scenarios and at larger scale
Stochastic geometry offers a powerful tool to analyze largescale networks through a few parameters and to understandthe role of these key parameters on the whole system Theother benefit of stochastic geometry is to consider realisticBase stations (BS) or eNodeB (evolved Node B) locationsIt uses random point processes rather than deterministic(grid or hexagonal patterns for instance) or predeterminedlocations of BSeNodeB For instance the Poisson Point
HindawiWireless Communications and Mobile ComputingVolume 2018 Article ID 9163783 15 pageshttpsdoiorg10115520189163783
2 Wireless Communications and Mobile Computing
Process (PPP) has been shown to be suitable to model thespatial location of BS [4ndash6] Nevertheless interference asexperienced by a user is not generated by all BS but onlyby the ones using the same radio resources The resourceallocation strategies have thus to be mapped to the pointprocess modeling BSeNodeB to determine which pointsBSare interfering with a given communication Consequentlythe traffic demand must also be taken into account as it setsthe number of resources used at a given time
In this work we propound a combination of severalassignment strategies realistic traffic demands and transmis-sion power control mechanisms into a stochastic geometrymodel We begin by reviewing related works and our contri-butions
11 Related Work In a downlink LTE system a resourceblock (RB) is the smallest radio resource unit that can beallocated to a user The LTE system has to schedule andassign RB to users as a function of the link qualities trafficdemands and potential quality of service requirements Inthis paper we focus on a system where a controller assignsRB for a set of eNodeB We do not overview RB assign-ment techniques in LTE network as they aim to optimizeRB assignments and modulationcoding rates for a giventopology and a traffic demand Instead this paper deals withthemacroscopic design of the network the impact of eNodeBdensity allocation scheme and power allocation on the globalperformance of a downlink LTE system Nevertheless thereader can refer to [7 8] for recent surveys Also interestingcontributions on the optimization of the downlink system fora given configuration are described in [3 9ndash12]
Stochastic geometry has emerged as an efficient toolto analyze the performance of cellular networks It offersthrough simple models a way to study wireless architec-tures at a large scale Recent surveys [13 14] summarizethe numerous wireless architectures and models for whichstochastic geometry has been applied One of the maindifficulties in the analysis of large wireless systems is tocharacterize interference This quantity does not dependonly on BS location and radio environment (path lossshadowingfading etc) but also on the way that radioresources (time frequency and power) are allocated Thepoint process modeling interfering nodes is thus of crucialimportance The PPP offers an accurate model to describeBS location [4ndash6] This process is tractable and it is possibleto derive closed formulas for some key performance metricsof the system interference coverage outage Signal overInterference plusNoise Ratio (SINR) etc But the PPPmodelsall BSeNodeB and not the subset of interfering eNodeB for agiven communication The process has thus to be thinned totake into account interference coordination (IC) techniquesand radio resources assignment for example leading toprocesses that are nomore Poisson In the next paragraph wefocus on recent contributions and on studies where resourcesallocation and more generally IC techniques are taken intoaccount
IC refers to techniques that aim to mitigate interferenceat the receivers Surveys on such techniques can be found in[2 15] A common IC approach consists in controlling the
allocated radio resources (frequencytimepower) in orderto alleviate the interference impact on communications In[16] the authors consider a random resources allocationstrategy where the BS are distributed as a PPP This simpleand tractable strategy allows model interfering BS as anindependent thinning of a PPP and deriving closed formulasfor the coverage probability They also deduce the minimalreuse factor achieving a given coverage probability Theperformance of strict FFR (Fractional Frequency Reuse)and SFR (Soft FFR) allocation strategies is evaluated usingstochastic geometry in [17] With these two techniquesdifferent radio resources are allocated to users that are at theedge of a cell (Voronoı cells here) with regard to the onesclose to the BS The criteria distinguishing core and edgeusers is based on the SINR at each user computed from theunderlying PPP modeling all BS For strict FFR the radioresources used at the edge and in the core are disjoint Insteadthe radio resources may be reused between the two regionsfor SFR For these two strategies the authors derive closedformulas for the coverage probability and discuss pros andcons of these approaches A superior interference reductionis observed for FFR but SFR benefits from a greater resourceefficiency This work is generalized in [18 19] to the contextof K-tier and heterogeneous networks considering differentpoint processes for each tier or network technology It isalso extended and studied in [20] with the dynamic strictFFR (DSFFR) where the edges of the cells are dynamicallydivided into sectors with the help of directional antennas In[21] a coordinated beamforming is employed to ensure thata set of closed BS ldquoa clusterrdquo will use different resources Auser associated with a BS is then not subject to interferencefrom BS belonging to the same cluster The authors deriveanalytical expressions for the Signal over Interference Ratio(SIR) for this strategy and discuss the impact of the clusterscardinality A similar approach is used in [22] where theset of coordinated BS corresponds to the most interferingones Interference level takes into account path loss andlong-term shadowing The interfering BS are outside this setThey are selected randomly and independently leading to athinned PPP For this model the authors study the coverageprobability for different scenarios In [23] a user is servedby its 1 or 2-closest BS according to the position of these BSwith regard to the user When the two BS are coordinatedthe transmission power is split into the two transmissionsThe total transmission power is thus the same with one ortwo coordinated BS Interference is generated by the otherBS without restriction which is assumed to be distributedthrough a PPP The authors derive a closed-form expressionfor the SIR distribution and the network coverage probabilityand discuss the benefit of this approach In [24] an ICtechnique is evaluated for a user at the edge of its cell Whenthe resource of this communication is used by neighboringcells they may not transmit any signal for a certain period tomitigate interference at this userThis coordination techniqueis analytically evaluated assuming that interfering nodes arestill distributed as a PPP
Besides the modeling of IC [22 25ndash27] propose spatialand tractablemodels that take into account the trafficdemandin the interference computation but they do not consider
Wireless Communications and Mobile Computing 3
concrete RB assignment algorithms In particular the authorsin [26] study SIR coverage for a cellular network based onPPP A queue is associated with each BS that determinesthe BS transmission activity as a function of the trafficConsidering the traffic at each BS is independent interferersat a given time are then an independent thinning of the initialPPP and are still Poisson This model differs with this paperas we do not take into account eNodeB activity as a functionof the traffic but instead the resource allocation as a functionof the number of associated users to each eNodeB Alsostochastic geometry models can be specific to certain powercontrol scheme [28] or radio technologies as in [29] wherethe authors consider a K-tier heterogeneous network withtransmissions operating on the millimeter wave band
12 Contributions The primary contribution of this work isto offer an analytical model based on stochastic geometryto evaluate the performance of a downlink LTE systemtaking into account RB allocation strategies power controland traffic demands All these mechanisms have never beencombined into a single stochastic geometry model Thenumber of allocated resources for an eNodeB is assumedto follow the distribution of the number of clients in anMMCC queue It models the number of communicationsin progress when both the interarrival of the communicationsand their duration follow an exponential distribution Suchassumptions are pertinent in cellular networks as it has beenrecently shown in [30] We associate to these traffic demandsseveral resource allocation strategies All these algorithms arecombined with a power control mechanism that depends onthe channels quality Allocation strategies lead to non-PPP ascorrelation appears between the locations of the interferingnodes It prevents the use of the convenient properties ofthe PPP to compute interference distribution Neverthelesswe propose approximations that allow us to deal with thesecorrelations and to obtain an analytical method that is shownvery accurate with regard to simulations
We compare our model to classical optimizationapproaches where for a given configurationsample theallocation is optimized with regard to an objective functionTo our knowledge such comparison has never been donebefore It shows that geometry stochastic based model maybe relevant to offer tight approximations on wireless systemperformance
Models are evaluated through a large set of simulationsthat highlights benefits of our approach to design some keyparameters of the wireless system Results show that theobtained values for SIR coding and modulation rates corre-spond to the reference values of the standards and technicalLTE documents empirically proving that our model is able toapproximate performance of real systemsThiswork has beenpartially presented in [31]
13 Paper Organization The remainder of this work is orga-nized as follows In the next section we present the systemmodel We expose the assignment strategies in Section 3In Section 4 we derive the first and second moments ofinterference for each allocation strategy SIR distribution isassessed in Section 5 Numerical results and simulations are
presented and discussed in Section 6 We conclude the paperin Section 7
2 System Model
21 eNodeB Location and Interference eNodeB location ismodeled by a point process 119873119890 = 119883119894119894isinN distributed in R2
with intensity 120582119890 Its distribution is detailed in Section 22The eNodeB are numbered with regard to their distanceto the origin eNodeB 0 at 1198830 being the closest one Weconsider a downlink system between a typical user and itsattached eNodeB Without loss of generality we assume thatthis user is located at the origin The users are assumed tobe associated with their closest eNodeB with regard to theEuclidean distance The channels between eNodeB and thetypical user are modeled through a sequence of iid randomvariables (ℎ119894)119894isinNThe transmission power between an eNodeBand one associated user is given by the function 119879119909(119903) Thisfunction models the power control algorithm implementedby the eNodeB and depends on the distance 119903 between a userand its attached eNodeBThe schedulercontroller manages aset of 119877119861119898119886119909 resource blocks that are common to all eNodeBThey are thus shared between eNodeBThe RB are numberedfrom 0 to 119877119861119898119886119909 minus 1 The scheduler assigns one RB for eachuser but the model can be easily extended with a randomnumber of RB for each demand Traffic demands exceeding119877119861119898119886119909 at an eNodeB are not served The RB with index 0 isallocated to the typical user We show in the appendix thatthis choice does not impact the computations and any otherindex could be chosen instead An eNodeB interferes with thetypical user if and only if it reuses this RB Interference at thetypical user can be expressed as
where 119897(sdot) is the path loss function The argument of inter-ference is the distance between the typical user and itseNodeB (eNodeB 0) 119868(1198830) expresses interference whenthis distance is random and depends on the rv 1198830 119868(119903)expresses interference when this distance is given and equalto 119903 This notation is motivated by the fact that the rv 119883119894
and 119883119894 minus 119880119894 are correlated to 1198830 Moreover the mean andthe variance of interference will be computed for both a givenvalue of 1198830 and with regard to its distribution The rv 119908119894
indicates whether eNodeB 119894 interferes with the typical user(119908119894 = 0 or 1)119880119894 is the random variable modeling the locationof a user attached to the eNodeB 119894 (at 119883119894) We assume that119880119894 is uniformly distributed in the Voronoı cell formed by theprocess 119873119890 and with nucleus 119883119894 The main notations usedthroughout this paper are given in Table 1
22 Point Process Modeling eNodeB The point process mod-eling eNodeB is a modified homogeneous PPP The process119873119890 1198830 = 119883119894119894gt0 is Poisson in R2 119861(0 1198830) where119861(0 1198830) is the ball centered at the origin and with radius1198830 We choose a distribution for 1198830 that makes theprocess119873119890 different of a PPP Indeed with a PPP the typicaluser at the origin lies in a Voronoı cell that is greater in
4 Wireless Communications and Mobile Computing
Table 1 Principal notation
119873119890 point process modeling eNodeB119868(1198830) interference at the typical user The distance between the typical user and eNodeB 0 is the rv 1198830119868(119903) interference at the typical user The distance between the typical user and eNodeB 0 is equal to 119903119908119894 rv indicating if eNodeB 119894 interferes with the typical user119863119894 number of RB allocated to eNodeB 119894119879119909(119903) transmission power between an eNodeB and its user at distance 119903Δ spatial reuse parameter (an RB is reused every Δ eNodeB in average)119891119883119894(sdot) PDF of 119883119894119891119883119894119883119895(sdot sdot) joint PDF of (119883119894 119883119895)1198911198941198830=119903
(sdot) conditional PDF of 119883119894 given that 1198830 = 119903119891119894119895
1198830=119903(sdot) conditional PDF of (119883119894 119883119895) given that X0 = 119903
11989101198830
(sdot) PDF of the distance between the typical user (at the origin) and its serving eNodeB 01198910119880119894minus119883119894
(sdot sdot) conditional PDF of the distance between a user and its attached nucleus119883119894 given 119883119894
average than the other cells Intuitively ldquobig cellsrdquo cover morespace than ldquosmall cellsrdquo and consequently the cell coveringthe origin has a greater size in average It is consistent witha modeling where users are homogeneously scattered in theplane but not with our assumption where the network hasbeen dimensioned to have the same load in average (the samenumber of users) in each cell It is more realistic as it hasbeen shown in [30] where a homogeneous load is observedfor the different cells independently of their sizes In thiscase the typical cell covering the typical user must have thesame distribution as the other cells Therefore we considerthe distribution of the distance 1198830 under Palm measureMore precisely this distribution corresponds to the distancebetween the nucleus of a typical cell under Palmmeasure anda point uniformly distributed in this cell The distributionof this distance is not known but we use the approximationpresented in [32] (page 133) We set the distribution of thedistance between the typical user and its closest eNodeB at1198830 as
with 119888 = 125 The angle between the lines (01198830) and theabscissa is uniformly distributed in [0 2120587)
The distribution of the distance between a user in agiven Voronoı cell and its nucleus (119883119894 minus 119880119894) follows thesame definition and consequently the same distribution Inthe next section we present the different RB assignmentstrategies evaluated in this paper
3 Assignment Strategies
We consider four different allocation strategies We beginby two simple RB allocation schemes independent andstatic allocations Then we develop a more realistic alloca-tion strategy using the 119872119872119877119861119898119886119909119877119861119898119886119909 queue named119872119872119877119861119898119886119909119877119861119898119886119909 allocation hereafter Also amore globalapproach where the RB are assigned in order to minimizethe sum of interference at each user is considered but for
which we do not propose a mathematical resolution All theallocation strategies are set in such a way that a given RB isreused every Δ (Δ ge 1) eNodeB in average The mean loadin each cell and equivalently the mean number of RB used byan eNodeB are then 119877119861119898119886119909Δ The reuse factor Δ reflects thenetwork load
31 Independent Allocation Thinning With this strategyeach eNodeB selects its resources independently of theother eNodeB Therefore we assume that an eNodeB hasa probability 1Δ to reuse the RB with index 0 The pointprocess describing the interfering eNodeB is then a thinnedPPP in R2 119861(119874 1198830) with intensity 120582119890Δ32 Static Allocation We assign a constant proportion119877119861119898119886119909Δ of the available resources to each eNodeBThey areallocated in their index order eNodeB 0 uses RB from 0 to119877119861119898119886119909Δ minus 1 (it includes the typical user) eNodeB 1 from119877119861119898119886119909Δ to 2(119877119861119898119886119909Δ) minus 1 etc We take the integer partof these fractions when 119877119861119898119886119909 is not a multiple of Δ Weloop when all resources have been used Consequently theeNodeB interfering with the typical user has an index withthe form 119896 sdot Δ with 119896 gt 033119872119872119877119861119898119886119909119877119861119898119886119909Allocation In an119872119872119862119862queuecustomers arrive according to a Poisson process (in R) andthe service times are exponentially distributed It modelsa system with 119862 resourcesservers and a capacity of thesame size A customer cannot enter in the system if allresourcesservers are busy We associate with each eNodeBan independent 119872119872119877119861119898119886119909119877119861119898119886119909 queue to model thenumber of RB in use The servers model the RB Upon thearrival of a requestuser an RBserver is used for a timeexponentially distributed If no RB is available the requestis rejected In order to have a mean reuse factor of Δ theparameter of the queue (the load) denoted 120588 is set in sucha way that the mean number of customers in the system orequivalently the mean number of busy resource blocks isequal to 119877119861119898119886119909Δ The distribution of the number of busy
Wireless Communications and Mobile Computing 5
(a) RB allocation (b) Impact of RB allocation on interferer
Figure 1 In Figure 1(a) an example of RB allocation is given eNodeB 0 uses 6 RB (the first one has index 14 and the other ones are indexedfrom 0 to 4) It uses the RB with index 0 to communicate with the typical user1198770 is the index of the last resource used by eNodeB 0 As it usesRB indexed from 14 to 4 1198770 = 4 hereThe number of used RB for the other eNodeB are1198631 = 41198632 = 3 etc An eNodeB interferes if and onlyif it reuses the resource 0 in this example eNodeB 3 and 6 In Figure 1(b) we plot a sample of the point process describing eNodeB locationwith the same RB assignmentThe eNodeB interfering with the typical user is then a dependent thinning of the original point process wherethe thinning involves eNodes 3 6 and 10 (this last interfering eNodeB was not shown in Figure 1(a))Their Voronoı cells are colored in green
resource blocks for a given eNodeB 119894 (119894 gt 0) denoted 119863119894 isthen given by
P (119863119894 = 119896) = 1205870 120588119896119896 (3)
where 1205870 = P(119863119894 = 0) and 119877119861119898119886119909 ge 119896 ge 0These RB are allocated in a cyclic order If the last119877119861 used
by eNodeB 119894 minus 1 has index 119896 eNodeB 119894 uses 119877119861 indexed from(119896 + 1)119898119900119889(119877119861119898119886119909) to (119896 + 119863119894) 119898119900119889(119877119861119898119886119909)For eNodeB 0 we do not consider the total number
of allocated RB (1198630) but instead a random variable 1198770 Itdescribes the index of the last RB used by eNodeB 0 Indeedfor this particular eNodeB the quantity used in practice tocompute the next allocation (RB indexes used by eNodeB 1)is 1198770 rather than 1198630 A formal definition of 1198770 is given inappendix (Appendix A) An example of allocation is given inFigure 1
The distribution of 1198770 is set according to the stationarydistribution of a Markov chain The transition probabilitiesof this Markov chain are
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (5)
where (119897 119898) isin 0 1 119877119861119898119886119909 minus 12 The motivation forthis particular construction is to keep the probability ofusing the resource 0 between eNodeB homogeneous Moreprecisely it is built in order to verify the property given inProposition 1 Details about the Markov chain constructionare given in appendix (Appendix A) It is worth noting thatother distributions for the resource demands (given by (3)in our case) can be considered as well As soon as thedistribution of1198770 verifies Proposition 1 themethod proposedin this paper holds
34 Property of These Assignment Strategies We define moreprecisely the sequence of rv (119908119894)119894isinN It indicates whicheNodeB interferes with the typical user It was already usedin (1)
119908119894 = 1 if eNodeB at 119883119894 uses RB with index 00 otherwise (6)
By convention we set 1199080 = 1 as In the following weshall thus assume that P(119908119894 = 1 | 1199080 = 1) = P(119908119894 = 1)Proposition 1 For the three allocation strategies defined inSections 31 32 and 33 the following property holds
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (7)
The proofs for the first two strategies are straightforwardFor the 119872119872119877119861119898119886119909119877119861119898119886119909 strategy the distribution of 1198770has been set to verify this property (see Appendix A)
35 Heuristics We compare these strategies to heuristics thataim to optimize interference or spatial reuse for a givenconfiguration It allows us to compare our RB assignmentand performs in a cyclic manner around a typical user tostrategies where a controller in charge of a set of eNodeBwill assign RB in order to optimize a certain objective func-tion For the heuristic minimizing the sum of interferencenamed ldquominimize interferencerdquo hereafter the consideredoptimization problem is similar to the one developed in[3] The problem has been shown NP-hard so we use agreedy algorithm to find a solution The number of usersassociated with each eNodeB follows the same distribution asthe119872119872119877119861119898119886119909119877119861119898119886119909 allocation Then we consider usersin a random order and apply Algorithm 1 to associate an RBto a user It chooses the resource block that minimizes thesum of interference Obviously we compute interference only
6 Wireless Communications and Mobile Computing
Data 119877119861119898119886119909 Total number of RBcost sum of interference for the current allocationbestCost sum of interference for the best allocation strategy initialized to -1Result Assign an RB to a user This algorithm is called by the controller for each
new user(1) for each 119903119887 isin 0 1 119877119861119898119886119909 minus 1 do(2) if rb is free then(3) assign rb to this user(4) costlarr sum Interference()(5) if bestCost lt0 or costltbestCost then(6) Save this allocation strategy(7) bestCost = cost(8) end(9) end(10) end(11) Assign the saved allocation strategy
Algorithm 1 Minimizing global interference
for the users already assigned The typical user is consideredin last when the system has reached the targeted load Thisalgorithm mimics an assignment strategy where the RB areassigned at the arrival of the users request without changingthe already assigned RB
The second heuristic ldquomaximize reuse distancerdquo maxi-mizes the distance at which the RB are reused Each user isconsidered in a random order Different RB are assigned tothe 119877119861119898119886119909 first users When assigning an RB to the otherusers the controller chooses the RB for which the reusedistance is maximum The typical user is considered in lastThis second heuristic may correspond to a case where thecontroller does not have information on channel conditionsand interference but knows the distances between eNodeB
4 Interference Characterization
We derive the mean and the variance of interference for thethree assignment strategies defined in the previous section
The point process modeling interferers is a dependentthinning of the original PPP Consequently conditions formean (respectively variance) to be finite with a PPP also holdfor our point process the path loss function 119897(sdot)must belongto 1198711 (respectively 1198712)41 Distribution of Distances between the Typical User andeNodeB (119883119894) As a preamble we give the PDF of thedistance between the typical user at the origin and eNodeBBoth PDF of 119883119894 and joint distribution of (119883119894 119883119895) arederived These PDF are used in the computation of the meanand the variance of interference
In the numerical evaluation we shall condition interfer-ence by the distance 1198830 It allows us to study interferencefor a given distance between the typical user and its attachedeNodeB It is also motivated by the computation of the SIRwhere both interference and the typical user signal strengthdepend on the distance 1198830
For our model the PDF of 119883119894with 119894 gt 0 given 1198830 = 119903is
1198911198941198830=119903 (119906 119903) = (120582119890120587)119894(119894 minus 1)2119906 (1199062 minus 1199032)119894minus1 119890minus120582119890120587(1199062minus1199032)1119906gt119903 (8)
The joint PDF of (119883119894 119883119895)with 119895 gt 119894 gt 0 given 1198830 =119903 is1198911198941198951198830=119903
(119906 V 119903)= (120582119890120587)119895(119894 minus 1) (119895 minus 119894 minus 1)4119906V (V2 minus 1199062)
119895minus119894minus1
times (1199062 minus 1199032)119894minus1 119890minus120582119890120587(V2minus1199032)1Vgt119906gt119903(9)
To obtain the PDF when 1198830 is not set (119891119883119894(sdot) and119891119883119894119883119895(sdot)) it suffices to integrate the two conditional PDFwith regard to the PDF of 1198830 given in (2)
42 Mean of Interference Themean is derived from (1)
E [119868 (100381710038171003817100381711988301003817100381710038171003817)]
= E [ℎ1] +infinsum119894=1
E [119908119894]E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) sdot 119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)] (10)
In this equation 119908119894 has been separated from the expecta-tion as it is independent of the process 119873119890 (according to thedefined strategies) We derive E[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)] andE[119908119894] in the two next sections
421 Computation of E[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)]No Power Control In absence of power control ie when119879119909(sdot) is constant or independent of the process 119873119890 a closedformula may be expressed for (10) E[119879119909(119883119894 minus119880119894) sdot 119897(119883119894)]
Wireless Communications and Mobile Computing 7
is then given by E[119879119909]E[119897(119883119894)] Expectation of 119897(119883119894) isobtained from the distribution of 119883119894 given in Section 41
E [119868 (100381710038171003817100381711988301003817100381710038171003817)]
= E [ℎ1] +infinsum119894=1
E [119908119894]E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817)]E [119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)] (11)
Power Control When the transmission power depends onthe distance between the receiver and its attached eNodeB(119879119909(119883119894 minus 119880119894)) the computations are more complex119864[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)] cannot be approximated byE[119879119909(119883119894minus119880119894)]119864[119897(119883119894)] as the size of the Voronoı cell withnucleus 119883119894 depends on its distance to the origin The jointdistribution of (119883119894 119883119894 minus119880119894) being unknown we proposethe following approximation
The PDF of 119880119894 minus 119883119894 is the same as 1198800 minus 1198830 givenby (2) Its parameter 119888119894(119883119894) depends on 119883119894 119888119894(119883119894) =14radic120582119890(120574119894119883119894 + 119887) with 120574119894 = (12 minus 119887)E[119883119894] and 119887 = 033The motivation and the computation details for this PDF aregiven in appendix (Appendix B) We obtain
E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) sdot 119897 (1003817100381710038171003817119883119894
Often in real systems the transmission power cannot beset arbitrarily and is limited to a set of predetermined valuesThe transmission power function can then be representedas a step function 119879119909(119903) = sum119873119879
119894=1 1199051198941119903isin[120572119894minus1120572119894] where 119873119879
is the number of possible transmission powers 119905119894 the 119894119905ℎtransmission power value and [120572119894minus1 120572119894] the distance intervalbetween a user and its eNodeB at which this transmissionpower is used An example of such setting is given in thenumerical evaluation section In this case (14) becomes
E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) 119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)]= 119873119879sum
119894=1
119905119895 int+infin
0(119890minus120582119890119888119894(119903)1205871205722119895 minus 119890minus120582119890119888119894(119903)1205871205722119895minus1)119891119883119894 (119903) 119889119903
(15)
When the computation is performed for a given distance1198830 the PDF 119891119883119894(sdot) in (12) and (15) must be replaced by1198911198941198830=119903(sdot sdot) (given in Section 41)
422 Computation of E[119908119894] Finally in order to compute(10) we need to expressE[119908119894] First note that E[119908119894] = P(119908119894 =1)Proposition 2 The probability for eNodeB 119894 to interfere withthe typical user is given by
(i) Independent allocation
P (119908119894 = 1) = 1Δ (16)
(ii) Static allocation
P (119908119894 = 1) = 1119894sdot119898119900119889(Δ)=0 (17)
E [119879119909 (1003817100381710038171003817119880119894 minus 1198831198941003817100381710038171003817) 119879119909 (10038171003817100381710038171003817119880119895 minus 119883119895
As for the mean complexity lies in the correlationbetween 119883119894 and 119880119894 minus 119883119894 The term E[119879119909(119880119894 minus 119883119894)2 sdot119897(119883119894)2] is computed with the same method as the firstmoment
Computation of E[119879119909(119880119894minus119883119894)119879119909(119880119895minus119883119895)119897(119883119894)119897(119883119895)]As 119880119894 minus 119883119894 (respectively 119880119895 minus 119883119895) depends on 119883119894(respectively 119883119895) we condition by the distribution of(119883119894 119883119895) given in Section 41 Given 119883119894 and 119883119895 we usethe same PDF as in (B1) assuming that 119880119894 minus 119883119894 and 119880119895 minus119883119895 are independent The considered joint distribution of(119883119894 119883119895 119880119894 minus 119883119894 119880119895 minus 119883119895) becomes
1198910119880119894minus119883119894 (119906 119903) 1198910119880119895minus119883119895 (V 119904) 119891119883119894119883119895 (119903 119904) (21)
When the distance 1198830 is fixed 119891119883119894119883119895(sdot sdot) must bereplaced by the PDF 119891119894119895
1198830=119903(sdot sdot sdot) given in Section 41
Computation of E[119908119894119908119895] It has been shown that the sequence(119908119894)119894gt0 verifies (7) for the three strategies It allows us toexpress E[119908119894119908119895] with 119894 gt 119895 as
E [119908119894119908119895] = P (119908119894minus119895 = 1)P (119908119895 = 1) (22)
8 Wireless Communications and Mobile Computing
Proposition 3 The joint probability for two eNodeB 119894 and 119895(119895 gt 119894) to interfere with the typical user is given by
(i) Independent allocation
E [119908119894119908119895] = 1Δ2 (23)
(ii) Static allocation
E [119908119894119908119895] = 1119894sdot119898119900119889(Δ)=0119895sdot119898119900119889(Δ)=0 (24)
where P(119908 = 1) is given by (18)5 Signal over Interference Ratio (SIR)
In our model there is a strong correlation between theinterfering eNodeB It is generated by the allocation strategiesand cannot be neglected Also a correlation exists betweenthe location of an eNodeB and the size of its Voronoı cellConsequently classical approach based on PPP which uses
Laplacian transform for instance cannot be applied hereand a formal derivation of interference distribution seemsintractable
Nevertheless the different simulations presented in thenext section will show that the PDF of interference can beapproximated by a log-normal distribution The parametersof this distribution mean and variance denoted by 119898119868119889119861
(sdot)and 120590119868119889119861(sdot) are directly derived from the previous analyticalcomputations The classical mapping between log-normaland normal parameters can be applied to derive parametersof the normal distribution when interferences are expressedin decibel In the following a variable is indexed by 119889119861 whenit is expressed in decibel
We get
P (119878119868119877119889119861 le 120573119889119861) = P (10sdot log10 (119875119905 (10038171003817100381710038171198830
1003817100381710038171003817) ℎ0119897 (100381710038171003817100381711988301003817100381710038171003817)) minus 119868119889119861 (10038171003817100381710038171198830
1003817100381710038171003817) le 120573119889119861) (26)
= P (119868119889119861 (100381710038171003817100381711988301003817100381710038171003817) ge minus120573119889119861 + 10
When ℎ0 is not constant the expectation with regard toits distribution must be taken into account in (28) In (29)ℎ0 is assumed to be constant that equals to 1 In these twoequations 119890119903119891(sdot) is the error function6 Numerical Results
We consider an E-UTRA channel with a bandwidth of5MHz with 119877119861119898119886119909 = 15 [33] The path loss function isthe same as [3] It is expressed in dB 119897(119903) = minus1281 minus376 sdot log10(119903) where 119903 is the distance (in km) 119879119909(sdot) isset in such a way to guarantee to each user a minimumreceiving power We set the transmission power function119879119909(sdot) to ensure a signal power greater than or equal tominus724dBm at the reception as specified in [33] For each 50meters (from 50 to 500 meters) we compute the minimumtransmitting power required to reach this threshold (119879119909(119903) sdot119897(119903) ge minus724 dBm for each interval of 50 meters leading to 10possible transmission powers)This step function models thecase where eNodeB has a finite set of predetermined powerThe process intensity modeling eNodeB is equal to 225 perkm2 It corresponds to the intensity of base stations in Paris(httpswwwantennesmobilesfr) Random variables ℎ119894 are
supposed constant equal to 1 This assumption facilitatesinterpretation of the results but any distribution can beconsidered as well It simply adds a factor in terms of variance(cf (20)) We simulate the different strategies through asimulator coded inC available here (httpwwwanthonybus-sonfrindexphppublications) In all simulations and nu-merical results we consider 50 eNodeB The different sumsin the equations (eg (10) and (20)) are then limited to 50For each set of parameters simulations have be run from10000 to several millions times depending on the evaluatedquantities The number of simulationssamples has been setin order to have negligible confidence intervals They areconsequently not shown in the different figures
Mean and Variance of Interference In Figure 2 we plot themean and the standard deviation of interference obtainedfrom simulations and computed from formulas (10) (19) and(20)when the distance 1198830 variesThe theoretical evaluationcloselymatches empirical estimators obtained by simulationsAs expected the highest interference level is observed forthe independent allocation and the lowest level for the staticallocation The static and the 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tions offer equivalent results with a multiplication factor
Figure 2 Interferencemean and standard deviation as a function of the distance between the typical user and its eNodeB (1198830) Simulationresults are given by points and solid lines and theoretical evaluations by the dotted lines
Table 2 Mean and standarddeviation for the119872119872119877119861119898119886119909119877119861m119886119909 allocation and the two heuristics
Assignment strategy Δ = 3 Δ = 6 Δ = 9mean std-dev mean std-dev mean std-dev
119872119872119877119861119898119886119909119877119861119898119886119909 178119890 minus 11 861119890 minus 11 366119890 minus 12 750119890 minus 12 155119890 minus 12 316119890 minus 12Minimize Interference 994119890 minus 12 307119890 minus 11 186119890 minus 12 188119890 minus 12 756119890 minus 13 626119890 minus 13Maximize reuse distance 526119890 minus 11 751119890 minus 10 118119890 minus 11 107119890 minus 10 258119890 minus 12 109119890 minus 11
varying from 11 to 87 for the mean interference Insteadthe independent allocation differs deeply Mean and standarddeviation are multiplied by a factor ranging from 17 to 700for the mean and can reach up to 26 times 104 for the standarddeviation (for 1198830 = 002 km) The use of the thinnedpoint process is thus questionable to approximate realisticassignment strategies This strategy is no more considered inthe following
Extrapolation of Interference Distribution We compareempirical distributions obtained from simulations to knowndistributions for the different strategies and the two heuristicswhen Δ = 3 6 and 9 Three of these empirical distributionsare shown in Figure 3 (in dB) The distribution parametershave been set according to the maximum likelihood Thedistributions that best fit simulations vary according to theallocation strategy and the reuse factorΔ For 4 cases over 12the distribution minimizing error is the normal distributionAs soon as an asymmetry is observed other distributionsare more accurate log-normal Weibull and Inverse GammaWe perform two hypothesis tests T-test and Smirnov-Kolmogorov for all these scenarios The alternative hypoth-esis is systematically ruled out For the 119872119872119877119861119898119886119909119877119861119898119886119909
strategy best fits are given by the normal (Δ = 3 and 6) andWeibull distributions (Δ = 9)
Nevertheless as mentioned in Section 5 when estimatingthe SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy weuse the normal law to model interference Indeed even if thisdistribution is not systematically the most accurate it offers agood approximation as shown inFigure 4 where the empiricaland normalCDF are plottedMoreover simulationswill showthat this assumption does not impact the accuracy of theanalytical model
Heuristics Simulations have shown that allocations resultingfrom the two heuristics correlate the transmission powersthe distance between eNodeB reusing the same resourceand the random variables 119908119894 Whenminimizing interferencecorrelations are caused by resource blocks allocated to usersthat require a low transmission power and that can be reusedat a short distance and inversely The transmission poweris then correlated to 119908119894 and to the distance of eNodeB thatreuse the same resource These complex phenomena impedethe proposal of an analytical model for these two heuristicsFor comparison purposes Table 2 reportsmean and standarddeviation for the two heuristics and the119872119872119877119861119898119886119909119877119861119898119886119909
Figure 3 Interference distribution empirical (histogram) and extrapolation (curve) Among the classical distributions the bestextrapolations are Gamma (m=1003 var=977) normal (m=2823 var=2216) and Weibull (a=2514b=575) for the heuristic MMRBallocation Δ = 6 and static allocation Δ = 9 respectively
Figure 4 CDF of interference for the 119872119872119877119861119898119886119909119877119861119898119886119909 strategy Empirical distributions are represented through the solid lines CDFof the Normal distribution are plotted with dotted lines Their parameters are obtained from the theoretical formulas The 119871+infin errors are338119890minus02 493119890minus02 and 565119890minus02 for Δ = 3 6 and 9 respectively
Figure 5 Mean interference when the number of RB variesThe load is adjusted to keep a constant reuse factor (Δ = 36 or 9) The distance 1198830 is set to 50 meters Dotted linescorrespond to the theoretical evaluation ofmean interference for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation and points to simulations
allocationTheheuristicminimizing interference presents thebest results This difference was expected as the objectivefunction minimizes interference and because the secondheuristic maximizes the reuse distance without taking intoaccount transmission powers 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tion offers interesting results with an intermediary interfer-ence level between the two heuristics Even if interference islower for the heuristic minimizing interference its complex-ity in terms of feedbackmeasures from users and eNodeBis significantly greater than the other strategies Indeed itassumes that the scheduler knows before an assignment theinterference contribution of each eNodeB on each user
Impact of the Number of RB We performed simulations andtheoretical estimations of interference for different values of119877119861119898119886119909 6 15 25 50 75 and 100 corresponding to differentbandwidth of E-UTRA [33] Results are shown in Figure 5 forthe 119872119872119877119861119898119886119909119877119861119898119886119909 allocation and the two heuristicsThe workload is adapted in order to keep the same Δreuse factor It appears that the mean interference is quiteinsensitive to the number of RB except when 119877119861119898119886119909 = 6The impacting factor is thus the reuse factor Δ rather thanthe number of RBAlso the same hierarchy between the threestrategies is observed except for 119877119861119898119886119909 = 6SIR Distribution We evaluate the SIR CDF according to(29) for the 119872119872119877119861119898119886119909119877119861119898119886119909 allocation This CDF iscompared to the one obtained by simulations in Figure 6(a)We clearly observe that the analytical model offers very
tight estimates of the SIR distribution The assumptionabout the normal distribution of interference does notintroduce a noticeable error The SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy and the two heuristics arecompared in Figure 6(b) The three CDF are similar and the119872119872119877119861119898119886119909119877119861119898119886119909 allocation still offers an intermediatedistribution between the two heuristics The 119871+infin normbetween the three CDF is 0192 0266 and 0221 for Δ = 36 and 9 respectivelyModulation and Coding Rate The SIR distribution allowsus to estimate modulations and coding rates that could beoffered to users We consider the thresholds between therequired SIR and coding rate given in [33] Applied to the SIRdistribution it gives the proportion of users that benefits froma certain modulation scheme and coding rate The couplemodulationcoding rate is referred to as transmission rate inthe following This proportion is given in Figure 7(a) for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation when the density of eNodeBincreases from 225 (the default intensity considered in theprevious evaluation) to 20 eNodeB per km2 It appears thatthe network densification even with a factor 10 does notsignificantly improve the transmission rate Only the besttransmission rate (64QAM-CR=45) really benefits from thisdensification with a proportion of users increasing from 35to 116
The spatial reuse has much more impact on the transmis-sion rates as it is shown in Figure 7(b) It clearly increasesthe transmission rates that are shifted from (QPSK CR=12-64QAM CR=23) for Δ = 3 to (16QAM CR=12-64QAMCR=45) for Δ = 9 The mean transmission rate increasesalmost of a factor 2 from 231 to 424 bitsbaud
Finally we compare in Figure 7(c) the transmission ratesfor the three allocation strategies As already observed inthe previous plots the heuristic minimizing interferenceoffers better performance but is comparable to the two otherstrategies For instance for Δ = 6 the mean transmis-sion rates are close 363 323 and 392 bitsbaud for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation the heuristic maximizingthe reuse distance and the one minimizing interferencerespectively
7 Conclusion
In recent LTE standards a server may assign RB to a set ofeNodeB to control resource usage and optimize performancein a global way Even if it exists solutions to address this prob-lem for a given configuration and topology literature lacksmodels that evaluate performance of these RB allocations fora wide range of scenarios and at large scale
To address this problem we propound a spatial stochasticmodel that takes into account RB assignment strategiesrealistic traffic demands and power control We proposeanalytical estimates for the two first moments of interferenceThis computation is based on two approximations the distri-bution of the distance between a point uniformly distributedin a cell that models the user-eNodeB distance and thejoint distribution of the distances eNodeB-origin and user-eNodeB For the latter simulations have shown that these
(a) SIR empirical and theoretical CDF for the119872119872119877119861119898119886119909119877119861119898119886119909allocation
MMRB Delta=3MMRB Delta=6MMRB Delta=9MI Delta=3 MI Delta=6
MI Delta=9 MD Delta=3 MD Delta=6 MD Delta=9
0 10 20 30 40minus10훽 (dB)
00e+00
20eminus01
40eminus01
60eminus01
80eminus01
10e+00
CDF
minus SI
NR
(b) SIR empirical CDF for the three strategies MI maximizeinterference MD maximize distance
Figure 6 (a) We compare the distribution of SIR obtained by simulations (solid lines) and given by (29) (dotted lines) Errors with regardto the 119871+infin norm are 247119890 minus 02 154119890 minus 02 and 158119890 minus 02 for Δ = 3 6 and 9 respectively (b) CDF for the three allocation strategies(119872119872119877119861119898119886119909119877119861119898119886119909 and Heuristics)
(c) 119872119872119877119861119898119886119909119877119861119898119886119909 and the two heuristics Δ = 6 120582119890 = 225
Figure 7 Modulation and coding rate for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation strategy when the density of eNodeB increases (Figure 7(a))for different reuse factor (Figure 7(b)) and comparison with the two heuristics (Figure 7(c))
variables are strongly correlated particularly for the points ofthe process close to the origin This correlation impacts thetransmission power used by the eNodeB and consequentlyinterference and SIR distributions The derivation of the SIRdistribution allows us to express the classical outage but alsothe transmission rates in terms of modulationcoding ratethat provide interesting insights on the throughput offers tothe users Also it appears that eNodeB densification does not
significantly improve the network performance in terms ofthroughput except for a small percentage of users for whichthe transmission rate is increased Spatial reuse expressedthrough the parameter Δ in our study has a much moreimpact on the performance as the transmission rates aresignificantly increased This spatial reuse must be expressedas the ratio between the total number of available resourcesover the number of usersrequests per cell Indeed we have
Wireless Communications and Mobile Computing 13
observed that the system performance is quite insensitive tovariation of these quantities when this ratio stays constant
Beside simulations show that the RB assignment strate-gies developed for the model give results close to the onesgiven by classical optimization problem which minimizesglobal interference or maximizes spatial reuse distances Itempirically shows that our analytical model is able to evaluateperformance of classical optimization approaches but at largescale in terms of number of nodes configurations andtopologies
Appendix
A Construction of the Markov Chain
In this section we introduce the Markov chain used to set thedistribution of 1198770 We consider that the RB used between thetypical user and its eNodeB has index 119903119890119904 (119903119890119904 isin 0 119877119861119898119886119909minus1)We shall prove that this index does not impact calculation(we have chosen 119903119890119904 = 0 in this paper by sack of simplicity)
We set the distribution of the number of used resourceblocks at eNodeB 0 in such a way that it leads to the followingproperty (119895 gt 119894 ge 0)
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (A1)
In order to obtain this property we first introduce somepreliminary notations and results An example of allocationwith the different notations is shown in Figure 1 We define asequence of rv (119877119899)119899ge0 If eNodeB 119894 is the 119899119905ℎ eNodeB using119903119890119904 and if 119864119899119889119899 is the index of the last RB used by this eNodeBthen
119877119899 = 119864119899119889119899 minus 119903119890119904 if 119864119899119889119899 gt 119903119890119904119864119899119889119899 + 119877119861119898119886119909 minus 119903119890119904 otherwise (A2)
If eNodeB 119894 is the 119899119905ℎ interfering eNodeB (using 119903119890119904) theneNodeB 119895 (119895 gt 119894) is the 119899 + 1119905ℎ if and only if
119863119896 ge 119877119861119898119886119909 (A4)
In this case 119877119899+1 is given by
119877119899+1 = 119877119899 +119895sum
119896=119894+1
119863119896 minus 119877119861119898119886119909 (A5)
As (119863119894)119894gt0 are iid the sequence (119877119899)119899ge0 is a homoge-neous Markov chain The transition probabilities for (119897 119898) isin0 1 119877119861119898119886119909 minus 12 are given by 119875119897119898 = P(119877119899+1 = 119898 | 119877119899 = 119897)
The event 119877119899+1 = 119898 given that 119877119899 = 119897 occurs if andonly if it exists 119895 ge 0 such that
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (A10)
We set the distribution of 1198770 with the correspondingstationary distribution Consequently the sequence (119877119899)119899ge0becomes identically distributed where 119877119899 follows the sta-tionary distribution for all 119899 ge 0 Moreover according to(A3) and (A4) the property given in (A1) holds Indeed 119877119899sum119895minus1
119896=119894+1119863119896 and sum119895
119896=119894+1119863119896 have the same distribution as 1198770sum119895minus119894minus1
119896=1119863119896 andsum119895minus119894
119896=1119863119896 respectively In the model evaluation
the stationary distribution is obtained numerically from thetransition probabilities As soon as 1198770 follows the stationarydistribution the index 119903119890119904 can be chosen arbitrarily
B Conditional Distribution of 119880119894minus119883119894Simulations have shown that given 119883119894 the distribution of119883119894 minus 119880119894 is still close to the one given by (2) but with adifferent parameter So we assume that the distribution of119883119894 minus 119880119894 given 119883119894 is equal to (2) but with a parameterfunction of 119883119894 denoted 119888(119883119894)
These simulations have also shown that E[119883119894 minus 119880119894 |119883119894 = 119903]may be approximated as an affine function 119886119894119903 + 119887119894We define 119888119894(sdot) accordingly It leads to
To set 119886119894 and 119887119894 we use the two following propertiesE [E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817 | 10038171003817100381710038171198831198941003817100381710038171003817 = 119903]] = E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817] (B3)
= 12radic120582119890 (B4)
E [E [1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817 | 1003817100381710038171003817119883119894
1003817100381710038171003817 = 119903]] = E [119886119894 10038171003817100381710038171198831198941003817100381710038171003817 + 119887119894] (B5)
Equation (B4) is derived from (2) and (B5) from ourassumption on 119888119894(119903) 119886119894 is then set as a function of 119887119894 E[119883119894]and 120582119890 119887119894 is obtained from our simulations For a givenintensity 119887119894 has been observed almost constant with regardto 119894 The best approximation as a function of the intensity isgiven by 119887119894 = 033radic120582119890 After a fewmanipulations we obtain
First note that by convention we have P(119908119894 = 1) = P(119908119894 =1 | 1199080 = 1) The event 119908119894 = 1 occurs if and only if it exists119895 ge 1 such that1198770 + 119894minus1sum
There is no data used in this paper Nevertheless the code ofour simulator is made available (the link is in the paper)
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] ldquoRequirements for further advancements for e-utra (lte-advanced) 3gpp technical specification ts 36913 release 13rdquo2015 httpwww3gpporg
[2] A S Hamza S S Khalifa H S Hamza and K Elsayed ldquoAsurvey on inter-cell interference coordination techniques inofdma-based cellular networksrdquo IEEE Communications Surveysamp Tutorials vol 15 no 4 pp 1642ndash1670 2013
[3] D Lopez-Perez A Ladanyi A JuttnerH Rivano and J ZhangldquoOptimization method for the joint allocation of modulationschemes coding rates resource blocks and power in self-organizing LTE networksrdquo in Proceedings of the IEEE INFO-COM 2011 pp 111ndash115 April 2011
[4] A Guo andM Haenggi ldquoSpatial stochasticmodels andmetricsfor the structure of base stations in cellular networksrdquo IEEE
Wireless Communications and Mobile Computing 15
Transactions on Wireless Communications vol 12 no 11 pp5800ndash5812 2013
[5] C-H Lee C-Y Shih and Y-S Chen ldquoStochastic geometrybased models for modeling cellular networks in urban areasrdquoWireless Networks vol 19 no 6 pp 1063ndash1072 2013
[6] W Lu and M D Renzo ldquoStochastic geometry modeling ofcellular networks analysis simulation and experimental valida-tionrdquo in Proceedings of the 18th ACM International Conferenceon Modelling Analysis and Simulation of Wireless and MobileSystems (MSWiM rsquo15) pp 179ndash188 New York NY USANovember 2015
[7] Y L Lee T C Chuah J Loo and A Vinel ldquoRecent advancesin radio resource management for heterogeneous LTELTE-Anetworksrdquo IEEE Communications Surveys amp Tutorials vol 16no 4 pp 2142ndash2180 2014
[8] F Mhiri K Sethom and R Bouallegue ldquoA survey on inter-ference management techniques in femtocell self-organizingnetworksrdquo Journal of Network and Computer Applications vol36 no 1 pp 58ndash65 2013
[9] X Chen L Li and X Xiang ldquoAnt colony learning methodfor joint MCS and resource block allocation in LTE Femtocelldownlink for multimedia applications with QoS guaranteesrdquoMultimedia Tools andApplications vol 76 no 3 pp 4035ndash40542017
[10] D Lopez-Perez X Chu A V Vasilakos and H Claussen ldquoOndistributed and coordinated resource allocation for interferencemitigation in self-organizing lte networksrdquo IEEEACMTransac-tions on Networking vol 21 no 4 pp 1145ndash1158 2013
[11] H Zhang H Liu J Cheng and V C M Leung ldquoDownlinkenergy efficiency of power allocation and wireless backhaulbandwidth allocation in heterogeneous small cell networksrdquoIEEE Transactions on Communications no 99 2017
[12] H Zhang W Zheng X Chu et al ldquoJoint subchannel andpower allocation in interference-limited OFDMA femtocellswith heterogeneous QoS guaranteerdquo in Proceedings of theIEEEGlobal Communications Conference (GLOBECOM rsquo12) pp4572ndash4577 December 2012
[13] H Elsawy E Hossain and M Haenggi ldquoStochastic geometryfor modeling analysis and design of multi-tier and cognitivecellular wireless networks a surveyrdquo IEEE CommunicationsSurveys amp Tutorials vol 15 no 3 pp 996ndash1019 2013
[14] H ElSawy A Sultan-Salem M-S Alouini and M Z WinldquoModeling and analysis of cellular networks using stochasticgeometry a tutorialrdquo IEEE Communications Surveys amp Tutori-als vol 19 no 1 pp 167ndash203 2017
[15] R Kwan and C Leung ldquoA survey of scheduling and inter-ference mitigation in lterdquo Journal of Electrical and ComputerEngineeringmdashSpecial Issue on LTELTE-AdvancedCellular Com-munication Networks pp 11ndash110 2010
[16] J G Andrews F Baccelli and R K Ganti ldquoA tractable approachto coverage and rate in cellular networksrdquo IEEE Transactions onCommunications vol 59 no 11 pp 3122ndash3134 2011
[17] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for OFDMA cellu-lar networksrdquo IEEE Transactions on Wireless Communicationsvol 10 no 12 pp 4294ndash4305 2011
[18] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for heterogeneouscellular networksrdquo IEEE Transactions on Communications vol60 no 7 pp 2029ndash2039 2012
[19] H Zhuang and T Ohtsuki ldquoA model based on poisson pointprocess for downlink K tiers fractional frequency reuse hetero-geneous networksrdquo Physical Communication vol 13 pp 3ndash122014 Special Issue on Heterogeneous and Small Cell Networks
[20] A A Gebremariam T Bao D Siracusa T Rasheed F Granelliand L Goratti ldquoDynamic strict fractional frequency reusefor software-defined 5G networksrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo16) pp 1ndash6May 2016
[21] N Lee D Morales-Jimenez A Lozano and R W HeathldquoSpectral efficiency of dynamic coordinated beamforming astochastic geometry approachrdquo IEEE Transactions on WirelessCommunications vol 14 no 1 pp 230ndash241 2015
[22] X Zhang and M Haenggi ldquoA Stochastic geometry analysisof inter-cell interference coordination and intra-cell diversityrdquoIEEE Transactions on Wireless Communications vol 13 no 12pp 6655ndash6669 2014
[23] F Baccelli andAGiovanidis ldquoA stochastic geometry frameworkfor analyzing pairwise-cooperative cellular networksrdquo IEEETransactions on Wireless Communications vol 14 no 2 pp794ndash808 2015
[24] J Yoon and G Hwang ldquoDistance-based inter-cell interferencecoordination in small cell networks stochastic geometry mod-eling and analysisrdquo IEEE Transactions on Wireless Communica-tions 2018
[25] B Błaszczyszyn M Jovanovicy and M K Karray ldquoHow userthroughput depends on the traffic demand in large cellularnetworksrdquo in Proceedings of the 12th International Symposiumon Modeling and Optimization in Mobile Ad Hoc and WirelessNetworks (WiOpt rsquo14) pp 611ndash619 May 2014
[26] H H Yang and T Q S Quek ldquoSIR coverage analysis incellular networks with temporal traffic a stochastic geometryapproachrdquo Corrossion 2018
[27] S M Yu and S-L Kim ldquoDownlink capacity and base stationdensity in cellular networksrdquo in Proceedings of the 11th Interna-tional SymposiumandWorkshops onModeling andOptimizationin Mobile Ad Hoc and Wireless Networks (WiOpt rsquo13) pp 119ndash124 May 2013
[28] YWang M Haenggi and Z Tan ldquoThemeta distribution of thesir for cellular networks with power controlrdquo Corrossion 2017
[29] M S Omar S A Hassan H Pervaiz et al ldquoMulti-objectiveoptimization in 5g hybrid networksrdquo IEEE Internet of ThingsJournal 2018
[30] Y Leo A Busson C Sarraute and E Fleury ldquoCall detail recordsto characterize usages and mobility events of phone usersrdquoComputer Communications vol 95 pp 43ndash53 2016
[31] I L Cherif Spectral and Energy Efficiency in 5G WirelessNetworks [PhD thesis] Univeristy Paris Saclay 2016
[32] S Mukherjee Analytical Modeling of Heterogeneous CellularNetworks Analytical Modeling of Heterogeneous Cellular Net-works Geometry Coverage and Capacity Cambridge Univer-sity Press 2014
[33] ldquoTechnical specification group radio access network base sta-tion (bs) radio transmission and reception 2012 Release 1123gpp ts 36104 v112rdquo 2012
Chemical EngineeringInternational Journal of Antennas and
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Submit your manuscripts atwwwhindawicom
2 Wireless Communications and Mobile Computing
Process (PPP) has been shown to be suitable to model thespatial location of BS [4ndash6] Nevertheless interference asexperienced by a user is not generated by all BS but onlyby the ones using the same radio resources The resourceallocation strategies have thus to be mapped to the pointprocess modeling BSeNodeB to determine which pointsBSare interfering with a given communication Consequentlythe traffic demand must also be taken into account as it setsthe number of resources used at a given time
In this work we propound a combination of severalassignment strategies realistic traffic demands and transmis-sion power control mechanisms into a stochastic geometrymodel We begin by reviewing related works and our contri-butions
11 Related Work In a downlink LTE system a resourceblock (RB) is the smallest radio resource unit that can beallocated to a user The LTE system has to schedule andassign RB to users as a function of the link qualities trafficdemands and potential quality of service requirements Inthis paper we focus on a system where a controller assignsRB for a set of eNodeB We do not overview RB assign-ment techniques in LTE network as they aim to optimizeRB assignments and modulationcoding rates for a giventopology and a traffic demand Instead this paper deals withthemacroscopic design of the network the impact of eNodeBdensity allocation scheme and power allocation on the globalperformance of a downlink LTE system Nevertheless thereader can refer to [7 8] for recent surveys Also interestingcontributions on the optimization of the downlink system fora given configuration are described in [3 9ndash12]
Stochastic geometry has emerged as an efficient toolto analyze the performance of cellular networks It offersthrough simple models a way to study wireless architec-tures at a large scale Recent surveys [13 14] summarizethe numerous wireless architectures and models for whichstochastic geometry has been applied One of the maindifficulties in the analysis of large wireless systems is tocharacterize interference This quantity does not dependonly on BS location and radio environment (path lossshadowingfading etc) but also on the way that radioresources (time frequency and power) are allocated Thepoint process modeling interfering nodes is thus of crucialimportance The PPP offers an accurate model to describeBS location [4ndash6] This process is tractable and it is possibleto derive closed formulas for some key performance metricsof the system interference coverage outage Signal overInterference plusNoise Ratio (SINR) etc But the PPPmodelsall BSeNodeB and not the subset of interfering eNodeB for agiven communication The process has thus to be thinned totake into account interference coordination (IC) techniquesand radio resources assignment for example leading toprocesses that are nomore Poisson In the next paragraph wefocus on recent contributions and on studies where resourcesallocation and more generally IC techniques are taken intoaccount
IC refers to techniques that aim to mitigate interferenceat the receivers Surveys on such techniques can be found in[2 15] A common IC approach consists in controlling the
allocated radio resources (frequencytimepower) in orderto alleviate the interference impact on communications In[16] the authors consider a random resources allocationstrategy where the BS are distributed as a PPP This simpleand tractable strategy allows model interfering BS as anindependent thinning of a PPP and deriving closed formulasfor the coverage probability They also deduce the minimalreuse factor achieving a given coverage probability Theperformance of strict FFR (Fractional Frequency Reuse)and SFR (Soft FFR) allocation strategies is evaluated usingstochastic geometry in [17] With these two techniquesdifferent radio resources are allocated to users that are at theedge of a cell (Voronoı cells here) with regard to the onesclose to the BS The criteria distinguishing core and edgeusers is based on the SINR at each user computed from theunderlying PPP modeling all BS For strict FFR the radioresources used at the edge and in the core are disjoint Insteadthe radio resources may be reused between the two regionsfor SFR For these two strategies the authors derive closedformulas for the coverage probability and discuss pros andcons of these approaches A superior interference reductionis observed for FFR but SFR benefits from a greater resourceefficiency This work is generalized in [18 19] to the contextof K-tier and heterogeneous networks considering differentpoint processes for each tier or network technology It isalso extended and studied in [20] with the dynamic strictFFR (DSFFR) where the edges of the cells are dynamicallydivided into sectors with the help of directional antennas In[21] a coordinated beamforming is employed to ensure thata set of closed BS ldquoa clusterrdquo will use different resources Auser associated with a BS is then not subject to interferencefrom BS belonging to the same cluster The authors deriveanalytical expressions for the Signal over Interference Ratio(SIR) for this strategy and discuss the impact of the clusterscardinality A similar approach is used in [22] where theset of coordinated BS corresponds to the most interferingones Interference level takes into account path loss andlong-term shadowing The interfering BS are outside this setThey are selected randomly and independently leading to athinned PPP For this model the authors study the coverageprobability for different scenarios In [23] a user is servedby its 1 or 2-closest BS according to the position of these BSwith regard to the user When the two BS are coordinatedthe transmission power is split into the two transmissionsThe total transmission power is thus the same with one ortwo coordinated BS Interference is generated by the otherBS without restriction which is assumed to be distributedthrough a PPP The authors derive a closed-form expressionfor the SIR distribution and the network coverage probabilityand discuss the benefit of this approach In [24] an ICtechnique is evaluated for a user at the edge of its cell Whenthe resource of this communication is used by neighboringcells they may not transmit any signal for a certain period tomitigate interference at this userThis coordination techniqueis analytically evaluated assuming that interfering nodes arestill distributed as a PPP
Besides the modeling of IC [22 25ndash27] propose spatialand tractablemodels that take into account the trafficdemandin the interference computation but they do not consider
Wireless Communications and Mobile Computing 3
concrete RB assignment algorithms In particular the authorsin [26] study SIR coverage for a cellular network based onPPP A queue is associated with each BS that determinesthe BS transmission activity as a function of the trafficConsidering the traffic at each BS is independent interferersat a given time are then an independent thinning of the initialPPP and are still Poisson This model differs with this paperas we do not take into account eNodeB activity as a functionof the traffic but instead the resource allocation as a functionof the number of associated users to each eNodeB Alsostochastic geometry models can be specific to certain powercontrol scheme [28] or radio technologies as in [29] wherethe authors consider a K-tier heterogeneous network withtransmissions operating on the millimeter wave band
12 Contributions The primary contribution of this work isto offer an analytical model based on stochastic geometryto evaluate the performance of a downlink LTE systemtaking into account RB allocation strategies power controland traffic demands All these mechanisms have never beencombined into a single stochastic geometry model Thenumber of allocated resources for an eNodeB is assumedto follow the distribution of the number of clients in anMMCC queue It models the number of communicationsin progress when both the interarrival of the communicationsand their duration follow an exponential distribution Suchassumptions are pertinent in cellular networks as it has beenrecently shown in [30] We associate to these traffic demandsseveral resource allocation strategies All these algorithms arecombined with a power control mechanism that depends onthe channels quality Allocation strategies lead to non-PPP ascorrelation appears between the locations of the interferingnodes It prevents the use of the convenient properties ofthe PPP to compute interference distribution Neverthelesswe propose approximations that allow us to deal with thesecorrelations and to obtain an analytical method that is shownvery accurate with regard to simulations
We compare our model to classical optimizationapproaches where for a given configurationsample theallocation is optimized with regard to an objective functionTo our knowledge such comparison has never been donebefore It shows that geometry stochastic based model maybe relevant to offer tight approximations on wireless systemperformance
Models are evaluated through a large set of simulationsthat highlights benefits of our approach to design some keyparameters of the wireless system Results show that theobtained values for SIR coding and modulation rates corre-spond to the reference values of the standards and technicalLTE documents empirically proving that our model is able toapproximate performance of real systemsThiswork has beenpartially presented in [31]
13 Paper Organization The remainder of this work is orga-nized as follows In the next section we present the systemmodel We expose the assignment strategies in Section 3In Section 4 we derive the first and second moments ofinterference for each allocation strategy SIR distribution isassessed in Section 5 Numerical results and simulations are
presented and discussed in Section 6 We conclude the paperin Section 7
2 System Model
21 eNodeB Location and Interference eNodeB location ismodeled by a point process 119873119890 = 119883119894119894isinN distributed in R2
with intensity 120582119890 Its distribution is detailed in Section 22The eNodeB are numbered with regard to their distanceto the origin eNodeB 0 at 1198830 being the closest one Weconsider a downlink system between a typical user and itsattached eNodeB Without loss of generality we assume thatthis user is located at the origin The users are assumed tobe associated with their closest eNodeB with regard to theEuclidean distance The channels between eNodeB and thetypical user are modeled through a sequence of iid randomvariables (ℎ119894)119894isinNThe transmission power between an eNodeBand one associated user is given by the function 119879119909(119903) Thisfunction models the power control algorithm implementedby the eNodeB and depends on the distance 119903 between a userand its attached eNodeBThe schedulercontroller manages aset of 119877119861119898119886119909 resource blocks that are common to all eNodeBThey are thus shared between eNodeBThe RB are numberedfrom 0 to 119877119861119898119886119909 minus 1 The scheduler assigns one RB for eachuser but the model can be easily extended with a randomnumber of RB for each demand Traffic demands exceeding119877119861119898119886119909 at an eNodeB are not served The RB with index 0 isallocated to the typical user We show in the appendix thatthis choice does not impact the computations and any otherindex could be chosen instead An eNodeB interferes with thetypical user if and only if it reuses this RB Interference at thetypical user can be expressed as
where 119897(sdot) is the path loss function The argument of inter-ference is the distance between the typical user and itseNodeB (eNodeB 0) 119868(1198830) expresses interference whenthis distance is random and depends on the rv 1198830 119868(119903)expresses interference when this distance is given and equalto 119903 This notation is motivated by the fact that the rv 119883119894
and 119883119894 minus 119880119894 are correlated to 1198830 Moreover the mean andthe variance of interference will be computed for both a givenvalue of 1198830 and with regard to its distribution The rv 119908119894
indicates whether eNodeB 119894 interferes with the typical user(119908119894 = 0 or 1)119880119894 is the random variable modeling the locationof a user attached to the eNodeB 119894 (at 119883119894) We assume that119880119894 is uniformly distributed in the Voronoı cell formed by theprocess 119873119890 and with nucleus 119883119894 The main notations usedthroughout this paper are given in Table 1
22 Point Process Modeling eNodeB The point process mod-eling eNodeB is a modified homogeneous PPP The process119873119890 1198830 = 119883119894119894gt0 is Poisson in R2 119861(0 1198830) where119861(0 1198830) is the ball centered at the origin and with radius1198830 We choose a distribution for 1198830 that makes theprocess119873119890 different of a PPP Indeed with a PPP the typicaluser at the origin lies in a Voronoı cell that is greater in
4 Wireless Communications and Mobile Computing
Table 1 Principal notation
119873119890 point process modeling eNodeB119868(1198830) interference at the typical user The distance between the typical user and eNodeB 0 is the rv 1198830119868(119903) interference at the typical user The distance between the typical user and eNodeB 0 is equal to 119903119908119894 rv indicating if eNodeB 119894 interferes with the typical user119863119894 number of RB allocated to eNodeB 119894119879119909(119903) transmission power between an eNodeB and its user at distance 119903Δ spatial reuse parameter (an RB is reused every Δ eNodeB in average)119891119883119894(sdot) PDF of 119883119894119891119883119894119883119895(sdot sdot) joint PDF of (119883119894 119883119895)1198911198941198830=119903
(sdot) conditional PDF of 119883119894 given that 1198830 = 119903119891119894119895
1198830=119903(sdot) conditional PDF of (119883119894 119883119895) given that X0 = 119903
11989101198830
(sdot) PDF of the distance between the typical user (at the origin) and its serving eNodeB 01198910119880119894minus119883119894
(sdot sdot) conditional PDF of the distance between a user and its attached nucleus119883119894 given 119883119894
average than the other cells Intuitively ldquobig cellsrdquo cover morespace than ldquosmall cellsrdquo and consequently the cell coveringthe origin has a greater size in average It is consistent witha modeling where users are homogeneously scattered in theplane but not with our assumption where the network hasbeen dimensioned to have the same load in average (the samenumber of users) in each cell It is more realistic as it hasbeen shown in [30] where a homogeneous load is observedfor the different cells independently of their sizes In thiscase the typical cell covering the typical user must have thesame distribution as the other cells Therefore we considerthe distribution of the distance 1198830 under Palm measureMore precisely this distribution corresponds to the distancebetween the nucleus of a typical cell under Palmmeasure anda point uniformly distributed in this cell The distributionof this distance is not known but we use the approximationpresented in [32] (page 133) We set the distribution of thedistance between the typical user and its closest eNodeB at1198830 as
with 119888 = 125 The angle between the lines (01198830) and theabscissa is uniformly distributed in [0 2120587)
The distribution of the distance between a user in agiven Voronoı cell and its nucleus (119883119894 minus 119880119894) follows thesame definition and consequently the same distribution Inthe next section we present the different RB assignmentstrategies evaluated in this paper
3 Assignment Strategies
We consider four different allocation strategies We beginby two simple RB allocation schemes independent andstatic allocations Then we develop a more realistic alloca-tion strategy using the 119872119872119877119861119898119886119909119877119861119898119886119909 queue named119872119872119877119861119898119886119909119877119861119898119886119909 allocation hereafter Also amore globalapproach where the RB are assigned in order to minimizethe sum of interference at each user is considered but for
which we do not propose a mathematical resolution All theallocation strategies are set in such a way that a given RB isreused every Δ (Δ ge 1) eNodeB in average The mean loadin each cell and equivalently the mean number of RB used byan eNodeB are then 119877119861119898119886119909Δ The reuse factor Δ reflects thenetwork load
31 Independent Allocation Thinning With this strategyeach eNodeB selects its resources independently of theother eNodeB Therefore we assume that an eNodeB hasa probability 1Δ to reuse the RB with index 0 The pointprocess describing the interfering eNodeB is then a thinnedPPP in R2 119861(119874 1198830) with intensity 120582119890Δ32 Static Allocation We assign a constant proportion119877119861119898119886119909Δ of the available resources to each eNodeBThey areallocated in their index order eNodeB 0 uses RB from 0 to119877119861119898119886119909Δ minus 1 (it includes the typical user) eNodeB 1 from119877119861119898119886119909Δ to 2(119877119861119898119886119909Δ) minus 1 etc We take the integer partof these fractions when 119877119861119898119886119909 is not a multiple of Δ Weloop when all resources have been used Consequently theeNodeB interfering with the typical user has an index withthe form 119896 sdot Δ with 119896 gt 033119872119872119877119861119898119886119909119877119861119898119886119909Allocation In an119872119872119862119862queuecustomers arrive according to a Poisson process (in R) andthe service times are exponentially distributed It modelsa system with 119862 resourcesservers and a capacity of thesame size A customer cannot enter in the system if allresourcesservers are busy We associate with each eNodeBan independent 119872119872119877119861119898119886119909119877119861119898119886119909 queue to model thenumber of RB in use The servers model the RB Upon thearrival of a requestuser an RBserver is used for a timeexponentially distributed If no RB is available the requestis rejected In order to have a mean reuse factor of Δ theparameter of the queue (the load) denoted 120588 is set in sucha way that the mean number of customers in the system orequivalently the mean number of busy resource blocks isequal to 119877119861119898119886119909Δ The distribution of the number of busy
Wireless Communications and Mobile Computing 5
(a) RB allocation (b) Impact of RB allocation on interferer
Figure 1 In Figure 1(a) an example of RB allocation is given eNodeB 0 uses 6 RB (the first one has index 14 and the other ones are indexedfrom 0 to 4) It uses the RB with index 0 to communicate with the typical user1198770 is the index of the last resource used by eNodeB 0 As it usesRB indexed from 14 to 4 1198770 = 4 hereThe number of used RB for the other eNodeB are1198631 = 41198632 = 3 etc An eNodeB interferes if and onlyif it reuses the resource 0 in this example eNodeB 3 and 6 In Figure 1(b) we plot a sample of the point process describing eNodeB locationwith the same RB assignmentThe eNodeB interfering with the typical user is then a dependent thinning of the original point process wherethe thinning involves eNodes 3 6 and 10 (this last interfering eNodeB was not shown in Figure 1(a))Their Voronoı cells are colored in green
resource blocks for a given eNodeB 119894 (119894 gt 0) denoted 119863119894 isthen given by
P (119863119894 = 119896) = 1205870 120588119896119896 (3)
where 1205870 = P(119863119894 = 0) and 119877119861119898119886119909 ge 119896 ge 0These RB are allocated in a cyclic order If the last119877119861 used
by eNodeB 119894 minus 1 has index 119896 eNodeB 119894 uses 119877119861 indexed from(119896 + 1)119898119900119889(119877119861119898119886119909) to (119896 + 119863119894) 119898119900119889(119877119861119898119886119909)For eNodeB 0 we do not consider the total number
of allocated RB (1198630) but instead a random variable 1198770 Itdescribes the index of the last RB used by eNodeB 0 Indeedfor this particular eNodeB the quantity used in practice tocompute the next allocation (RB indexes used by eNodeB 1)is 1198770 rather than 1198630 A formal definition of 1198770 is given inappendix (Appendix A) An example of allocation is given inFigure 1
The distribution of 1198770 is set according to the stationarydistribution of a Markov chain The transition probabilitiesof this Markov chain are
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (5)
where (119897 119898) isin 0 1 119877119861119898119886119909 minus 12 The motivation forthis particular construction is to keep the probability ofusing the resource 0 between eNodeB homogeneous Moreprecisely it is built in order to verify the property given inProposition 1 Details about the Markov chain constructionare given in appendix (Appendix A) It is worth noting thatother distributions for the resource demands (given by (3)in our case) can be considered as well As soon as thedistribution of1198770 verifies Proposition 1 themethod proposedin this paper holds
34 Property of These Assignment Strategies We define moreprecisely the sequence of rv (119908119894)119894isinN It indicates whicheNodeB interferes with the typical user It was already usedin (1)
119908119894 = 1 if eNodeB at 119883119894 uses RB with index 00 otherwise (6)
By convention we set 1199080 = 1 as In the following weshall thus assume that P(119908119894 = 1 | 1199080 = 1) = P(119908119894 = 1)Proposition 1 For the three allocation strategies defined inSections 31 32 and 33 the following property holds
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (7)
The proofs for the first two strategies are straightforwardFor the 119872119872119877119861119898119886119909119877119861119898119886119909 strategy the distribution of 1198770has been set to verify this property (see Appendix A)
35 Heuristics We compare these strategies to heuristics thataim to optimize interference or spatial reuse for a givenconfiguration It allows us to compare our RB assignmentand performs in a cyclic manner around a typical user tostrategies where a controller in charge of a set of eNodeBwill assign RB in order to optimize a certain objective func-tion For the heuristic minimizing the sum of interferencenamed ldquominimize interferencerdquo hereafter the consideredoptimization problem is similar to the one developed in[3] The problem has been shown NP-hard so we use agreedy algorithm to find a solution The number of usersassociated with each eNodeB follows the same distribution asthe119872119872119877119861119898119886119909119877119861119898119886119909 allocation Then we consider usersin a random order and apply Algorithm 1 to associate an RBto a user It chooses the resource block that minimizes thesum of interference Obviously we compute interference only
6 Wireless Communications and Mobile Computing
Data 119877119861119898119886119909 Total number of RBcost sum of interference for the current allocationbestCost sum of interference for the best allocation strategy initialized to -1Result Assign an RB to a user This algorithm is called by the controller for each
new user(1) for each 119903119887 isin 0 1 119877119861119898119886119909 minus 1 do(2) if rb is free then(3) assign rb to this user(4) costlarr sum Interference()(5) if bestCost lt0 or costltbestCost then(6) Save this allocation strategy(7) bestCost = cost(8) end(9) end(10) end(11) Assign the saved allocation strategy
Algorithm 1 Minimizing global interference
for the users already assigned The typical user is consideredin last when the system has reached the targeted load Thisalgorithm mimics an assignment strategy where the RB areassigned at the arrival of the users request without changingthe already assigned RB
The second heuristic ldquomaximize reuse distancerdquo maxi-mizes the distance at which the RB are reused Each user isconsidered in a random order Different RB are assigned tothe 119877119861119898119886119909 first users When assigning an RB to the otherusers the controller chooses the RB for which the reusedistance is maximum The typical user is considered in lastThis second heuristic may correspond to a case where thecontroller does not have information on channel conditionsand interference but knows the distances between eNodeB
4 Interference Characterization
We derive the mean and the variance of interference for thethree assignment strategies defined in the previous section
The point process modeling interferers is a dependentthinning of the original PPP Consequently conditions formean (respectively variance) to be finite with a PPP also holdfor our point process the path loss function 119897(sdot)must belongto 1198711 (respectively 1198712)41 Distribution of Distances between the Typical User andeNodeB (119883119894) As a preamble we give the PDF of thedistance between the typical user at the origin and eNodeBBoth PDF of 119883119894 and joint distribution of (119883119894 119883119895) arederived These PDF are used in the computation of the meanand the variance of interference
In the numerical evaluation we shall condition interfer-ence by the distance 1198830 It allows us to study interferencefor a given distance between the typical user and its attachedeNodeB It is also motivated by the computation of the SIRwhere both interference and the typical user signal strengthdepend on the distance 1198830
For our model the PDF of 119883119894with 119894 gt 0 given 1198830 = 119903is
1198911198941198830=119903 (119906 119903) = (120582119890120587)119894(119894 minus 1)2119906 (1199062 minus 1199032)119894minus1 119890minus120582119890120587(1199062minus1199032)1119906gt119903 (8)
The joint PDF of (119883119894 119883119895)with 119895 gt 119894 gt 0 given 1198830 =119903 is1198911198941198951198830=119903
(119906 V 119903)= (120582119890120587)119895(119894 minus 1) (119895 minus 119894 minus 1)4119906V (V2 minus 1199062)
119895minus119894minus1
times (1199062 minus 1199032)119894minus1 119890minus120582119890120587(V2minus1199032)1Vgt119906gt119903(9)
To obtain the PDF when 1198830 is not set (119891119883119894(sdot) and119891119883119894119883119895(sdot)) it suffices to integrate the two conditional PDFwith regard to the PDF of 1198830 given in (2)
42 Mean of Interference Themean is derived from (1)
E [119868 (100381710038171003817100381711988301003817100381710038171003817)]
= E [ℎ1] +infinsum119894=1
E [119908119894]E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) sdot 119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)] (10)
In this equation 119908119894 has been separated from the expecta-tion as it is independent of the process 119873119890 (according to thedefined strategies) We derive E[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)] andE[119908119894] in the two next sections
421 Computation of E[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)]No Power Control In absence of power control ie when119879119909(sdot) is constant or independent of the process 119873119890 a closedformula may be expressed for (10) E[119879119909(119883119894 minus119880119894) sdot 119897(119883119894)]
Wireless Communications and Mobile Computing 7
is then given by E[119879119909]E[119897(119883119894)] Expectation of 119897(119883119894) isobtained from the distribution of 119883119894 given in Section 41
E [119868 (100381710038171003817100381711988301003817100381710038171003817)]
= E [ℎ1] +infinsum119894=1
E [119908119894]E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817)]E [119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)] (11)
Power Control When the transmission power depends onthe distance between the receiver and its attached eNodeB(119879119909(119883119894 minus 119880119894)) the computations are more complex119864[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)] cannot be approximated byE[119879119909(119883119894minus119880119894)]119864[119897(119883119894)] as the size of the Voronoı cell withnucleus 119883119894 depends on its distance to the origin The jointdistribution of (119883119894 119883119894 minus119880119894) being unknown we proposethe following approximation
The PDF of 119880119894 minus 119883119894 is the same as 1198800 minus 1198830 givenby (2) Its parameter 119888119894(119883119894) depends on 119883119894 119888119894(119883119894) =14radic120582119890(120574119894119883119894 + 119887) with 120574119894 = (12 minus 119887)E[119883119894] and 119887 = 033The motivation and the computation details for this PDF aregiven in appendix (Appendix B) We obtain
E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) sdot 119897 (1003817100381710038171003817119883119894
Often in real systems the transmission power cannot beset arbitrarily and is limited to a set of predetermined valuesThe transmission power function can then be representedas a step function 119879119909(119903) = sum119873119879
119894=1 1199051198941119903isin[120572119894minus1120572119894] where 119873119879
is the number of possible transmission powers 119905119894 the 119894119905ℎtransmission power value and [120572119894minus1 120572119894] the distance intervalbetween a user and its eNodeB at which this transmissionpower is used An example of such setting is given in thenumerical evaluation section In this case (14) becomes
E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) 119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)]= 119873119879sum
119894=1
119905119895 int+infin
0(119890minus120582119890119888119894(119903)1205871205722119895 minus 119890minus120582119890119888119894(119903)1205871205722119895minus1)119891119883119894 (119903) 119889119903
(15)
When the computation is performed for a given distance1198830 the PDF 119891119883119894(sdot) in (12) and (15) must be replaced by1198911198941198830=119903(sdot sdot) (given in Section 41)
422 Computation of E[119908119894] Finally in order to compute(10) we need to expressE[119908119894] First note that E[119908119894] = P(119908119894 =1)Proposition 2 The probability for eNodeB 119894 to interfere withthe typical user is given by
(i) Independent allocation
P (119908119894 = 1) = 1Δ (16)
(ii) Static allocation
P (119908119894 = 1) = 1119894sdot119898119900119889(Δ)=0 (17)
E [119879119909 (1003817100381710038171003817119880119894 minus 1198831198941003817100381710038171003817) 119879119909 (10038171003817100381710038171003817119880119895 minus 119883119895
As for the mean complexity lies in the correlationbetween 119883119894 and 119880119894 minus 119883119894 The term E[119879119909(119880119894 minus 119883119894)2 sdot119897(119883119894)2] is computed with the same method as the firstmoment
Computation of E[119879119909(119880119894minus119883119894)119879119909(119880119895minus119883119895)119897(119883119894)119897(119883119895)]As 119880119894 minus 119883119894 (respectively 119880119895 minus 119883119895) depends on 119883119894(respectively 119883119895) we condition by the distribution of(119883119894 119883119895) given in Section 41 Given 119883119894 and 119883119895 we usethe same PDF as in (B1) assuming that 119880119894 minus 119883119894 and 119880119895 minus119883119895 are independent The considered joint distribution of(119883119894 119883119895 119880119894 minus 119883119894 119880119895 minus 119883119895) becomes
1198910119880119894minus119883119894 (119906 119903) 1198910119880119895minus119883119895 (V 119904) 119891119883119894119883119895 (119903 119904) (21)
When the distance 1198830 is fixed 119891119883119894119883119895(sdot sdot) must bereplaced by the PDF 119891119894119895
1198830=119903(sdot sdot sdot) given in Section 41
Computation of E[119908119894119908119895] It has been shown that the sequence(119908119894)119894gt0 verifies (7) for the three strategies It allows us toexpress E[119908119894119908119895] with 119894 gt 119895 as
E [119908119894119908119895] = P (119908119894minus119895 = 1)P (119908119895 = 1) (22)
8 Wireless Communications and Mobile Computing
Proposition 3 The joint probability for two eNodeB 119894 and 119895(119895 gt 119894) to interfere with the typical user is given by
(i) Independent allocation
E [119908119894119908119895] = 1Δ2 (23)
(ii) Static allocation
E [119908119894119908119895] = 1119894sdot119898119900119889(Δ)=0119895sdot119898119900119889(Δ)=0 (24)
where P(119908 = 1) is given by (18)5 Signal over Interference Ratio (SIR)
In our model there is a strong correlation between theinterfering eNodeB It is generated by the allocation strategiesand cannot be neglected Also a correlation exists betweenthe location of an eNodeB and the size of its Voronoı cellConsequently classical approach based on PPP which uses
Laplacian transform for instance cannot be applied hereand a formal derivation of interference distribution seemsintractable
Nevertheless the different simulations presented in thenext section will show that the PDF of interference can beapproximated by a log-normal distribution The parametersof this distribution mean and variance denoted by 119898119868119889119861
(sdot)and 120590119868119889119861(sdot) are directly derived from the previous analyticalcomputations The classical mapping between log-normaland normal parameters can be applied to derive parametersof the normal distribution when interferences are expressedin decibel In the following a variable is indexed by 119889119861 whenit is expressed in decibel
We get
P (119878119868119877119889119861 le 120573119889119861) = P (10sdot log10 (119875119905 (10038171003817100381710038171198830
1003817100381710038171003817) ℎ0119897 (100381710038171003817100381711988301003817100381710038171003817)) minus 119868119889119861 (10038171003817100381710038171198830
1003817100381710038171003817) le 120573119889119861) (26)
= P (119868119889119861 (100381710038171003817100381711988301003817100381710038171003817) ge minus120573119889119861 + 10
When ℎ0 is not constant the expectation with regard toits distribution must be taken into account in (28) In (29)ℎ0 is assumed to be constant that equals to 1 In these twoequations 119890119903119891(sdot) is the error function6 Numerical Results
We consider an E-UTRA channel with a bandwidth of5MHz with 119877119861119898119886119909 = 15 [33] The path loss function isthe same as [3] It is expressed in dB 119897(119903) = minus1281 minus376 sdot log10(119903) where 119903 is the distance (in km) 119879119909(sdot) isset in such a way to guarantee to each user a minimumreceiving power We set the transmission power function119879119909(sdot) to ensure a signal power greater than or equal tominus724dBm at the reception as specified in [33] For each 50meters (from 50 to 500 meters) we compute the minimumtransmitting power required to reach this threshold (119879119909(119903) sdot119897(119903) ge minus724 dBm for each interval of 50 meters leading to 10possible transmission powers)This step function models thecase where eNodeB has a finite set of predetermined powerThe process intensity modeling eNodeB is equal to 225 perkm2 It corresponds to the intensity of base stations in Paris(httpswwwantennesmobilesfr) Random variables ℎ119894 are
supposed constant equal to 1 This assumption facilitatesinterpretation of the results but any distribution can beconsidered as well It simply adds a factor in terms of variance(cf (20)) We simulate the different strategies through asimulator coded inC available here (httpwwwanthonybus-sonfrindexphppublications) In all simulations and nu-merical results we consider 50 eNodeB The different sumsin the equations (eg (10) and (20)) are then limited to 50For each set of parameters simulations have be run from10000 to several millions times depending on the evaluatedquantities The number of simulationssamples has been setin order to have negligible confidence intervals They areconsequently not shown in the different figures
Mean and Variance of Interference In Figure 2 we plot themean and the standard deviation of interference obtainedfrom simulations and computed from formulas (10) (19) and(20)when the distance 1198830 variesThe theoretical evaluationcloselymatches empirical estimators obtained by simulationsAs expected the highest interference level is observed forthe independent allocation and the lowest level for the staticallocation The static and the 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tions offer equivalent results with a multiplication factor
Figure 2 Interferencemean and standard deviation as a function of the distance between the typical user and its eNodeB (1198830) Simulationresults are given by points and solid lines and theoretical evaluations by the dotted lines
Table 2 Mean and standarddeviation for the119872119872119877119861119898119886119909119877119861m119886119909 allocation and the two heuristics
Assignment strategy Δ = 3 Δ = 6 Δ = 9mean std-dev mean std-dev mean std-dev
119872119872119877119861119898119886119909119877119861119898119886119909 178119890 minus 11 861119890 minus 11 366119890 minus 12 750119890 minus 12 155119890 minus 12 316119890 minus 12Minimize Interference 994119890 minus 12 307119890 minus 11 186119890 minus 12 188119890 minus 12 756119890 minus 13 626119890 minus 13Maximize reuse distance 526119890 minus 11 751119890 minus 10 118119890 minus 11 107119890 minus 10 258119890 minus 12 109119890 minus 11
varying from 11 to 87 for the mean interference Insteadthe independent allocation differs deeply Mean and standarddeviation are multiplied by a factor ranging from 17 to 700for the mean and can reach up to 26 times 104 for the standarddeviation (for 1198830 = 002 km) The use of the thinnedpoint process is thus questionable to approximate realisticassignment strategies This strategy is no more considered inthe following
Extrapolation of Interference Distribution We compareempirical distributions obtained from simulations to knowndistributions for the different strategies and the two heuristicswhen Δ = 3 6 and 9 Three of these empirical distributionsare shown in Figure 3 (in dB) The distribution parametershave been set according to the maximum likelihood Thedistributions that best fit simulations vary according to theallocation strategy and the reuse factorΔ For 4 cases over 12the distribution minimizing error is the normal distributionAs soon as an asymmetry is observed other distributionsare more accurate log-normal Weibull and Inverse GammaWe perform two hypothesis tests T-test and Smirnov-Kolmogorov for all these scenarios The alternative hypoth-esis is systematically ruled out For the 119872119872119877119861119898119886119909119877119861119898119886119909
strategy best fits are given by the normal (Δ = 3 and 6) andWeibull distributions (Δ = 9)
Nevertheless as mentioned in Section 5 when estimatingthe SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy weuse the normal law to model interference Indeed even if thisdistribution is not systematically the most accurate it offers agood approximation as shown inFigure 4 where the empiricaland normalCDF are plottedMoreover simulationswill showthat this assumption does not impact the accuracy of theanalytical model
Heuristics Simulations have shown that allocations resultingfrom the two heuristics correlate the transmission powersthe distance between eNodeB reusing the same resourceand the random variables 119908119894 Whenminimizing interferencecorrelations are caused by resource blocks allocated to usersthat require a low transmission power and that can be reusedat a short distance and inversely The transmission poweris then correlated to 119908119894 and to the distance of eNodeB thatreuse the same resource These complex phenomena impedethe proposal of an analytical model for these two heuristicsFor comparison purposes Table 2 reportsmean and standarddeviation for the two heuristics and the119872119872119877119861119898119886119909119877119861119898119886119909
Figure 3 Interference distribution empirical (histogram) and extrapolation (curve) Among the classical distributions the bestextrapolations are Gamma (m=1003 var=977) normal (m=2823 var=2216) and Weibull (a=2514b=575) for the heuristic MMRBallocation Δ = 6 and static allocation Δ = 9 respectively
Figure 4 CDF of interference for the 119872119872119877119861119898119886119909119877119861119898119886119909 strategy Empirical distributions are represented through the solid lines CDFof the Normal distribution are plotted with dotted lines Their parameters are obtained from the theoretical formulas The 119871+infin errors are338119890minus02 493119890minus02 and 565119890minus02 for Δ = 3 6 and 9 respectively
Figure 5 Mean interference when the number of RB variesThe load is adjusted to keep a constant reuse factor (Δ = 36 or 9) The distance 1198830 is set to 50 meters Dotted linescorrespond to the theoretical evaluation ofmean interference for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation and points to simulations
allocationTheheuristicminimizing interference presents thebest results This difference was expected as the objectivefunction minimizes interference and because the secondheuristic maximizes the reuse distance without taking intoaccount transmission powers 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tion offers interesting results with an intermediary interfer-ence level between the two heuristics Even if interference islower for the heuristic minimizing interference its complex-ity in terms of feedbackmeasures from users and eNodeBis significantly greater than the other strategies Indeed itassumes that the scheduler knows before an assignment theinterference contribution of each eNodeB on each user
Impact of the Number of RB We performed simulations andtheoretical estimations of interference for different values of119877119861119898119886119909 6 15 25 50 75 and 100 corresponding to differentbandwidth of E-UTRA [33] Results are shown in Figure 5 forthe 119872119872119877119861119898119886119909119877119861119898119886119909 allocation and the two heuristicsThe workload is adapted in order to keep the same Δreuse factor It appears that the mean interference is quiteinsensitive to the number of RB except when 119877119861119898119886119909 = 6The impacting factor is thus the reuse factor Δ rather thanthe number of RBAlso the same hierarchy between the threestrategies is observed except for 119877119861119898119886119909 = 6SIR Distribution We evaluate the SIR CDF according to(29) for the 119872119872119877119861119898119886119909119877119861119898119886119909 allocation This CDF iscompared to the one obtained by simulations in Figure 6(a)We clearly observe that the analytical model offers very
tight estimates of the SIR distribution The assumptionabout the normal distribution of interference does notintroduce a noticeable error The SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy and the two heuristics arecompared in Figure 6(b) The three CDF are similar and the119872119872119877119861119898119886119909119877119861119898119886119909 allocation still offers an intermediatedistribution between the two heuristics The 119871+infin normbetween the three CDF is 0192 0266 and 0221 for Δ = 36 and 9 respectivelyModulation and Coding Rate The SIR distribution allowsus to estimate modulations and coding rates that could beoffered to users We consider the thresholds between therequired SIR and coding rate given in [33] Applied to the SIRdistribution it gives the proportion of users that benefits froma certain modulation scheme and coding rate The couplemodulationcoding rate is referred to as transmission rate inthe following This proportion is given in Figure 7(a) for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation when the density of eNodeBincreases from 225 (the default intensity considered in theprevious evaluation) to 20 eNodeB per km2 It appears thatthe network densification even with a factor 10 does notsignificantly improve the transmission rate Only the besttransmission rate (64QAM-CR=45) really benefits from thisdensification with a proportion of users increasing from 35to 116
The spatial reuse has much more impact on the transmis-sion rates as it is shown in Figure 7(b) It clearly increasesthe transmission rates that are shifted from (QPSK CR=12-64QAM CR=23) for Δ = 3 to (16QAM CR=12-64QAMCR=45) for Δ = 9 The mean transmission rate increasesalmost of a factor 2 from 231 to 424 bitsbaud
Finally we compare in Figure 7(c) the transmission ratesfor the three allocation strategies As already observed inthe previous plots the heuristic minimizing interferenceoffers better performance but is comparable to the two otherstrategies For instance for Δ = 6 the mean transmis-sion rates are close 363 323 and 392 bitsbaud for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation the heuristic maximizingthe reuse distance and the one minimizing interferencerespectively
7 Conclusion
In recent LTE standards a server may assign RB to a set ofeNodeB to control resource usage and optimize performancein a global way Even if it exists solutions to address this prob-lem for a given configuration and topology literature lacksmodels that evaluate performance of these RB allocations fora wide range of scenarios and at large scale
To address this problem we propound a spatial stochasticmodel that takes into account RB assignment strategiesrealistic traffic demands and power control We proposeanalytical estimates for the two first moments of interferenceThis computation is based on two approximations the distri-bution of the distance between a point uniformly distributedin a cell that models the user-eNodeB distance and thejoint distribution of the distances eNodeB-origin and user-eNodeB For the latter simulations have shown that these
(a) SIR empirical and theoretical CDF for the119872119872119877119861119898119886119909119877119861119898119886119909allocation
MMRB Delta=3MMRB Delta=6MMRB Delta=9MI Delta=3 MI Delta=6
MI Delta=9 MD Delta=3 MD Delta=6 MD Delta=9
0 10 20 30 40minus10훽 (dB)
00e+00
20eminus01
40eminus01
60eminus01
80eminus01
10e+00
CDF
minus SI
NR
(b) SIR empirical CDF for the three strategies MI maximizeinterference MD maximize distance
Figure 6 (a) We compare the distribution of SIR obtained by simulations (solid lines) and given by (29) (dotted lines) Errors with regardto the 119871+infin norm are 247119890 minus 02 154119890 minus 02 and 158119890 minus 02 for Δ = 3 6 and 9 respectively (b) CDF for the three allocation strategies(119872119872119877119861119898119886119909119877119861119898119886119909 and Heuristics)
(c) 119872119872119877119861119898119886119909119877119861119898119886119909 and the two heuristics Δ = 6 120582119890 = 225
Figure 7 Modulation and coding rate for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation strategy when the density of eNodeB increases (Figure 7(a))for different reuse factor (Figure 7(b)) and comparison with the two heuristics (Figure 7(c))
variables are strongly correlated particularly for the points ofthe process close to the origin This correlation impacts thetransmission power used by the eNodeB and consequentlyinterference and SIR distributions The derivation of the SIRdistribution allows us to express the classical outage but alsothe transmission rates in terms of modulationcoding ratethat provide interesting insights on the throughput offers tothe users Also it appears that eNodeB densification does not
significantly improve the network performance in terms ofthroughput except for a small percentage of users for whichthe transmission rate is increased Spatial reuse expressedthrough the parameter Δ in our study has a much moreimpact on the performance as the transmission rates aresignificantly increased This spatial reuse must be expressedas the ratio between the total number of available resourcesover the number of usersrequests per cell Indeed we have
Wireless Communications and Mobile Computing 13
observed that the system performance is quite insensitive tovariation of these quantities when this ratio stays constant
Beside simulations show that the RB assignment strate-gies developed for the model give results close to the onesgiven by classical optimization problem which minimizesglobal interference or maximizes spatial reuse distances Itempirically shows that our analytical model is able to evaluateperformance of classical optimization approaches but at largescale in terms of number of nodes configurations andtopologies
Appendix
A Construction of the Markov Chain
In this section we introduce the Markov chain used to set thedistribution of 1198770 We consider that the RB used between thetypical user and its eNodeB has index 119903119890119904 (119903119890119904 isin 0 119877119861119898119886119909minus1)We shall prove that this index does not impact calculation(we have chosen 119903119890119904 = 0 in this paper by sack of simplicity)
We set the distribution of the number of used resourceblocks at eNodeB 0 in such a way that it leads to the followingproperty (119895 gt 119894 ge 0)
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (A1)
In order to obtain this property we first introduce somepreliminary notations and results An example of allocationwith the different notations is shown in Figure 1 We define asequence of rv (119877119899)119899ge0 If eNodeB 119894 is the 119899119905ℎ eNodeB using119903119890119904 and if 119864119899119889119899 is the index of the last RB used by this eNodeBthen
119877119899 = 119864119899119889119899 minus 119903119890119904 if 119864119899119889119899 gt 119903119890119904119864119899119889119899 + 119877119861119898119886119909 minus 119903119890119904 otherwise (A2)
If eNodeB 119894 is the 119899119905ℎ interfering eNodeB (using 119903119890119904) theneNodeB 119895 (119895 gt 119894) is the 119899 + 1119905ℎ if and only if
119863119896 ge 119877119861119898119886119909 (A4)
In this case 119877119899+1 is given by
119877119899+1 = 119877119899 +119895sum
119896=119894+1
119863119896 minus 119877119861119898119886119909 (A5)
As (119863119894)119894gt0 are iid the sequence (119877119899)119899ge0 is a homoge-neous Markov chain The transition probabilities for (119897 119898) isin0 1 119877119861119898119886119909 minus 12 are given by 119875119897119898 = P(119877119899+1 = 119898 | 119877119899 = 119897)
The event 119877119899+1 = 119898 given that 119877119899 = 119897 occurs if andonly if it exists 119895 ge 0 such that
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (A10)
We set the distribution of 1198770 with the correspondingstationary distribution Consequently the sequence (119877119899)119899ge0becomes identically distributed where 119877119899 follows the sta-tionary distribution for all 119899 ge 0 Moreover according to(A3) and (A4) the property given in (A1) holds Indeed 119877119899sum119895minus1
119896=119894+1119863119896 and sum119895
119896=119894+1119863119896 have the same distribution as 1198770sum119895minus119894minus1
119896=1119863119896 andsum119895minus119894
119896=1119863119896 respectively In the model evaluation
the stationary distribution is obtained numerically from thetransition probabilities As soon as 1198770 follows the stationarydistribution the index 119903119890119904 can be chosen arbitrarily
B Conditional Distribution of 119880119894minus119883119894Simulations have shown that given 119883119894 the distribution of119883119894 minus 119880119894 is still close to the one given by (2) but with adifferent parameter So we assume that the distribution of119883119894 minus 119880119894 given 119883119894 is equal to (2) but with a parameterfunction of 119883119894 denoted 119888(119883119894)
These simulations have also shown that E[119883119894 minus 119880119894 |119883119894 = 119903]may be approximated as an affine function 119886119894119903 + 119887119894We define 119888119894(sdot) accordingly It leads to
To set 119886119894 and 119887119894 we use the two following propertiesE [E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817 | 10038171003817100381710038171198831198941003817100381710038171003817 = 119903]] = E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817] (B3)
= 12radic120582119890 (B4)
E [E [1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817 | 1003817100381710038171003817119883119894
1003817100381710038171003817 = 119903]] = E [119886119894 10038171003817100381710038171198831198941003817100381710038171003817 + 119887119894] (B5)
Equation (B4) is derived from (2) and (B5) from ourassumption on 119888119894(119903) 119886119894 is then set as a function of 119887119894 E[119883119894]and 120582119890 119887119894 is obtained from our simulations For a givenintensity 119887119894 has been observed almost constant with regardto 119894 The best approximation as a function of the intensity isgiven by 119887119894 = 033radic120582119890 After a fewmanipulations we obtain
First note that by convention we have P(119908119894 = 1) = P(119908119894 =1 | 1199080 = 1) The event 119908119894 = 1 occurs if and only if it exists119895 ge 1 such that1198770 + 119894minus1sum
There is no data used in this paper Nevertheless the code ofour simulator is made available (the link is in the paper)
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] ldquoRequirements for further advancements for e-utra (lte-advanced) 3gpp technical specification ts 36913 release 13rdquo2015 httpwww3gpporg
[2] A S Hamza S S Khalifa H S Hamza and K Elsayed ldquoAsurvey on inter-cell interference coordination techniques inofdma-based cellular networksrdquo IEEE Communications Surveysamp Tutorials vol 15 no 4 pp 1642ndash1670 2013
[3] D Lopez-Perez A Ladanyi A JuttnerH Rivano and J ZhangldquoOptimization method for the joint allocation of modulationschemes coding rates resource blocks and power in self-organizing LTE networksrdquo in Proceedings of the IEEE INFO-COM 2011 pp 111ndash115 April 2011
[4] A Guo andM Haenggi ldquoSpatial stochasticmodels andmetricsfor the structure of base stations in cellular networksrdquo IEEE
Wireless Communications and Mobile Computing 15
Transactions on Wireless Communications vol 12 no 11 pp5800ndash5812 2013
[5] C-H Lee C-Y Shih and Y-S Chen ldquoStochastic geometrybased models for modeling cellular networks in urban areasrdquoWireless Networks vol 19 no 6 pp 1063ndash1072 2013
[6] W Lu and M D Renzo ldquoStochastic geometry modeling ofcellular networks analysis simulation and experimental valida-tionrdquo in Proceedings of the 18th ACM International Conferenceon Modelling Analysis and Simulation of Wireless and MobileSystems (MSWiM rsquo15) pp 179ndash188 New York NY USANovember 2015
[7] Y L Lee T C Chuah J Loo and A Vinel ldquoRecent advancesin radio resource management for heterogeneous LTELTE-Anetworksrdquo IEEE Communications Surveys amp Tutorials vol 16no 4 pp 2142ndash2180 2014
[8] F Mhiri K Sethom and R Bouallegue ldquoA survey on inter-ference management techniques in femtocell self-organizingnetworksrdquo Journal of Network and Computer Applications vol36 no 1 pp 58ndash65 2013
[9] X Chen L Li and X Xiang ldquoAnt colony learning methodfor joint MCS and resource block allocation in LTE Femtocelldownlink for multimedia applications with QoS guaranteesrdquoMultimedia Tools andApplications vol 76 no 3 pp 4035ndash40542017
[10] D Lopez-Perez X Chu A V Vasilakos and H Claussen ldquoOndistributed and coordinated resource allocation for interferencemitigation in self-organizing lte networksrdquo IEEEACMTransac-tions on Networking vol 21 no 4 pp 1145ndash1158 2013
[11] H Zhang H Liu J Cheng and V C M Leung ldquoDownlinkenergy efficiency of power allocation and wireless backhaulbandwidth allocation in heterogeneous small cell networksrdquoIEEE Transactions on Communications no 99 2017
[12] H Zhang W Zheng X Chu et al ldquoJoint subchannel andpower allocation in interference-limited OFDMA femtocellswith heterogeneous QoS guaranteerdquo in Proceedings of theIEEEGlobal Communications Conference (GLOBECOM rsquo12) pp4572ndash4577 December 2012
[13] H Elsawy E Hossain and M Haenggi ldquoStochastic geometryfor modeling analysis and design of multi-tier and cognitivecellular wireless networks a surveyrdquo IEEE CommunicationsSurveys amp Tutorials vol 15 no 3 pp 996ndash1019 2013
[14] H ElSawy A Sultan-Salem M-S Alouini and M Z WinldquoModeling and analysis of cellular networks using stochasticgeometry a tutorialrdquo IEEE Communications Surveys amp Tutori-als vol 19 no 1 pp 167ndash203 2017
[15] R Kwan and C Leung ldquoA survey of scheduling and inter-ference mitigation in lterdquo Journal of Electrical and ComputerEngineeringmdashSpecial Issue on LTELTE-AdvancedCellular Com-munication Networks pp 11ndash110 2010
[16] J G Andrews F Baccelli and R K Ganti ldquoA tractable approachto coverage and rate in cellular networksrdquo IEEE Transactions onCommunications vol 59 no 11 pp 3122ndash3134 2011
[17] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for OFDMA cellu-lar networksrdquo IEEE Transactions on Wireless Communicationsvol 10 no 12 pp 4294ndash4305 2011
[18] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for heterogeneouscellular networksrdquo IEEE Transactions on Communications vol60 no 7 pp 2029ndash2039 2012
[19] H Zhuang and T Ohtsuki ldquoA model based on poisson pointprocess for downlink K tiers fractional frequency reuse hetero-geneous networksrdquo Physical Communication vol 13 pp 3ndash122014 Special Issue on Heterogeneous and Small Cell Networks
[20] A A Gebremariam T Bao D Siracusa T Rasheed F Granelliand L Goratti ldquoDynamic strict fractional frequency reusefor software-defined 5G networksrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo16) pp 1ndash6May 2016
[21] N Lee D Morales-Jimenez A Lozano and R W HeathldquoSpectral efficiency of dynamic coordinated beamforming astochastic geometry approachrdquo IEEE Transactions on WirelessCommunications vol 14 no 1 pp 230ndash241 2015
[22] X Zhang and M Haenggi ldquoA Stochastic geometry analysisof inter-cell interference coordination and intra-cell diversityrdquoIEEE Transactions on Wireless Communications vol 13 no 12pp 6655ndash6669 2014
[23] F Baccelli andAGiovanidis ldquoA stochastic geometry frameworkfor analyzing pairwise-cooperative cellular networksrdquo IEEETransactions on Wireless Communications vol 14 no 2 pp794ndash808 2015
[24] J Yoon and G Hwang ldquoDistance-based inter-cell interferencecoordination in small cell networks stochastic geometry mod-eling and analysisrdquo IEEE Transactions on Wireless Communica-tions 2018
[25] B Błaszczyszyn M Jovanovicy and M K Karray ldquoHow userthroughput depends on the traffic demand in large cellularnetworksrdquo in Proceedings of the 12th International Symposiumon Modeling and Optimization in Mobile Ad Hoc and WirelessNetworks (WiOpt rsquo14) pp 611ndash619 May 2014
[26] H H Yang and T Q S Quek ldquoSIR coverage analysis incellular networks with temporal traffic a stochastic geometryapproachrdquo Corrossion 2018
[27] S M Yu and S-L Kim ldquoDownlink capacity and base stationdensity in cellular networksrdquo in Proceedings of the 11th Interna-tional SymposiumandWorkshops onModeling andOptimizationin Mobile Ad Hoc and Wireless Networks (WiOpt rsquo13) pp 119ndash124 May 2013
[28] YWang M Haenggi and Z Tan ldquoThemeta distribution of thesir for cellular networks with power controlrdquo Corrossion 2017
[29] M S Omar S A Hassan H Pervaiz et al ldquoMulti-objectiveoptimization in 5g hybrid networksrdquo IEEE Internet of ThingsJournal 2018
[30] Y Leo A Busson C Sarraute and E Fleury ldquoCall detail recordsto characterize usages and mobility events of phone usersrdquoComputer Communications vol 95 pp 43ndash53 2016
[31] I L Cherif Spectral and Energy Efficiency in 5G WirelessNetworks [PhD thesis] Univeristy Paris Saclay 2016
[32] S Mukherjee Analytical Modeling of Heterogeneous CellularNetworks Analytical Modeling of Heterogeneous Cellular Net-works Geometry Coverage and Capacity Cambridge Univer-sity Press 2014
[33] ldquoTechnical specification group radio access network base sta-tion (bs) radio transmission and reception 2012 Release 1123gpp ts 36104 v112rdquo 2012
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Wireless Communications and Mobile Computing 3
concrete RB assignment algorithms In particular the authorsin [26] study SIR coverage for a cellular network based onPPP A queue is associated with each BS that determinesthe BS transmission activity as a function of the trafficConsidering the traffic at each BS is independent interferersat a given time are then an independent thinning of the initialPPP and are still Poisson This model differs with this paperas we do not take into account eNodeB activity as a functionof the traffic but instead the resource allocation as a functionof the number of associated users to each eNodeB Alsostochastic geometry models can be specific to certain powercontrol scheme [28] or radio technologies as in [29] wherethe authors consider a K-tier heterogeneous network withtransmissions operating on the millimeter wave band
12 Contributions The primary contribution of this work isto offer an analytical model based on stochastic geometryto evaluate the performance of a downlink LTE systemtaking into account RB allocation strategies power controland traffic demands All these mechanisms have never beencombined into a single stochastic geometry model Thenumber of allocated resources for an eNodeB is assumedto follow the distribution of the number of clients in anMMCC queue It models the number of communicationsin progress when both the interarrival of the communicationsand their duration follow an exponential distribution Suchassumptions are pertinent in cellular networks as it has beenrecently shown in [30] We associate to these traffic demandsseveral resource allocation strategies All these algorithms arecombined with a power control mechanism that depends onthe channels quality Allocation strategies lead to non-PPP ascorrelation appears between the locations of the interferingnodes It prevents the use of the convenient properties ofthe PPP to compute interference distribution Neverthelesswe propose approximations that allow us to deal with thesecorrelations and to obtain an analytical method that is shownvery accurate with regard to simulations
We compare our model to classical optimizationapproaches where for a given configurationsample theallocation is optimized with regard to an objective functionTo our knowledge such comparison has never been donebefore It shows that geometry stochastic based model maybe relevant to offer tight approximations on wireless systemperformance
Models are evaluated through a large set of simulationsthat highlights benefits of our approach to design some keyparameters of the wireless system Results show that theobtained values for SIR coding and modulation rates corre-spond to the reference values of the standards and technicalLTE documents empirically proving that our model is able toapproximate performance of real systemsThiswork has beenpartially presented in [31]
13 Paper Organization The remainder of this work is orga-nized as follows In the next section we present the systemmodel We expose the assignment strategies in Section 3In Section 4 we derive the first and second moments ofinterference for each allocation strategy SIR distribution isassessed in Section 5 Numerical results and simulations are
presented and discussed in Section 6 We conclude the paperin Section 7
2 System Model
21 eNodeB Location and Interference eNodeB location ismodeled by a point process 119873119890 = 119883119894119894isinN distributed in R2
with intensity 120582119890 Its distribution is detailed in Section 22The eNodeB are numbered with regard to their distanceto the origin eNodeB 0 at 1198830 being the closest one Weconsider a downlink system between a typical user and itsattached eNodeB Without loss of generality we assume thatthis user is located at the origin The users are assumed tobe associated with their closest eNodeB with regard to theEuclidean distance The channels between eNodeB and thetypical user are modeled through a sequence of iid randomvariables (ℎ119894)119894isinNThe transmission power between an eNodeBand one associated user is given by the function 119879119909(119903) Thisfunction models the power control algorithm implementedby the eNodeB and depends on the distance 119903 between a userand its attached eNodeBThe schedulercontroller manages aset of 119877119861119898119886119909 resource blocks that are common to all eNodeBThey are thus shared between eNodeBThe RB are numberedfrom 0 to 119877119861119898119886119909 minus 1 The scheduler assigns one RB for eachuser but the model can be easily extended with a randomnumber of RB for each demand Traffic demands exceeding119877119861119898119886119909 at an eNodeB are not served The RB with index 0 isallocated to the typical user We show in the appendix thatthis choice does not impact the computations and any otherindex could be chosen instead An eNodeB interferes with thetypical user if and only if it reuses this RB Interference at thetypical user can be expressed as
where 119897(sdot) is the path loss function The argument of inter-ference is the distance between the typical user and itseNodeB (eNodeB 0) 119868(1198830) expresses interference whenthis distance is random and depends on the rv 1198830 119868(119903)expresses interference when this distance is given and equalto 119903 This notation is motivated by the fact that the rv 119883119894
and 119883119894 minus 119880119894 are correlated to 1198830 Moreover the mean andthe variance of interference will be computed for both a givenvalue of 1198830 and with regard to its distribution The rv 119908119894
indicates whether eNodeB 119894 interferes with the typical user(119908119894 = 0 or 1)119880119894 is the random variable modeling the locationof a user attached to the eNodeB 119894 (at 119883119894) We assume that119880119894 is uniformly distributed in the Voronoı cell formed by theprocess 119873119890 and with nucleus 119883119894 The main notations usedthroughout this paper are given in Table 1
22 Point Process Modeling eNodeB The point process mod-eling eNodeB is a modified homogeneous PPP The process119873119890 1198830 = 119883119894119894gt0 is Poisson in R2 119861(0 1198830) where119861(0 1198830) is the ball centered at the origin and with radius1198830 We choose a distribution for 1198830 that makes theprocess119873119890 different of a PPP Indeed with a PPP the typicaluser at the origin lies in a Voronoı cell that is greater in
4 Wireless Communications and Mobile Computing
Table 1 Principal notation
119873119890 point process modeling eNodeB119868(1198830) interference at the typical user The distance between the typical user and eNodeB 0 is the rv 1198830119868(119903) interference at the typical user The distance between the typical user and eNodeB 0 is equal to 119903119908119894 rv indicating if eNodeB 119894 interferes with the typical user119863119894 number of RB allocated to eNodeB 119894119879119909(119903) transmission power between an eNodeB and its user at distance 119903Δ spatial reuse parameter (an RB is reused every Δ eNodeB in average)119891119883119894(sdot) PDF of 119883119894119891119883119894119883119895(sdot sdot) joint PDF of (119883119894 119883119895)1198911198941198830=119903
(sdot) conditional PDF of 119883119894 given that 1198830 = 119903119891119894119895
1198830=119903(sdot) conditional PDF of (119883119894 119883119895) given that X0 = 119903
11989101198830
(sdot) PDF of the distance between the typical user (at the origin) and its serving eNodeB 01198910119880119894minus119883119894
(sdot sdot) conditional PDF of the distance between a user and its attached nucleus119883119894 given 119883119894
average than the other cells Intuitively ldquobig cellsrdquo cover morespace than ldquosmall cellsrdquo and consequently the cell coveringthe origin has a greater size in average It is consistent witha modeling where users are homogeneously scattered in theplane but not with our assumption where the network hasbeen dimensioned to have the same load in average (the samenumber of users) in each cell It is more realistic as it hasbeen shown in [30] where a homogeneous load is observedfor the different cells independently of their sizes In thiscase the typical cell covering the typical user must have thesame distribution as the other cells Therefore we considerthe distribution of the distance 1198830 under Palm measureMore precisely this distribution corresponds to the distancebetween the nucleus of a typical cell under Palmmeasure anda point uniformly distributed in this cell The distributionof this distance is not known but we use the approximationpresented in [32] (page 133) We set the distribution of thedistance between the typical user and its closest eNodeB at1198830 as
with 119888 = 125 The angle between the lines (01198830) and theabscissa is uniformly distributed in [0 2120587)
The distribution of the distance between a user in agiven Voronoı cell and its nucleus (119883119894 minus 119880119894) follows thesame definition and consequently the same distribution Inthe next section we present the different RB assignmentstrategies evaluated in this paper
3 Assignment Strategies
We consider four different allocation strategies We beginby two simple RB allocation schemes independent andstatic allocations Then we develop a more realistic alloca-tion strategy using the 119872119872119877119861119898119886119909119877119861119898119886119909 queue named119872119872119877119861119898119886119909119877119861119898119886119909 allocation hereafter Also amore globalapproach where the RB are assigned in order to minimizethe sum of interference at each user is considered but for
which we do not propose a mathematical resolution All theallocation strategies are set in such a way that a given RB isreused every Δ (Δ ge 1) eNodeB in average The mean loadin each cell and equivalently the mean number of RB used byan eNodeB are then 119877119861119898119886119909Δ The reuse factor Δ reflects thenetwork load
31 Independent Allocation Thinning With this strategyeach eNodeB selects its resources independently of theother eNodeB Therefore we assume that an eNodeB hasa probability 1Δ to reuse the RB with index 0 The pointprocess describing the interfering eNodeB is then a thinnedPPP in R2 119861(119874 1198830) with intensity 120582119890Δ32 Static Allocation We assign a constant proportion119877119861119898119886119909Δ of the available resources to each eNodeBThey areallocated in their index order eNodeB 0 uses RB from 0 to119877119861119898119886119909Δ minus 1 (it includes the typical user) eNodeB 1 from119877119861119898119886119909Δ to 2(119877119861119898119886119909Δ) minus 1 etc We take the integer partof these fractions when 119877119861119898119886119909 is not a multiple of Δ Weloop when all resources have been used Consequently theeNodeB interfering with the typical user has an index withthe form 119896 sdot Δ with 119896 gt 033119872119872119877119861119898119886119909119877119861119898119886119909Allocation In an119872119872119862119862queuecustomers arrive according to a Poisson process (in R) andthe service times are exponentially distributed It modelsa system with 119862 resourcesservers and a capacity of thesame size A customer cannot enter in the system if allresourcesservers are busy We associate with each eNodeBan independent 119872119872119877119861119898119886119909119877119861119898119886119909 queue to model thenumber of RB in use The servers model the RB Upon thearrival of a requestuser an RBserver is used for a timeexponentially distributed If no RB is available the requestis rejected In order to have a mean reuse factor of Δ theparameter of the queue (the load) denoted 120588 is set in sucha way that the mean number of customers in the system orequivalently the mean number of busy resource blocks isequal to 119877119861119898119886119909Δ The distribution of the number of busy
Wireless Communications and Mobile Computing 5
(a) RB allocation (b) Impact of RB allocation on interferer
Figure 1 In Figure 1(a) an example of RB allocation is given eNodeB 0 uses 6 RB (the first one has index 14 and the other ones are indexedfrom 0 to 4) It uses the RB with index 0 to communicate with the typical user1198770 is the index of the last resource used by eNodeB 0 As it usesRB indexed from 14 to 4 1198770 = 4 hereThe number of used RB for the other eNodeB are1198631 = 41198632 = 3 etc An eNodeB interferes if and onlyif it reuses the resource 0 in this example eNodeB 3 and 6 In Figure 1(b) we plot a sample of the point process describing eNodeB locationwith the same RB assignmentThe eNodeB interfering with the typical user is then a dependent thinning of the original point process wherethe thinning involves eNodes 3 6 and 10 (this last interfering eNodeB was not shown in Figure 1(a))Their Voronoı cells are colored in green
resource blocks for a given eNodeB 119894 (119894 gt 0) denoted 119863119894 isthen given by
P (119863119894 = 119896) = 1205870 120588119896119896 (3)
where 1205870 = P(119863119894 = 0) and 119877119861119898119886119909 ge 119896 ge 0These RB are allocated in a cyclic order If the last119877119861 used
by eNodeB 119894 minus 1 has index 119896 eNodeB 119894 uses 119877119861 indexed from(119896 + 1)119898119900119889(119877119861119898119886119909) to (119896 + 119863119894) 119898119900119889(119877119861119898119886119909)For eNodeB 0 we do not consider the total number
of allocated RB (1198630) but instead a random variable 1198770 Itdescribes the index of the last RB used by eNodeB 0 Indeedfor this particular eNodeB the quantity used in practice tocompute the next allocation (RB indexes used by eNodeB 1)is 1198770 rather than 1198630 A formal definition of 1198770 is given inappendix (Appendix A) An example of allocation is given inFigure 1
The distribution of 1198770 is set according to the stationarydistribution of a Markov chain The transition probabilitiesof this Markov chain are
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (5)
where (119897 119898) isin 0 1 119877119861119898119886119909 minus 12 The motivation forthis particular construction is to keep the probability ofusing the resource 0 between eNodeB homogeneous Moreprecisely it is built in order to verify the property given inProposition 1 Details about the Markov chain constructionare given in appendix (Appendix A) It is worth noting thatother distributions for the resource demands (given by (3)in our case) can be considered as well As soon as thedistribution of1198770 verifies Proposition 1 themethod proposedin this paper holds
34 Property of These Assignment Strategies We define moreprecisely the sequence of rv (119908119894)119894isinN It indicates whicheNodeB interferes with the typical user It was already usedin (1)
119908119894 = 1 if eNodeB at 119883119894 uses RB with index 00 otherwise (6)
By convention we set 1199080 = 1 as In the following weshall thus assume that P(119908119894 = 1 | 1199080 = 1) = P(119908119894 = 1)Proposition 1 For the three allocation strategies defined inSections 31 32 and 33 the following property holds
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (7)
The proofs for the first two strategies are straightforwardFor the 119872119872119877119861119898119886119909119877119861119898119886119909 strategy the distribution of 1198770has been set to verify this property (see Appendix A)
35 Heuristics We compare these strategies to heuristics thataim to optimize interference or spatial reuse for a givenconfiguration It allows us to compare our RB assignmentand performs in a cyclic manner around a typical user tostrategies where a controller in charge of a set of eNodeBwill assign RB in order to optimize a certain objective func-tion For the heuristic minimizing the sum of interferencenamed ldquominimize interferencerdquo hereafter the consideredoptimization problem is similar to the one developed in[3] The problem has been shown NP-hard so we use agreedy algorithm to find a solution The number of usersassociated with each eNodeB follows the same distribution asthe119872119872119877119861119898119886119909119877119861119898119886119909 allocation Then we consider usersin a random order and apply Algorithm 1 to associate an RBto a user It chooses the resource block that minimizes thesum of interference Obviously we compute interference only
6 Wireless Communications and Mobile Computing
Data 119877119861119898119886119909 Total number of RBcost sum of interference for the current allocationbestCost sum of interference for the best allocation strategy initialized to -1Result Assign an RB to a user This algorithm is called by the controller for each
new user(1) for each 119903119887 isin 0 1 119877119861119898119886119909 minus 1 do(2) if rb is free then(3) assign rb to this user(4) costlarr sum Interference()(5) if bestCost lt0 or costltbestCost then(6) Save this allocation strategy(7) bestCost = cost(8) end(9) end(10) end(11) Assign the saved allocation strategy
Algorithm 1 Minimizing global interference
for the users already assigned The typical user is consideredin last when the system has reached the targeted load Thisalgorithm mimics an assignment strategy where the RB areassigned at the arrival of the users request without changingthe already assigned RB
The second heuristic ldquomaximize reuse distancerdquo maxi-mizes the distance at which the RB are reused Each user isconsidered in a random order Different RB are assigned tothe 119877119861119898119886119909 first users When assigning an RB to the otherusers the controller chooses the RB for which the reusedistance is maximum The typical user is considered in lastThis second heuristic may correspond to a case where thecontroller does not have information on channel conditionsand interference but knows the distances between eNodeB
4 Interference Characterization
We derive the mean and the variance of interference for thethree assignment strategies defined in the previous section
The point process modeling interferers is a dependentthinning of the original PPP Consequently conditions formean (respectively variance) to be finite with a PPP also holdfor our point process the path loss function 119897(sdot)must belongto 1198711 (respectively 1198712)41 Distribution of Distances between the Typical User andeNodeB (119883119894) As a preamble we give the PDF of thedistance between the typical user at the origin and eNodeBBoth PDF of 119883119894 and joint distribution of (119883119894 119883119895) arederived These PDF are used in the computation of the meanand the variance of interference
In the numerical evaluation we shall condition interfer-ence by the distance 1198830 It allows us to study interferencefor a given distance between the typical user and its attachedeNodeB It is also motivated by the computation of the SIRwhere both interference and the typical user signal strengthdepend on the distance 1198830
For our model the PDF of 119883119894with 119894 gt 0 given 1198830 = 119903is
1198911198941198830=119903 (119906 119903) = (120582119890120587)119894(119894 minus 1)2119906 (1199062 minus 1199032)119894minus1 119890minus120582119890120587(1199062minus1199032)1119906gt119903 (8)
The joint PDF of (119883119894 119883119895)with 119895 gt 119894 gt 0 given 1198830 =119903 is1198911198941198951198830=119903
(119906 V 119903)= (120582119890120587)119895(119894 minus 1) (119895 minus 119894 minus 1)4119906V (V2 minus 1199062)
119895minus119894minus1
times (1199062 minus 1199032)119894minus1 119890minus120582119890120587(V2minus1199032)1Vgt119906gt119903(9)
To obtain the PDF when 1198830 is not set (119891119883119894(sdot) and119891119883119894119883119895(sdot)) it suffices to integrate the two conditional PDFwith regard to the PDF of 1198830 given in (2)
42 Mean of Interference Themean is derived from (1)
E [119868 (100381710038171003817100381711988301003817100381710038171003817)]
= E [ℎ1] +infinsum119894=1
E [119908119894]E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) sdot 119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)] (10)
In this equation 119908119894 has been separated from the expecta-tion as it is independent of the process 119873119890 (according to thedefined strategies) We derive E[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)] andE[119908119894] in the two next sections
421 Computation of E[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)]No Power Control In absence of power control ie when119879119909(sdot) is constant or independent of the process 119873119890 a closedformula may be expressed for (10) E[119879119909(119883119894 minus119880119894) sdot 119897(119883119894)]
Wireless Communications and Mobile Computing 7
is then given by E[119879119909]E[119897(119883119894)] Expectation of 119897(119883119894) isobtained from the distribution of 119883119894 given in Section 41
E [119868 (100381710038171003817100381711988301003817100381710038171003817)]
= E [ℎ1] +infinsum119894=1
E [119908119894]E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817)]E [119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)] (11)
Power Control When the transmission power depends onthe distance between the receiver and its attached eNodeB(119879119909(119883119894 minus 119880119894)) the computations are more complex119864[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)] cannot be approximated byE[119879119909(119883119894minus119880119894)]119864[119897(119883119894)] as the size of the Voronoı cell withnucleus 119883119894 depends on its distance to the origin The jointdistribution of (119883119894 119883119894 minus119880119894) being unknown we proposethe following approximation
The PDF of 119880119894 minus 119883119894 is the same as 1198800 minus 1198830 givenby (2) Its parameter 119888119894(119883119894) depends on 119883119894 119888119894(119883119894) =14radic120582119890(120574119894119883119894 + 119887) with 120574119894 = (12 minus 119887)E[119883119894] and 119887 = 033The motivation and the computation details for this PDF aregiven in appendix (Appendix B) We obtain
E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) sdot 119897 (1003817100381710038171003817119883119894
Often in real systems the transmission power cannot beset arbitrarily and is limited to a set of predetermined valuesThe transmission power function can then be representedas a step function 119879119909(119903) = sum119873119879
119894=1 1199051198941119903isin[120572119894minus1120572119894] where 119873119879
is the number of possible transmission powers 119905119894 the 119894119905ℎtransmission power value and [120572119894minus1 120572119894] the distance intervalbetween a user and its eNodeB at which this transmissionpower is used An example of such setting is given in thenumerical evaluation section In this case (14) becomes
E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) 119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)]= 119873119879sum
119894=1
119905119895 int+infin
0(119890minus120582119890119888119894(119903)1205871205722119895 minus 119890minus120582119890119888119894(119903)1205871205722119895minus1)119891119883119894 (119903) 119889119903
(15)
When the computation is performed for a given distance1198830 the PDF 119891119883119894(sdot) in (12) and (15) must be replaced by1198911198941198830=119903(sdot sdot) (given in Section 41)
422 Computation of E[119908119894] Finally in order to compute(10) we need to expressE[119908119894] First note that E[119908119894] = P(119908119894 =1)Proposition 2 The probability for eNodeB 119894 to interfere withthe typical user is given by
(i) Independent allocation
P (119908119894 = 1) = 1Δ (16)
(ii) Static allocation
P (119908119894 = 1) = 1119894sdot119898119900119889(Δ)=0 (17)
E [119879119909 (1003817100381710038171003817119880119894 minus 1198831198941003817100381710038171003817) 119879119909 (10038171003817100381710038171003817119880119895 minus 119883119895
As for the mean complexity lies in the correlationbetween 119883119894 and 119880119894 minus 119883119894 The term E[119879119909(119880119894 minus 119883119894)2 sdot119897(119883119894)2] is computed with the same method as the firstmoment
Computation of E[119879119909(119880119894minus119883119894)119879119909(119880119895minus119883119895)119897(119883119894)119897(119883119895)]As 119880119894 minus 119883119894 (respectively 119880119895 minus 119883119895) depends on 119883119894(respectively 119883119895) we condition by the distribution of(119883119894 119883119895) given in Section 41 Given 119883119894 and 119883119895 we usethe same PDF as in (B1) assuming that 119880119894 minus 119883119894 and 119880119895 minus119883119895 are independent The considered joint distribution of(119883119894 119883119895 119880119894 minus 119883119894 119880119895 minus 119883119895) becomes
1198910119880119894minus119883119894 (119906 119903) 1198910119880119895minus119883119895 (V 119904) 119891119883119894119883119895 (119903 119904) (21)
When the distance 1198830 is fixed 119891119883119894119883119895(sdot sdot) must bereplaced by the PDF 119891119894119895
1198830=119903(sdot sdot sdot) given in Section 41
Computation of E[119908119894119908119895] It has been shown that the sequence(119908119894)119894gt0 verifies (7) for the three strategies It allows us toexpress E[119908119894119908119895] with 119894 gt 119895 as
E [119908119894119908119895] = P (119908119894minus119895 = 1)P (119908119895 = 1) (22)
8 Wireless Communications and Mobile Computing
Proposition 3 The joint probability for two eNodeB 119894 and 119895(119895 gt 119894) to interfere with the typical user is given by
(i) Independent allocation
E [119908119894119908119895] = 1Δ2 (23)
(ii) Static allocation
E [119908119894119908119895] = 1119894sdot119898119900119889(Δ)=0119895sdot119898119900119889(Δ)=0 (24)
where P(119908 = 1) is given by (18)5 Signal over Interference Ratio (SIR)
In our model there is a strong correlation between theinterfering eNodeB It is generated by the allocation strategiesand cannot be neglected Also a correlation exists betweenthe location of an eNodeB and the size of its Voronoı cellConsequently classical approach based on PPP which uses
Laplacian transform for instance cannot be applied hereand a formal derivation of interference distribution seemsintractable
Nevertheless the different simulations presented in thenext section will show that the PDF of interference can beapproximated by a log-normal distribution The parametersof this distribution mean and variance denoted by 119898119868119889119861
(sdot)and 120590119868119889119861(sdot) are directly derived from the previous analyticalcomputations The classical mapping between log-normaland normal parameters can be applied to derive parametersof the normal distribution when interferences are expressedin decibel In the following a variable is indexed by 119889119861 whenit is expressed in decibel
We get
P (119878119868119877119889119861 le 120573119889119861) = P (10sdot log10 (119875119905 (10038171003817100381710038171198830
1003817100381710038171003817) ℎ0119897 (100381710038171003817100381711988301003817100381710038171003817)) minus 119868119889119861 (10038171003817100381710038171198830
1003817100381710038171003817) le 120573119889119861) (26)
= P (119868119889119861 (100381710038171003817100381711988301003817100381710038171003817) ge minus120573119889119861 + 10
When ℎ0 is not constant the expectation with regard toits distribution must be taken into account in (28) In (29)ℎ0 is assumed to be constant that equals to 1 In these twoequations 119890119903119891(sdot) is the error function6 Numerical Results
We consider an E-UTRA channel with a bandwidth of5MHz with 119877119861119898119886119909 = 15 [33] The path loss function isthe same as [3] It is expressed in dB 119897(119903) = minus1281 minus376 sdot log10(119903) where 119903 is the distance (in km) 119879119909(sdot) isset in such a way to guarantee to each user a minimumreceiving power We set the transmission power function119879119909(sdot) to ensure a signal power greater than or equal tominus724dBm at the reception as specified in [33] For each 50meters (from 50 to 500 meters) we compute the minimumtransmitting power required to reach this threshold (119879119909(119903) sdot119897(119903) ge minus724 dBm for each interval of 50 meters leading to 10possible transmission powers)This step function models thecase where eNodeB has a finite set of predetermined powerThe process intensity modeling eNodeB is equal to 225 perkm2 It corresponds to the intensity of base stations in Paris(httpswwwantennesmobilesfr) Random variables ℎ119894 are
supposed constant equal to 1 This assumption facilitatesinterpretation of the results but any distribution can beconsidered as well It simply adds a factor in terms of variance(cf (20)) We simulate the different strategies through asimulator coded inC available here (httpwwwanthonybus-sonfrindexphppublications) In all simulations and nu-merical results we consider 50 eNodeB The different sumsin the equations (eg (10) and (20)) are then limited to 50For each set of parameters simulations have be run from10000 to several millions times depending on the evaluatedquantities The number of simulationssamples has been setin order to have negligible confidence intervals They areconsequently not shown in the different figures
Mean and Variance of Interference In Figure 2 we plot themean and the standard deviation of interference obtainedfrom simulations and computed from formulas (10) (19) and(20)when the distance 1198830 variesThe theoretical evaluationcloselymatches empirical estimators obtained by simulationsAs expected the highest interference level is observed forthe independent allocation and the lowest level for the staticallocation The static and the 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tions offer equivalent results with a multiplication factor
Figure 2 Interferencemean and standard deviation as a function of the distance between the typical user and its eNodeB (1198830) Simulationresults are given by points and solid lines and theoretical evaluations by the dotted lines
Table 2 Mean and standarddeviation for the119872119872119877119861119898119886119909119877119861m119886119909 allocation and the two heuristics
Assignment strategy Δ = 3 Δ = 6 Δ = 9mean std-dev mean std-dev mean std-dev
119872119872119877119861119898119886119909119877119861119898119886119909 178119890 minus 11 861119890 minus 11 366119890 minus 12 750119890 minus 12 155119890 minus 12 316119890 minus 12Minimize Interference 994119890 minus 12 307119890 minus 11 186119890 minus 12 188119890 minus 12 756119890 minus 13 626119890 minus 13Maximize reuse distance 526119890 minus 11 751119890 minus 10 118119890 minus 11 107119890 minus 10 258119890 minus 12 109119890 minus 11
varying from 11 to 87 for the mean interference Insteadthe independent allocation differs deeply Mean and standarddeviation are multiplied by a factor ranging from 17 to 700for the mean and can reach up to 26 times 104 for the standarddeviation (for 1198830 = 002 km) The use of the thinnedpoint process is thus questionable to approximate realisticassignment strategies This strategy is no more considered inthe following
Extrapolation of Interference Distribution We compareempirical distributions obtained from simulations to knowndistributions for the different strategies and the two heuristicswhen Δ = 3 6 and 9 Three of these empirical distributionsare shown in Figure 3 (in dB) The distribution parametershave been set according to the maximum likelihood Thedistributions that best fit simulations vary according to theallocation strategy and the reuse factorΔ For 4 cases over 12the distribution minimizing error is the normal distributionAs soon as an asymmetry is observed other distributionsare more accurate log-normal Weibull and Inverse GammaWe perform two hypothesis tests T-test and Smirnov-Kolmogorov for all these scenarios The alternative hypoth-esis is systematically ruled out For the 119872119872119877119861119898119886119909119877119861119898119886119909
strategy best fits are given by the normal (Δ = 3 and 6) andWeibull distributions (Δ = 9)
Nevertheless as mentioned in Section 5 when estimatingthe SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy weuse the normal law to model interference Indeed even if thisdistribution is not systematically the most accurate it offers agood approximation as shown inFigure 4 where the empiricaland normalCDF are plottedMoreover simulationswill showthat this assumption does not impact the accuracy of theanalytical model
Heuristics Simulations have shown that allocations resultingfrom the two heuristics correlate the transmission powersthe distance between eNodeB reusing the same resourceand the random variables 119908119894 Whenminimizing interferencecorrelations are caused by resource blocks allocated to usersthat require a low transmission power and that can be reusedat a short distance and inversely The transmission poweris then correlated to 119908119894 and to the distance of eNodeB thatreuse the same resource These complex phenomena impedethe proposal of an analytical model for these two heuristicsFor comparison purposes Table 2 reportsmean and standarddeviation for the two heuristics and the119872119872119877119861119898119886119909119877119861119898119886119909
Figure 3 Interference distribution empirical (histogram) and extrapolation (curve) Among the classical distributions the bestextrapolations are Gamma (m=1003 var=977) normal (m=2823 var=2216) and Weibull (a=2514b=575) for the heuristic MMRBallocation Δ = 6 and static allocation Δ = 9 respectively
Figure 4 CDF of interference for the 119872119872119877119861119898119886119909119877119861119898119886119909 strategy Empirical distributions are represented through the solid lines CDFof the Normal distribution are plotted with dotted lines Their parameters are obtained from the theoretical formulas The 119871+infin errors are338119890minus02 493119890minus02 and 565119890minus02 for Δ = 3 6 and 9 respectively
Figure 5 Mean interference when the number of RB variesThe load is adjusted to keep a constant reuse factor (Δ = 36 or 9) The distance 1198830 is set to 50 meters Dotted linescorrespond to the theoretical evaluation ofmean interference for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation and points to simulations
allocationTheheuristicminimizing interference presents thebest results This difference was expected as the objectivefunction minimizes interference and because the secondheuristic maximizes the reuse distance without taking intoaccount transmission powers 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tion offers interesting results with an intermediary interfer-ence level between the two heuristics Even if interference islower for the heuristic minimizing interference its complex-ity in terms of feedbackmeasures from users and eNodeBis significantly greater than the other strategies Indeed itassumes that the scheduler knows before an assignment theinterference contribution of each eNodeB on each user
Impact of the Number of RB We performed simulations andtheoretical estimations of interference for different values of119877119861119898119886119909 6 15 25 50 75 and 100 corresponding to differentbandwidth of E-UTRA [33] Results are shown in Figure 5 forthe 119872119872119877119861119898119886119909119877119861119898119886119909 allocation and the two heuristicsThe workload is adapted in order to keep the same Δreuse factor It appears that the mean interference is quiteinsensitive to the number of RB except when 119877119861119898119886119909 = 6The impacting factor is thus the reuse factor Δ rather thanthe number of RBAlso the same hierarchy between the threestrategies is observed except for 119877119861119898119886119909 = 6SIR Distribution We evaluate the SIR CDF according to(29) for the 119872119872119877119861119898119886119909119877119861119898119886119909 allocation This CDF iscompared to the one obtained by simulations in Figure 6(a)We clearly observe that the analytical model offers very
tight estimates of the SIR distribution The assumptionabout the normal distribution of interference does notintroduce a noticeable error The SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy and the two heuristics arecompared in Figure 6(b) The three CDF are similar and the119872119872119877119861119898119886119909119877119861119898119886119909 allocation still offers an intermediatedistribution between the two heuristics The 119871+infin normbetween the three CDF is 0192 0266 and 0221 for Δ = 36 and 9 respectivelyModulation and Coding Rate The SIR distribution allowsus to estimate modulations and coding rates that could beoffered to users We consider the thresholds between therequired SIR and coding rate given in [33] Applied to the SIRdistribution it gives the proportion of users that benefits froma certain modulation scheme and coding rate The couplemodulationcoding rate is referred to as transmission rate inthe following This proportion is given in Figure 7(a) for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation when the density of eNodeBincreases from 225 (the default intensity considered in theprevious evaluation) to 20 eNodeB per km2 It appears thatthe network densification even with a factor 10 does notsignificantly improve the transmission rate Only the besttransmission rate (64QAM-CR=45) really benefits from thisdensification with a proportion of users increasing from 35to 116
The spatial reuse has much more impact on the transmis-sion rates as it is shown in Figure 7(b) It clearly increasesthe transmission rates that are shifted from (QPSK CR=12-64QAM CR=23) for Δ = 3 to (16QAM CR=12-64QAMCR=45) for Δ = 9 The mean transmission rate increasesalmost of a factor 2 from 231 to 424 bitsbaud
Finally we compare in Figure 7(c) the transmission ratesfor the three allocation strategies As already observed inthe previous plots the heuristic minimizing interferenceoffers better performance but is comparable to the two otherstrategies For instance for Δ = 6 the mean transmis-sion rates are close 363 323 and 392 bitsbaud for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation the heuristic maximizingthe reuse distance and the one minimizing interferencerespectively
7 Conclusion
In recent LTE standards a server may assign RB to a set ofeNodeB to control resource usage and optimize performancein a global way Even if it exists solutions to address this prob-lem for a given configuration and topology literature lacksmodels that evaluate performance of these RB allocations fora wide range of scenarios and at large scale
To address this problem we propound a spatial stochasticmodel that takes into account RB assignment strategiesrealistic traffic demands and power control We proposeanalytical estimates for the two first moments of interferenceThis computation is based on two approximations the distri-bution of the distance between a point uniformly distributedin a cell that models the user-eNodeB distance and thejoint distribution of the distances eNodeB-origin and user-eNodeB For the latter simulations have shown that these
(a) SIR empirical and theoretical CDF for the119872119872119877119861119898119886119909119877119861119898119886119909allocation
MMRB Delta=3MMRB Delta=6MMRB Delta=9MI Delta=3 MI Delta=6
MI Delta=9 MD Delta=3 MD Delta=6 MD Delta=9
0 10 20 30 40minus10훽 (dB)
00e+00
20eminus01
40eminus01
60eminus01
80eminus01
10e+00
CDF
minus SI
NR
(b) SIR empirical CDF for the three strategies MI maximizeinterference MD maximize distance
Figure 6 (a) We compare the distribution of SIR obtained by simulations (solid lines) and given by (29) (dotted lines) Errors with regardto the 119871+infin norm are 247119890 minus 02 154119890 minus 02 and 158119890 minus 02 for Δ = 3 6 and 9 respectively (b) CDF for the three allocation strategies(119872119872119877119861119898119886119909119877119861119898119886119909 and Heuristics)
(c) 119872119872119877119861119898119886119909119877119861119898119886119909 and the two heuristics Δ = 6 120582119890 = 225
Figure 7 Modulation and coding rate for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation strategy when the density of eNodeB increases (Figure 7(a))for different reuse factor (Figure 7(b)) and comparison with the two heuristics (Figure 7(c))
variables are strongly correlated particularly for the points ofthe process close to the origin This correlation impacts thetransmission power used by the eNodeB and consequentlyinterference and SIR distributions The derivation of the SIRdistribution allows us to express the classical outage but alsothe transmission rates in terms of modulationcoding ratethat provide interesting insights on the throughput offers tothe users Also it appears that eNodeB densification does not
significantly improve the network performance in terms ofthroughput except for a small percentage of users for whichthe transmission rate is increased Spatial reuse expressedthrough the parameter Δ in our study has a much moreimpact on the performance as the transmission rates aresignificantly increased This spatial reuse must be expressedas the ratio between the total number of available resourcesover the number of usersrequests per cell Indeed we have
Wireless Communications and Mobile Computing 13
observed that the system performance is quite insensitive tovariation of these quantities when this ratio stays constant
Beside simulations show that the RB assignment strate-gies developed for the model give results close to the onesgiven by classical optimization problem which minimizesglobal interference or maximizes spatial reuse distances Itempirically shows that our analytical model is able to evaluateperformance of classical optimization approaches but at largescale in terms of number of nodes configurations andtopologies
Appendix
A Construction of the Markov Chain
In this section we introduce the Markov chain used to set thedistribution of 1198770 We consider that the RB used between thetypical user and its eNodeB has index 119903119890119904 (119903119890119904 isin 0 119877119861119898119886119909minus1)We shall prove that this index does not impact calculation(we have chosen 119903119890119904 = 0 in this paper by sack of simplicity)
We set the distribution of the number of used resourceblocks at eNodeB 0 in such a way that it leads to the followingproperty (119895 gt 119894 ge 0)
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (A1)
In order to obtain this property we first introduce somepreliminary notations and results An example of allocationwith the different notations is shown in Figure 1 We define asequence of rv (119877119899)119899ge0 If eNodeB 119894 is the 119899119905ℎ eNodeB using119903119890119904 and if 119864119899119889119899 is the index of the last RB used by this eNodeBthen
119877119899 = 119864119899119889119899 minus 119903119890119904 if 119864119899119889119899 gt 119903119890119904119864119899119889119899 + 119877119861119898119886119909 minus 119903119890119904 otherwise (A2)
If eNodeB 119894 is the 119899119905ℎ interfering eNodeB (using 119903119890119904) theneNodeB 119895 (119895 gt 119894) is the 119899 + 1119905ℎ if and only if
119863119896 ge 119877119861119898119886119909 (A4)
In this case 119877119899+1 is given by
119877119899+1 = 119877119899 +119895sum
119896=119894+1
119863119896 minus 119877119861119898119886119909 (A5)
As (119863119894)119894gt0 are iid the sequence (119877119899)119899ge0 is a homoge-neous Markov chain The transition probabilities for (119897 119898) isin0 1 119877119861119898119886119909 minus 12 are given by 119875119897119898 = P(119877119899+1 = 119898 | 119877119899 = 119897)
The event 119877119899+1 = 119898 given that 119877119899 = 119897 occurs if andonly if it exists 119895 ge 0 such that
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (A10)
We set the distribution of 1198770 with the correspondingstationary distribution Consequently the sequence (119877119899)119899ge0becomes identically distributed where 119877119899 follows the sta-tionary distribution for all 119899 ge 0 Moreover according to(A3) and (A4) the property given in (A1) holds Indeed 119877119899sum119895minus1
119896=119894+1119863119896 and sum119895
119896=119894+1119863119896 have the same distribution as 1198770sum119895minus119894minus1
119896=1119863119896 andsum119895minus119894
119896=1119863119896 respectively In the model evaluation
the stationary distribution is obtained numerically from thetransition probabilities As soon as 1198770 follows the stationarydistribution the index 119903119890119904 can be chosen arbitrarily
B Conditional Distribution of 119880119894minus119883119894Simulations have shown that given 119883119894 the distribution of119883119894 minus 119880119894 is still close to the one given by (2) but with adifferent parameter So we assume that the distribution of119883119894 minus 119880119894 given 119883119894 is equal to (2) but with a parameterfunction of 119883119894 denoted 119888(119883119894)
These simulations have also shown that E[119883119894 minus 119880119894 |119883119894 = 119903]may be approximated as an affine function 119886119894119903 + 119887119894We define 119888119894(sdot) accordingly It leads to
To set 119886119894 and 119887119894 we use the two following propertiesE [E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817 | 10038171003817100381710038171198831198941003817100381710038171003817 = 119903]] = E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817] (B3)
= 12radic120582119890 (B4)
E [E [1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817 | 1003817100381710038171003817119883119894
1003817100381710038171003817 = 119903]] = E [119886119894 10038171003817100381710038171198831198941003817100381710038171003817 + 119887119894] (B5)
Equation (B4) is derived from (2) and (B5) from ourassumption on 119888119894(119903) 119886119894 is then set as a function of 119887119894 E[119883119894]and 120582119890 119887119894 is obtained from our simulations For a givenintensity 119887119894 has been observed almost constant with regardto 119894 The best approximation as a function of the intensity isgiven by 119887119894 = 033radic120582119890 After a fewmanipulations we obtain
First note that by convention we have P(119908119894 = 1) = P(119908119894 =1 | 1199080 = 1) The event 119908119894 = 1 occurs if and only if it exists119895 ge 1 such that1198770 + 119894minus1sum
There is no data used in this paper Nevertheless the code ofour simulator is made available (the link is in the paper)
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] ldquoRequirements for further advancements for e-utra (lte-advanced) 3gpp technical specification ts 36913 release 13rdquo2015 httpwww3gpporg
[2] A S Hamza S S Khalifa H S Hamza and K Elsayed ldquoAsurvey on inter-cell interference coordination techniques inofdma-based cellular networksrdquo IEEE Communications Surveysamp Tutorials vol 15 no 4 pp 1642ndash1670 2013
[3] D Lopez-Perez A Ladanyi A JuttnerH Rivano and J ZhangldquoOptimization method for the joint allocation of modulationschemes coding rates resource blocks and power in self-organizing LTE networksrdquo in Proceedings of the IEEE INFO-COM 2011 pp 111ndash115 April 2011
[4] A Guo andM Haenggi ldquoSpatial stochasticmodels andmetricsfor the structure of base stations in cellular networksrdquo IEEE
Wireless Communications and Mobile Computing 15
Transactions on Wireless Communications vol 12 no 11 pp5800ndash5812 2013
[5] C-H Lee C-Y Shih and Y-S Chen ldquoStochastic geometrybased models for modeling cellular networks in urban areasrdquoWireless Networks vol 19 no 6 pp 1063ndash1072 2013
[6] W Lu and M D Renzo ldquoStochastic geometry modeling ofcellular networks analysis simulation and experimental valida-tionrdquo in Proceedings of the 18th ACM International Conferenceon Modelling Analysis and Simulation of Wireless and MobileSystems (MSWiM rsquo15) pp 179ndash188 New York NY USANovember 2015
[7] Y L Lee T C Chuah J Loo and A Vinel ldquoRecent advancesin radio resource management for heterogeneous LTELTE-Anetworksrdquo IEEE Communications Surveys amp Tutorials vol 16no 4 pp 2142ndash2180 2014
[8] F Mhiri K Sethom and R Bouallegue ldquoA survey on inter-ference management techniques in femtocell self-organizingnetworksrdquo Journal of Network and Computer Applications vol36 no 1 pp 58ndash65 2013
[9] X Chen L Li and X Xiang ldquoAnt colony learning methodfor joint MCS and resource block allocation in LTE Femtocelldownlink for multimedia applications with QoS guaranteesrdquoMultimedia Tools andApplications vol 76 no 3 pp 4035ndash40542017
[10] D Lopez-Perez X Chu A V Vasilakos and H Claussen ldquoOndistributed and coordinated resource allocation for interferencemitigation in self-organizing lte networksrdquo IEEEACMTransac-tions on Networking vol 21 no 4 pp 1145ndash1158 2013
[11] H Zhang H Liu J Cheng and V C M Leung ldquoDownlinkenergy efficiency of power allocation and wireless backhaulbandwidth allocation in heterogeneous small cell networksrdquoIEEE Transactions on Communications no 99 2017
[12] H Zhang W Zheng X Chu et al ldquoJoint subchannel andpower allocation in interference-limited OFDMA femtocellswith heterogeneous QoS guaranteerdquo in Proceedings of theIEEEGlobal Communications Conference (GLOBECOM rsquo12) pp4572ndash4577 December 2012
[13] H Elsawy E Hossain and M Haenggi ldquoStochastic geometryfor modeling analysis and design of multi-tier and cognitivecellular wireless networks a surveyrdquo IEEE CommunicationsSurveys amp Tutorials vol 15 no 3 pp 996ndash1019 2013
[14] H ElSawy A Sultan-Salem M-S Alouini and M Z WinldquoModeling and analysis of cellular networks using stochasticgeometry a tutorialrdquo IEEE Communications Surveys amp Tutori-als vol 19 no 1 pp 167ndash203 2017
[15] R Kwan and C Leung ldquoA survey of scheduling and inter-ference mitigation in lterdquo Journal of Electrical and ComputerEngineeringmdashSpecial Issue on LTELTE-AdvancedCellular Com-munication Networks pp 11ndash110 2010
[16] J G Andrews F Baccelli and R K Ganti ldquoA tractable approachto coverage and rate in cellular networksrdquo IEEE Transactions onCommunications vol 59 no 11 pp 3122ndash3134 2011
[17] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for OFDMA cellu-lar networksrdquo IEEE Transactions on Wireless Communicationsvol 10 no 12 pp 4294ndash4305 2011
[18] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for heterogeneouscellular networksrdquo IEEE Transactions on Communications vol60 no 7 pp 2029ndash2039 2012
[19] H Zhuang and T Ohtsuki ldquoA model based on poisson pointprocess for downlink K tiers fractional frequency reuse hetero-geneous networksrdquo Physical Communication vol 13 pp 3ndash122014 Special Issue on Heterogeneous and Small Cell Networks
[20] A A Gebremariam T Bao D Siracusa T Rasheed F Granelliand L Goratti ldquoDynamic strict fractional frequency reusefor software-defined 5G networksrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo16) pp 1ndash6May 2016
[21] N Lee D Morales-Jimenez A Lozano and R W HeathldquoSpectral efficiency of dynamic coordinated beamforming astochastic geometry approachrdquo IEEE Transactions on WirelessCommunications vol 14 no 1 pp 230ndash241 2015
[22] X Zhang and M Haenggi ldquoA Stochastic geometry analysisof inter-cell interference coordination and intra-cell diversityrdquoIEEE Transactions on Wireless Communications vol 13 no 12pp 6655ndash6669 2014
[23] F Baccelli andAGiovanidis ldquoA stochastic geometry frameworkfor analyzing pairwise-cooperative cellular networksrdquo IEEETransactions on Wireless Communications vol 14 no 2 pp794ndash808 2015
[24] J Yoon and G Hwang ldquoDistance-based inter-cell interferencecoordination in small cell networks stochastic geometry mod-eling and analysisrdquo IEEE Transactions on Wireless Communica-tions 2018
[25] B Błaszczyszyn M Jovanovicy and M K Karray ldquoHow userthroughput depends on the traffic demand in large cellularnetworksrdquo in Proceedings of the 12th International Symposiumon Modeling and Optimization in Mobile Ad Hoc and WirelessNetworks (WiOpt rsquo14) pp 611ndash619 May 2014
[26] H H Yang and T Q S Quek ldquoSIR coverage analysis incellular networks with temporal traffic a stochastic geometryapproachrdquo Corrossion 2018
[27] S M Yu and S-L Kim ldquoDownlink capacity and base stationdensity in cellular networksrdquo in Proceedings of the 11th Interna-tional SymposiumandWorkshops onModeling andOptimizationin Mobile Ad Hoc and Wireless Networks (WiOpt rsquo13) pp 119ndash124 May 2013
[28] YWang M Haenggi and Z Tan ldquoThemeta distribution of thesir for cellular networks with power controlrdquo Corrossion 2017
[29] M S Omar S A Hassan H Pervaiz et al ldquoMulti-objectiveoptimization in 5g hybrid networksrdquo IEEE Internet of ThingsJournal 2018
[30] Y Leo A Busson C Sarraute and E Fleury ldquoCall detail recordsto characterize usages and mobility events of phone usersrdquoComputer Communications vol 95 pp 43ndash53 2016
[31] I L Cherif Spectral and Energy Efficiency in 5G WirelessNetworks [PhD thesis] Univeristy Paris Saclay 2016
[32] S Mukherjee Analytical Modeling of Heterogeneous CellularNetworks Analytical Modeling of Heterogeneous Cellular Net-works Geometry Coverage and Capacity Cambridge Univer-sity Press 2014
[33] ldquoTechnical specification group radio access network base sta-tion (bs) radio transmission and reception 2012 Release 1123gpp ts 36104 v112rdquo 2012
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
4 Wireless Communications and Mobile Computing
Table 1 Principal notation
119873119890 point process modeling eNodeB119868(1198830) interference at the typical user The distance between the typical user and eNodeB 0 is the rv 1198830119868(119903) interference at the typical user The distance between the typical user and eNodeB 0 is equal to 119903119908119894 rv indicating if eNodeB 119894 interferes with the typical user119863119894 number of RB allocated to eNodeB 119894119879119909(119903) transmission power between an eNodeB and its user at distance 119903Δ spatial reuse parameter (an RB is reused every Δ eNodeB in average)119891119883119894(sdot) PDF of 119883119894119891119883119894119883119895(sdot sdot) joint PDF of (119883119894 119883119895)1198911198941198830=119903
(sdot) conditional PDF of 119883119894 given that 1198830 = 119903119891119894119895
1198830=119903(sdot) conditional PDF of (119883119894 119883119895) given that X0 = 119903
11989101198830
(sdot) PDF of the distance between the typical user (at the origin) and its serving eNodeB 01198910119880119894minus119883119894
(sdot sdot) conditional PDF of the distance between a user and its attached nucleus119883119894 given 119883119894
average than the other cells Intuitively ldquobig cellsrdquo cover morespace than ldquosmall cellsrdquo and consequently the cell coveringthe origin has a greater size in average It is consistent witha modeling where users are homogeneously scattered in theplane but not with our assumption where the network hasbeen dimensioned to have the same load in average (the samenumber of users) in each cell It is more realistic as it hasbeen shown in [30] where a homogeneous load is observedfor the different cells independently of their sizes In thiscase the typical cell covering the typical user must have thesame distribution as the other cells Therefore we considerthe distribution of the distance 1198830 under Palm measureMore precisely this distribution corresponds to the distancebetween the nucleus of a typical cell under Palmmeasure anda point uniformly distributed in this cell The distributionof this distance is not known but we use the approximationpresented in [32] (page 133) We set the distribution of thedistance between the typical user and its closest eNodeB at1198830 as
with 119888 = 125 The angle between the lines (01198830) and theabscissa is uniformly distributed in [0 2120587)
The distribution of the distance between a user in agiven Voronoı cell and its nucleus (119883119894 minus 119880119894) follows thesame definition and consequently the same distribution Inthe next section we present the different RB assignmentstrategies evaluated in this paper
3 Assignment Strategies
We consider four different allocation strategies We beginby two simple RB allocation schemes independent andstatic allocations Then we develop a more realistic alloca-tion strategy using the 119872119872119877119861119898119886119909119877119861119898119886119909 queue named119872119872119877119861119898119886119909119877119861119898119886119909 allocation hereafter Also amore globalapproach where the RB are assigned in order to minimizethe sum of interference at each user is considered but for
which we do not propose a mathematical resolution All theallocation strategies are set in such a way that a given RB isreused every Δ (Δ ge 1) eNodeB in average The mean loadin each cell and equivalently the mean number of RB used byan eNodeB are then 119877119861119898119886119909Δ The reuse factor Δ reflects thenetwork load
31 Independent Allocation Thinning With this strategyeach eNodeB selects its resources independently of theother eNodeB Therefore we assume that an eNodeB hasa probability 1Δ to reuse the RB with index 0 The pointprocess describing the interfering eNodeB is then a thinnedPPP in R2 119861(119874 1198830) with intensity 120582119890Δ32 Static Allocation We assign a constant proportion119877119861119898119886119909Δ of the available resources to each eNodeBThey areallocated in their index order eNodeB 0 uses RB from 0 to119877119861119898119886119909Δ minus 1 (it includes the typical user) eNodeB 1 from119877119861119898119886119909Δ to 2(119877119861119898119886119909Δ) minus 1 etc We take the integer partof these fractions when 119877119861119898119886119909 is not a multiple of Δ Weloop when all resources have been used Consequently theeNodeB interfering with the typical user has an index withthe form 119896 sdot Δ with 119896 gt 033119872119872119877119861119898119886119909119877119861119898119886119909Allocation In an119872119872119862119862queuecustomers arrive according to a Poisson process (in R) andthe service times are exponentially distributed It modelsa system with 119862 resourcesservers and a capacity of thesame size A customer cannot enter in the system if allresourcesservers are busy We associate with each eNodeBan independent 119872119872119877119861119898119886119909119877119861119898119886119909 queue to model thenumber of RB in use The servers model the RB Upon thearrival of a requestuser an RBserver is used for a timeexponentially distributed If no RB is available the requestis rejected In order to have a mean reuse factor of Δ theparameter of the queue (the load) denoted 120588 is set in sucha way that the mean number of customers in the system orequivalently the mean number of busy resource blocks isequal to 119877119861119898119886119909Δ The distribution of the number of busy
Wireless Communications and Mobile Computing 5
(a) RB allocation (b) Impact of RB allocation on interferer
Figure 1 In Figure 1(a) an example of RB allocation is given eNodeB 0 uses 6 RB (the first one has index 14 and the other ones are indexedfrom 0 to 4) It uses the RB with index 0 to communicate with the typical user1198770 is the index of the last resource used by eNodeB 0 As it usesRB indexed from 14 to 4 1198770 = 4 hereThe number of used RB for the other eNodeB are1198631 = 41198632 = 3 etc An eNodeB interferes if and onlyif it reuses the resource 0 in this example eNodeB 3 and 6 In Figure 1(b) we plot a sample of the point process describing eNodeB locationwith the same RB assignmentThe eNodeB interfering with the typical user is then a dependent thinning of the original point process wherethe thinning involves eNodes 3 6 and 10 (this last interfering eNodeB was not shown in Figure 1(a))Their Voronoı cells are colored in green
resource blocks for a given eNodeB 119894 (119894 gt 0) denoted 119863119894 isthen given by
P (119863119894 = 119896) = 1205870 120588119896119896 (3)
where 1205870 = P(119863119894 = 0) and 119877119861119898119886119909 ge 119896 ge 0These RB are allocated in a cyclic order If the last119877119861 used
by eNodeB 119894 minus 1 has index 119896 eNodeB 119894 uses 119877119861 indexed from(119896 + 1)119898119900119889(119877119861119898119886119909) to (119896 + 119863119894) 119898119900119889(119877119861119898119886119909)For eNodeB 0 we do not consider the total number
of allocated RB (1198630) but instead a random variable 1198770 Itdescribes the index of the last RB used by eNodeB 0 Indeedfor this particular eNodeB the quantity used in practice tocompute the next allocation (RB indexes used by eNodeB 1)is 1198770 rather than 1198630 A formal definition of 1198770 is given inappendix (Appendix A) An example of allocation is given inFigure 1
The distribution of 1198770 is set according to the stationarydistribution of a Markov chain The transition probabilitiesof this Markov chain are
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (5)
where (119897 119898) isin 0 1 119877119861119898119886119909 minus 12 The motivation forthis particular construction is to keep the probability ofusing the resource 0 between eNodeB homogeneous Moreprecisely it is built in order to verify the property given inProposition 1 Details about the Markov chain constructionare given in appendix (Appendix A) It is worth noting thatother distributions for the resource demands (given by (3)in our case) can be considered as well As soon as thedistribution of1198770 verifies Proposition 1 themethod proposedin this paper holds
34 Property of These Assignment Strategies We define moreprecisely the sequence of rv (119908119894)119894isinN It indicates whicheNodeB interferes with the typical user It was already usedin (1)
119908119894 = 1 if eNodeB at 119883119894 uses RB with index 00 otherwise (6)
By convention we set 1199080 = 1 as In the following weshall thus assume that P(119908119894 = 1 | 1199080 = 1) = P(119908119894 = 1)Proposition 1 For the three allocation strategies defined inSections 31 32 and 33 the following property holds
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (7)
The proofs for the first two strategies are straightforwardFor the 119872119872119877119861119898119886119909119877119861119898119886119909 strategy the distribution of 1198770has been set to verify this property (see Appendix A)
35 Heuristics We compare these strategies to heuristics thataim to optimize interference or spatial reuse for a givenconfiguration It allows us to compare our RB assignmentand performs in a cyclic manner around a typical user tostrategies where a controller in charge of a set of eNodeBwill assign RB in order to optimize a certain objective func-tion For the heuristic minimizing the sum of interferencenamed ldquominimize interferencerdquo hereafter the consideredoptimization problem is similar to the one developed in[3] The problem has been shown NP-hard so we use agreedy algorithm to find a solution The number of usersassociated with each eNodeB follows the same distribution asthe119872119872119877119861119898119886119909119877119861119898119886119909 allocation Then we consider usersin a random order and apply Algorithm 1 to associate an RBto a user It chooses the resource block that minimizes thesum of interference Obviously we compute interference only
6 Wireless Communications and Mobile Computing
Data 119877119861119898119886119909 Total number of RBcost sum of interference for the current allocationbestCost sum of interference for the best allocation strategy initialized to -1Result Assign an RB to a user This algorithm is called by the controller for each
new user(1) for each 119903119887 isin 0 1 119877119861119898119886119909 minus 1 do(2) if rb is free then(3) assign rb to this user(4) costlarr sum Interference()(5) if bestCost lt0 or costltbestCost then(6) Save this allocation strategy(7) bestCost = cost(8) end(9) end(10) end(11) Assign the saved allocation strategy
Algorithm 1 Minimizing global interference
for the users already assigned The typical user is consideredin last when the system has reached the targeted load Thisalgorithm mimics an assignment strategy where the RB areassigned at the arrival of the users request without changingthe already assigned RB
The second heuristic ldquomaximize reuse distancerdquo maxi-mizes the distance at which the RB are reused Each user isconsidered in a random order Different RB are assigned tothe 119877119861119898119886119909 first users When assigning an RB to the otherusers the controller chooses the RB for which the reusedistance is maximum The typical user is considered in lastThis second heuristic may correspond to a case where thecontroller does not have information on channel conditionsand interference but knows the distances between eNodeB
4 Interference Characterization
We derive the mean and the variance of interference for thethree assignment strategies defined in the previous section
The point process modeling interferers is a dependentthinning of the original PPP Consequently conditions formean (respectively variance) to be finite with a PPP also holdfor our point process the path loss function 119897(sdot)must belongto 1198711 (respectively 1198712)41 Distribution of Distances between the Typical User andeNodeB (119883119894) As a preamble we give the PDF of thedistance between the typical user at the origin and eNodeBBoth PDF of 119883119894 and joint distribution of (119883119894 119883119895) arederived These PDF are used in the computation of the meanand the variance of interference
In the numerical evaluation we shall condition interfer-ence by the distance 1198830 It allows us to study interferencefor a given distance between the typical user and its attachedeNodeB It is also motivated by the computation of the SIRwhere both interference and the typical user signal strengthdepend on the distance 1198830
For our model the PDF of 119883119894with 119894 gt 0 given 1198830 = 119903is
1198911198941198830=119903 (119906 119903) = (120582119890120587)119894(119894 minus 1)2119906 (1199062 minus 1199032)119894minus1 119890minus120582119890120587(1199062minus1199032)1119906gt119903 (8)
The joint PDF of (119883119894 119883119895)with 119895 gt 119894 gt 0 given 1198830 =119903 is1198911198941198951198830=119903
(119906 V 119903)= (120582119890120587)119895(119894 minus 1) (119895 minus 119894 minus 1)4119906V (V2 minus 1199062)
119895minus119894minus1
times (1199062 minus 1199032)119894minus1 119890minus120582119890120587(V2minus1199032)1Vgt119906gt119903(9)
To obtain the PDF when 1198830 is not set (119891119883119894(sdot) and119891119883119894119883119895(sdot)) it suffices to integrate the two conditional PDFwith regard to the PDF of 1198830 given in (2)
42 Mean of Interference Themean is derived from (1)
E [119868 (100381710038171003817100381711988301003817100381710038171003817)]
= E [ℎ1] +infinsum119894=1
E [119908119894]E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) sdot 119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)] (10)
In this equation 119908119894 has been separated from the expecta-tion as it is independent of the process 119873119890 (according to thedefined strategies) We derive E[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)] andE[119908119894] in the two next sections
421 Computation of E[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)]No Power Control In absence of power control ie when119879119909(sdot) is constant or independent of the process 119873119890 a closedformula may be expressed for (10) E[119879119909(119883119894 minus119880119894) sdot 119897(119883119894)]
Wireless Communications and Mobile Computing 7
is then given by E[119879119909]E[119897(119883119894)] Expectation of 119897(119883119894) isobtained from the distribution of 119883119894 given in Section 41
E [119868 (100381710038171003817100381711988301003817100381710038171003817)]
= E [ℎ1] +infinsum119894=1
E [119908119894]E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817)]E [119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)] (11)
Power Control When the transmission power depends onthe distance between the receiver and its attached eNodeB(119879119909(119883119894 minus 119880119894)) the computations are more complex119864[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)] cannot be approximated byE[119879119909(119883119894minus119880119894)]119864[119897(119883119894)] as the size of the Voronoı cell withnucleus 119883119894 depends on its distance to the origin The jointdistribution of (119883119894 119883119894 minus119880119894) being unknown we proposethe following approximation
The PDF of 119880119894 minus 119883119894 is the same as 1198800 minus 1198830 givenby (2) Its parameter 119888119894(119883119894) depends on 119883119894 119888119894(119883119894) =14radic120582119890(120574119894119883119894 + 119887) with 120574119894 = (12 minus 119887)E[119883119894] and 119887 = 033The motivation and the computation details for this PDF aregiven in appendix (Appendix B) We obtain
E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) sdot 119897 (1003817100381710038171003817119883119894
Often in real systems the transmission power cannot beset arbitrarily and is limited to a set of predetermined valuesThe transmission power function can then be representedas a step function 119879119909(119903) = sum119873119879
119894=1 1199051198941119903isin[120572119894minus1120572119894] where 119873119879
is the number of possible transmission powers 119905119894 the 119894119905ℎtransmission power value and [120572119894minus1 120572119894] the distance intervalbetween a user and its eNodeB at which this transmissionpower is used An example of such setting is given in thenumerical evaluation section In this case (14) becomes
E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) 119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)]= 119873119879sum
119894=1
119905119895 int+infin
0(119890minus120582119890119888119894(119903)1205871205722119895 minus 119890minus120582119890119888119894(119903)1205871205722119895minus1)119891119883119894 (119903) 119889119903
(15)
When the computation is performed for a given distance1198830 the PDF 119891119883119894(sdot) in (12) and (15) must be replaced by1198911198941198830=119903(sdot sdot) (given in Section 41)
422 Computation of E[119908119894] Finally in order to compute(10) we need to expressE[119908119894] First note that E[119908119894] = P(119908119894 =1)Proposition 2 The probability for eNodeB 119894 to interfere withthe typical user is given by
(i) Independent allocation
P (119908119894 = 1) = 1Δ (16)
(ii) Static allocation
P (119908119894 = 1) = 1119894sdot119898119900119889(Δ)=0 (17)
E [119879119909 (1003817100381710038171003817119880119894 minus 1198831198941003817100381710038171003817) 119879119909 (10038171003817100381710038171003817119880119895 minus 119883119895
As for the mean complexity lies in the correlationbetween 119883119894 and 119880119894 minus 119883119894 The term E[119879119909(119880119894 minus 119883119894)2 sdot119897(119883119894)2] is computed with the same method as the firstmoment
Computation of E[119879119909(119880119894minus119883119894)119879119909(119880119895minus119883119895)119897(119883119894)119897(119883119895)]As 119880119894 minus 119883119894 (respectively 119880119895 minus 119883119895) depends on 119883119894(respectively 119883119895) we condition by the distribution of(119883119894 119883119895) given in Section 41 Given 119883119894 and 119883119895 we usethe same PDF as in (B1) assuming that 119880119894 minus 119883119894 and 119880119895 minus119883119895 are independent The considered joint distribution of(119883119894 119883119895 119880119894 minus 119883119894 119880119895 minus 119883119895) becomes
1198910119880119894minus119883119894 (119906 119903) 1198910119880119895minus119883119895 (V 119904) 119891119883119894119883119895 (119903 119904) (21)
When the distance 1198830 is fixed 119891119883119894119883119895(sdot sdot) must bereplaced by the PDF 119891119894119895
1198830=119903(sdot sdot sdot) given in Section 41
Computation of E[119908119894119908119895] It has been shown that the sequence(119908119894)119894gt0 verifies (7) for the three strategies It allows us toexpress E[119908119894119908119895] with 119894 gt 119895 as
E [119908119894119908119895] = P (119908119894minus119895 = 1)P (119908119895 = 1) (22)
8 Wireless Communications and Mobile Computing
Proposition 3 The joint probability for two eNodeB 119894 and 119895(119895 gt 119894) to interfere with the typical user is given by
(i) Independent allocation
E [119908119894119908119895] = 1Δ2 (23)
(ii) Static allocation
E [119908119894119908119895] = 1119894sdot119898119900119889(Δ)=0119895sdot119898119900119889(Δ)=0 (24)
where P(119908 = 1) is given by (18)5 Signal over Interference Ratio (SIR)
In our model there is a strong correlation between theinterfering eNodeB It is generated by the allocation strategiesand cannot be neglected Also a correlation exists betweenthe location of an eNodeB and the size of its Voronoı cellConsequently classical approach based on PPP which uses
Laplacian transform for instance cannot be applied hereand a formal derivation of interference distribution seemsintractable
Nevertheless the different simulations presented in thenext section will show that the PDF of interference can beapproximated by a log-normal distribution The parametersof this distribution mean and variance denoted by 119898119868119889119861
(sdot)and 120590119868119889119861(sdot) are directly derived from the previous analyticalcomputations The classical mapping between log-normaland normal parameters can be applied to derive parametersof the normal distribution when interferences are expressedin decibel In the following a variable is indexed by 119889119861 whenit is expressed in decibel
We get
P (119878119868119877119889119861 le 120573119889119861) = P (10sdot log10 (119875119905 (10038171003817100381710038171198830
1003817100381710038171003817) ℎ0119897 (100381710038171003817100381711988301003817100381710038171003817)) minus 119868119889119861 (10038171003817100381710038171198830
1003817100381710038171003817) le 120573119889119861) (26)
= P (119868119889119861 (100381710038171003817100381711988301003817100381710038171003817) ge minus120573119889119861 + 10
When ℎ0 is not constant the expectation with regard toits distribution must be taken into account in (28) In (29)ℎ0 is assumed to be constant that equals to 1 In these twoequations 119890119903119891(sdot) is the error function6 Numerical Results
We consider an E-UTRA channel with a bandwidth of5MHz with 119877119861119898119886119909 = 15 [33] The path loss function isthe same as [3] It is expressed in dB 119897(119903) = minus1281 minus376 sdot log10(119903) where 119903 is the distance (in km) 119879119909(sdot) isset in such a way to guarantee to each user a minimumreceiving power We set the transmission power function119879119909(sdot) to ensure a signal power greater than or equal tominus724dBm at the reception as specified in [33] For each 50meters (from 50 to 500 meters) we compute the minimumtransmitting power required to reach this threshold (119879119909(119903) sdot119897(119903) ge minus724 dBm for each interval of 50 meters leading to 10possible transmission powers)This step function models thecase where eNodeB has a finite set of predetermined powerThe process intensity modeling eNodeB is equal to 225 perkm2 It corresponds to the intensity of base stations in Paris(httpswwwantennesmobilesfr) Random variables ℎ119894 are
supposed constant equal to 1 This assumption facilitatesinterpretation of the results but any distribution can beconsidered as well It simply adds a factor in terms of variance(cf (20)) We simulate the different strategies through asimulator coded inC available here (httpwwwanthonybus-sonfrindexphppublications) In all simulations and nu-merical results we consider 50 eNodeB The different sumsin the equations (eg (10) and (20)) are then limited to 50For each set of parameters simulations have be run from10000 to several millions times depending on the evaluatedquantities The number of simulationssamples has been setin order to have negligible confidence intervals They areconsequently not shown in the different figures
Mean and Variance of Interference In Figure 2 we plot themean and the standard deviation of interference obtainedfrom simulations and computed from formulas (10) (19) and(20)when the distance 1198830 variesThe theoretical evaluationcloselymatches empirical estimators obtained by simulationsAs expected the highest interference level is observed forthe independent allocation and the lowest level for the staticallocation The static and the 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tions offer equivalent results with a multiplication factor
Figure 2 Interferencemean and standard deviation as a function of the distance between the typical user and its eNodeB (1198830) Simulationresults are given by points and solid lines and theoretical evaluations by the dotted lines
Table 2 Mean and standarddeviation for the119872119872119877119861119898119886119909119877119861m119886119909 allocation and the two heuristics
Assignment strategy Δ = 3 Δ = 6 Δ = 9mean std-dev mean std-dev mean std-dev
119872119872119877119861119898119886119909119877119861119898119886119909 178119890 minus 11 861119890 minus 11 366119890 minus 12 750119890 minus 12 155119890 minus 12 316119890 minus 12Minimize Interference 994119890 minus 12 307119890 minus 11 186119890 minus 12 188119890 minus 12 756119890 minus 13 626119890 minus 13Maximize reuse distance 526119890 minus 11 751119890 minus 10 118119890 minus 11 107119890 minus 10 258119890 minus 12 109119890 minus 11
varying from 11 to 87 for the mean interference Insteadthe independent allocation differs deeply Mean and standarddeviation are multiplied by a factor ranging from 17 to 700for the mean and can reach up to 26 times 104 for the standarddeviation (for 1198830 = 002 km) The use of the thinnedpoint process is thus questionable to approximate realisticassignment strategies This strategy is no more considered inthe following
Extrapolation of Interference Distribution We compareempirical distributions obtained from simulations to knowndistributions for the different strategies and the two heuristicswhen Δ = 3 6 and 9 Three of these empirical distributionsare shown in Figure 3 (in dB) The distribution parametershave been set according to the maximum likelihood Thedistributions that best fit simulations vary according to theallocation strategy and the reuse factorΔ For 4 cases over 12the distribution minimizing error is the normal distributionAs soon as an asymmetry is observed other distributionsare more accurate log-normal Weibull and Inverse GammaWe perform two hypothesis tests T-test and Smirnov-Kolmogorov for all these scenarios The alternative hypoth-esis is systematically ruled out For the 119872119872119877119861119898119886119909119877119861119898119886119909
strategy best fits are given by the normal (Δ = 3 and 6) andWeibull distributions (Δ = 9)
Nevertheless as mentioned in Section 5 when estimatingthe SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy weuse the normal law to model interference Indeed even if thisdistribution is not systematically the most accurate it offers agood approximation as shown inFigure 4 where the empiricaland normalCDF are plottedMoreover simulationswill showthat this assumption does not impact the accuracy of theanalytical model
Heuristics Simulations have shown that allocations resultingfrom the two heuristics correlate the transmission powersthe distance between eNodeB reusing the same resourceand the random variables 119908119894 Whenminimizing interferencecorrelations are caused by resource blocks allocated to usersthat require a low transmission power and that can be reusedat a short distance and inversely The transmission poweris then correlated to 119908119894 and to the distance of eNodeB thatreuse the same resource These complex phenomena impedethe proposal of an analytical model for these two heuristicsFor comparison purposes Table 2 reportsmean and standarddeviation for the two heuristics and the119872119872119877119861119898119886119909119877119861119898119886119909
Figure 3 Interference distribution empirical (histogram) and extrapolation (curve) Among the classical distributions the bestextrapolations are Gamma (m=1003 var=977) normal (m=2823 var=2216) and Weibull (a=2514b=575) for the heuristic MMRBallocation Δ = 6 and static allocation Δ = 9 respectively
Figure 4 CDF of interference for the 119872119872119877119861119898119886119909119877119861119898119886119909 strategy Empirical distributions are represented through the solid lines CDFof the Normal distribution are plotted with dotted lines Their parameters are obtained from the theoretical formulas The 119871+infin errors are338119890minus02 493119890minus02 and 565119890minus02 for Δ = 3 6 and 9 respectively
Figure 5 Mean interference when the number of RB variesThe load is adjusted to keep a constant reuse factor (Δ = 36 or 9) The distance 1198830 is set to 50 meters Dotted linescorrespond to the theoretical evaluation ofmean interference for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation and points to simulations
allocationTheheuristicminimizing interference presents thebest results This difference was expected as the objectivefunction minimizes interference and because the secondheuristic maximizes the reuse distance without taking intoaccount transmission powers 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tion offers interesting results with an intermediary interfer-ence level between the two heuristics Even if interference islower for the heuristic minimizing interference its complex-ity in terms of feedbackmeasures from users and eNodeBis significantly greater than the other strategies Indeed itassumes that the scheduler knows before an assignment theinterference contribution of each eNodeB on each user
Impact of the Number of RB We performed simulations andtheoretical estimations of interference for different values of119877119861119898119886119909 6 15 25 50 75 and 100 corresponding to differentbandwidth of E-UTRA [33] Results are shown in Figure 5 forthe 119872119872119877119861119898119886119909119877119861119898119886119909 allocation and the two heuristicsThe workload is adapted in order to keep the same Δreuse factor It appears that the mean interference is quiteinsensitive to the number of RB except when 119877119861119898119886119909 = 6The impacting factor is thus the reuse factor Δ rather thanthe number of RBAlso the same hierarchy between the threestrategies is observed except for 119877119861119898119886119909 = 6SIR Distribution We evaluate the SIR CDF according to(29) for the 119872119872119877119861119898119886119909119877119861119898119886119909 allocation This CDF iscompared to the one obtained by simulations in Figure 6(a)We clearly observe that the analytical model offers very
tight estimates of the SIR distribution The assumptionabout the normal distribution of interference does notintroduce a noticeable error The SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy and the two heuristics arecompared in Figure 6(b) The three CDF are similar and the119872119872119877119861119898119886119909119877119861119898119886119909 allocation still offers an intermediatedistribution between the two heuristics The 119871+infin normbetween the three CDF is 0192 0266 and 0221 for Δ = 36 and 9 respectivelyModulation and Coding Rate The SIR distribution allowsus to estimate modulations and coding rates that could beoffered to users We consider the thresholds between therequired SIR and coding rate given in [33] Applied to the SIRdistribution it gives the proportion of users that benefits froma certain modulation scheme and coding rate The couplemodulationcoding rate is referred to as transmission rate inthe following This proportion is given in Figure 7(a) for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation when the density of eNodeBincreases from 225 (the default intensity considered in theprevious evaluation) to 20 eNodeB per km2 It appears thatthe network densification even with a factor 10 does notsignificantly improve the transmission rate Only the besttransmission rate (64QAM-CR=45) really benefits from thisdensification with a proportion of users increasing from 35to 116
The spatial reuse has much more impact on the transmis-sion rates as it is shown in Figure 7(b) It clearly increasesthe transmission rates that are shifted from (QPSK CR=12-64QAM CR=23) for Δ = 3 to (16QAM CR=12-64QAMCR=45) for Δ = 9 The mean transmission rate increasesalmost of a factor 2 from 231 to 424 bitsbaud
Finally we compare in Figure 7(c) the transmission ratesfor the three allocation strategies As already observed inthe previous plots the heuristic minimizing interferenceoffers better performance but is comparable to the two otherstrategies For instance for Δ = 6 the mean transmis-sion rates are close 363 323 and 392 bitsbaud for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation the heuristic maximizingthe reuse distance and the one minimizing interferencerespectively
7 Conclusion
In recent LTE standards a server may assign RB to a set ofeNodeB to control resource usage and optimize performancein a global way Even if it exists solutions to address this prob-lem for a given configuration and topology literature lacksmodels that evaluate performance of these RB allocations fora wide range of scenarios and at large scale
To address this problem we propound a spatial stochasticmodel that takes into account RB assignment strategiesrealistic traffic demands and power control We proposeanalytical estimates for the two first moments of interferenceThis computation is based on two approximations the distri-bution of the distance between a point uniformly distributedin a cell that models the user-eNodeB distance and thejoint distribution of the distances eNodeB-origin and user-eNodeB For the latter simulations have shown that these
(a) SIR empirical and theoretical CDF for the119872119872119877119861119898119886119909119877119861119898119886119909allocation
MMRB Delta=3MMRB Delta=6MMRB Delta=9MI Delta=3 MI Delta=6
MI Delta=9 MD Delta=3 MD Delta=6 MD Delta=9
0 10 20 30 40minus10훽 (dB)
00e+00
20eminus01
40eminus01
60eminus01
80eminus01
10e+00
CDF
minus SI
NR
(b) SIR empirical CDF for the three strategies MI maximizeinterference MD maximize distance
Figure 6 (a) We compare the distribution of SIR obtained by simulations (solid lines) and given by (29) (dotted lines) Errors with regardto the 119871+infin norm are 247119890 minus 02 154119890 minus 02 and 158119890 minus 02 for Δ = 3 6 and 9 respectively (b) CDF for the three allocation strategies(119872119872119877119861119898119886119909119877119861119898119886119909 and Heuristics)
(c) 119872119872119877119861119898119886119909119877119861119898119886119909 and the two heuristics Δ = 6 120582119890 = 225
Figure 7 Modulation and coding rate for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation strategy when the density of eNodeB increases (Figure 7(a))for different reuse factor (Figure 7(b)) and comparison with the two heuristics (Figure 7(c))
variables are strongly correlated particularly for the points ofthe process close to the origin This correlation impacts thetransmission power used by the eNodeB and consequentlyinterference and SIR distributions The derivation of the SIRdistribution allows us to express the classical outage but alsothe transmission rates in terms of modulationcoding ratethat provide interesting insights on the throughput offers tothe users Also it appears that eNodeB densification does not
significantly improve the network performance in terms ofthroughput except for a small percentage of users for whichthe transmission rate is increased Spatial reuse expressedthrough the parameter Δ in our study has a much moreimpact on the performance as the transmission rates aresignificantly increased This spatial reuse must be expressedas the ratio between the total number of available resourcesover the number of usersrequests per cell Indeed we have
Wireless Communications and Mobile Computing 13
observed that the system performance is quite insensitive tovariation of these quantities when this ratio stays constant
Beside simulations show that the RB assignment strate-gies developed for the model give results close to the onesgiven by classical optimization problem which minimizesglobal interference or maximizes spatial reuse distances Itempirically shows that our analytical model is able to evaluateperformance of classical optimization approaches but at largescale in terms of number of nodes configurations andtopologies
Appendix
A Construction of the Markov Chain
In this section we introduce the Markov chain used to set thedistribution of 1198770 We consider that the RB used between thetypical user and its eNodeB has index 119903119890119904 (119903119890119904 isin 0 119877119861119898119886119909minus1)We shall prove that this index does not impact calculation(we have chosen 119903119890119904 = 0 in this paper by sack of simplicity)
We set the distribution of the number of used resourceblocks at eNodeB 0 in such a way that it leads to the followingproperty (119895 gt 119894 ge 0)
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (A1)
In order to obtain this property we first introduce somepreliminary notations and results An example of allocationwith the different notations is shown in Figure 1 We define asequence of rv (119877119899)119899ge0 If eNodeB 119894 is the 119899119905ℎ eNodeB using119903119890119904 and if 119864119899119889119899 is the index of the last RB used by this eNodeBthen
119877119899 = 119864119899119889119899 minus 119903119890119904 if 119864119899119889119899 gt 119903119890119904119864119899119889119899 + 119877119861119898119886119909 minus 119903119890119904 otherwise (A2)
If eNodeB 119894 is the 119899119905ℎ interfering eNodeB (using 119903119890119904) theneNodeB 119895 (119895 gt 119894) is the 119899 + 1119905ℎ if and only if
119863119896 ge 119877119861119898119886119909 (A4)
In this case 119877119899+1 is given by
119877119899+1 = 119877119899 +119895sum
119896=119894+1
119863119896 minus 119877119861119898119886119909 (A5)
As (119863119894)119894gt0 are iid the sequence (119877119899)119899ge0 is a homoge-neous Markov chain The transition probabilities for (119897 119898) isin0 1 119877119861119898119886119909 minus 12 are given by 119875119897119898 = P(119877119899+1 = 119898 | 119877119899 = 119897)
The event 119877119899+1 = 119898 given that 119877119899 = 119897 occurs if andonly if it exists 119895 ge 0 such that
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (A10)
We set the distribution of 1198770 with the correspondingstationary distribution Consequently the sequence (119877119899)119899ge0becomes identically distributed where 119877119899 follows the sta-tionary distribution for all 119899 ge 0 Moreover according to(A3) and (A4) the property given in (A1) holds Indeed 119877119899sum119895minus1
119896=119894+1119863119896 and sum119895
119896=119894+1119863119896 have the same distribution as 1198770sum119895minus119894minus1
119896=1119863119896 andsum119895minus119894
119896=1119863119896 respectively In the model evaluation
the stationary distribution is obtained numerically from thetransition probabilities As soon as 1198770 follows the stationarydistribution the index 119903119890119904 can be chosen arbitrarily
B Conditional Distribution of 119880119894minus119883119894Simulations have shown that given 119883119894 the distribution of119883119894 minus 119880119894 is still close to the one given by (2) but with adifferent parameter So we assume that the distribution of119883119894 minus 119880119894 given 119883119894 is equal to (2) but with a parameterfunction of 119883119894 denoted 119888(119883119894)
These simulations have also shown that E[119883119894 minus 119880119894 |119883119894 = 119903]may be approximated as an affine function 119886119894119903 + 119887119894We define 119888119894(sdot) accordingly It leads to
To set 119886119894 and 119887119894 we use the two following propertiesE [E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817 | 10038171003817100381710038171198831198941003817100381710038171003817 = 119903]] = E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817] (B3)
= 12radic120582119890 (B4)
E [E [1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817 | 1003817100381710038171003817119883119894
1003817100381710038171003817 = 119903]] = E [119886119894 10038171003817100381710038171198831198941003817100381710038171003817 + 119887119894] (B5)
Equation (B4) is derived from (2) and (B5) from ourassumption on 119888119894(119903) 119886119894 is then set as a function of 119887119894 E[119883119894]and 120582119890 119887119894 is obtained from our simulations For a givenintensity 119887119894 has been observed almost constant with regardto 119894 The best approximation as a function of the intensity isgiven by 119887119894 = 033radic120582119890 After a fewmanipulations we obtain
First note that by convention we have P(119908119894 = 1) = P(119908119894 =1 | 1199080 = 1) The event 119908119894 = 1 occurs if and only if it exists119895 ge 1 such that1198770 + 119894minus1sum
There is no data used in this paper Nevertheless the code ofour simulator is made available (the link is in the paper)
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] ldquoRequirements for further advancements for e-utra (lte-advanced) 3gpp technical specification ts 36913 release 13rdquo2015 httpwww3gpporg
[2] A S Hamza S S Khalifa H S Hamza and K Elsayed ldquoAsurvey on inter-cell interference coordination techniques inofdma-based cellular networksrdquo IEEE Communications Surveysamp Tutorials vol 15 no 4 pp 1642ndash1670 2013
[3] D Lopez-Perez A Ladanyi A JuttnerH Rivano and J ZhangldquoOptimization method for the joint allocation of modulationschemes coding rates resource blocks and power in self-organizing LTE networksrdquo in Proceedings of the IEEE INFO-COM 2011 pp 111ndash115 April 2011
[4] A Guo andM Haenggi ldquoSpatial stochasticmodels andmetricsfor the structure of base stations in cellular networksrdquo IEEE
Wireless Communications and Mobile Computing 15
Transactions on Wireless Communications vol 12 no 11 pp5800ndash5812 2013
[5] C-H Lee C-Y Shih and Y-S Chen ldquoStochastic geometrybased models for modeling cellular networks in urban areasrdquoWireless Networks vol 19 no 6 pp 1063ndash1072 2013
[6] W Lu and M D Renzo ldquoStochastic geometry modeling ofcellular networks analysis simulation and experimental valida-tionrdquo in Proceedings of the 18th ACM International Conferenceon Modelling Analysis and Simulation of Wireless and MobileSystems (MSWiM rsquo15) pp 179ndash188 New York NY USANovember 2015
[7] Y L Lee T C Chuah J Loo and A Vinel ldquoRecent advancesin radio resource management for heterogeneous LTELTE-Anetworksrdquo IEEE Communications Surveys amp Tutorials vol 16no 4 pp 2142ndash2180 2014
[8] F Mhiri K Sethom and R Bouallegue ldquoA survey on inter-ference management techniques in femtocell self-organizingnetworksrdquo Journal of Network and Computer Applications vol36 no 1 pp 58ndash65 2013
[9] X Chen L Li and X Xiang ldquoAnt colony learning methodfor joint MCS and resource block allocation in LTE Femtocelldownlink for multimedia applications with QoS guaranteesrdquoMultimedia Tools andApplications vol 76 no 3 pp 4035ndash40542017
[10] D Lopez-Perez X Chu A V Vasilakos and H Claussen ldquoOndistributed and coordinated resource allocation for interferencemitigation in self-organizing lte networksrdquo IEEEACMTransac-tions on Networking vol 21 no 4 pp 1145ndash1158 2013
[11] H Zhang H Liu J Cheng and V C M Leung ldquoDownlinkenergy efficiency of power allocation and wireless backhaulbandwidth allocation in heterogeneous small cell networksrdquoIEEE Transactions on Communications no 99 2017
[12] H Zhang W Zheng X Chu et al ldquoJoint subchannel andpower allocation in interference-limited OFDMA femtocellswith heterogeneous QoS guaranteerdquo in Proceedings of theIEEEGlobal Communications Conference (GLOBECOM rsquo12) pp4572ndash4577 December 2012
[13] H Elsawy E Hossain and M Haenggi ldquoStochastic geometryfor modeling analysis and design of multi-tier and cognitivecellular wireless networks a surveyrdquo IEEE CommunicationsSurveys amp Tutorials vol 15 no 3 pp 996ndash1019 2013
[14] H ElSawy A Sultan-Salem M-S Alouini and M Z WinldquoModeling and analysis of cellular networks using stochasticgeometry a tutorialrdquo IEEE Communications Surveys amp Tutori-als vol 19 no 1 pp 167ndash203 2017
[15] R Kwan and C Leung ldquoA survey of scheduling and inter-ference mitigation in lterdquo Journal of Electrical and ComputerEngineeringmdashSpecial Issue on LTELTE-AdvancedCellular Com-munication Networks pp 11ndash110 2010
[16] J G Andrews F Baccelli and R K Ganti ldquoA tractable approachto coverage and rate in cellular networksrdquo IEEE Transactions onCommunications vol 59 no 11 pp 3122ndash3134 2011
[17] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for OFDMA cellu-lar networksrdquo IEEE Transactions on Wireless Communicationsvol 10 no 12 pp 4294ndash4305 2011
[18] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for heterogeneouscellular networksrdquo IEEE Transactions on Communications vol60 no 7 pp 2029ndash2039 2012
[19] H Zhuang and T Ohtsuki ldquoA model based on poisson pointprocess for downlink K tiers fractional frequency reuse hetero-geneous networksrdquo Physical Communication vol 13 pp 3ndash122014 Special Issue on Heterogeneous and Small Cell Networks
[20] A A Gebremariam T Bao D Siracusa T Rasheed F Granelliand L Goratti ldquoDynamic strict fractional frequency reusefor software-defined 5G networksrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo16) pp 1ndash6May 2016
[21] N Lee D Morales-Jimenez A Lozano and R W HeathldquoSpectral efficiency of dynamic coordinated beamforming astochastic geometry approachrdquo IEEE Transactions on WirelessCommunications vol 14 no 1 pp 230ndash241 2015
[22] X Zhang and M Haenggi ldquoA Stochastic geometry analysisof inter-cell interference coordination and intra-cell diversityrdquoIEEE Transactions on Wireless Communications vol 13 no 12pp 6655ndash6669 2014
[23] F Baccelli andAGiovanidis ldquoA stochastic geometry frameworkfor analyzing pairwise-cooperative cellular networksrdquo IEEETransactions on Wireless Communications vol 14 no 2 pp794ndash808 2015
[24] J Yoon and G Hwang ldquoDistance-based inter-cell interferencecoordination in small cell networks stochastic geometry mod-eling and analysisrdquo IEEE Transactions on Wireless Communica-tions 2018
[25] B Błaszczyszyn M Jovanovicy and M K Karray ldquoHow userthroughput depends on the traffic demand in large cellularnetworksrdquo in Proceedings of the 12th International Symposiumon Modeling and Optimization in Mobile Ad Hoc and WirelessNetworks (WiOpt rsquo14) pp 611ndash619 May 2014
[26] H H Yang and T Q S Quek ldquoSIR coverage analysis incellular networks with temporal traffic a stochastic geometryapproachrdquo Corrossion 2018
[27] S M Yu and S-L Kim ldquoDownlink capacity and base stationdensity in cellular networksrdquo in Proceedings of the 11th Interna-tional SymposiumandWorkshops onModeling andOptimizationin Mobile Ad Hoc and Wireless Networks (WiOpt rsquo13) pp 119ndash124 May 2013
[28] YWang M Haenggi and Z Tan ldquoThemeta distribution of thesir for cellular networks with power controlrdquo Corrossion 2017
[29] M S Omar S A Hassan H Pervaiz et al ldquoMulti-objectiveoptimization in 5g hybrid networksrdquo IEEE Internet of ThingsJournal 2018
[30] Y Leo A Busson C Sarraute and E Fleury ldquoCall detail recordsto characterize usages and mobility events of phone usersrdquoComputer Communications vol 95 pp 43ndash53 2016
[31] I L Cherif Spectral and Energy Efficiency in 5G WirelessNetworks [PhD thesis] Univeristy Paris Saclay 2016
[32] S Mukherjee Analytical Modeling of Heterogeneous CellularNetworks Analytical Modeling of Heterogeneous Cellular Net-works Geometry Coverage and Capacity Cambridge Univer-sity Press 2014
[33] ldquoTechnical specification group radio access network base sta-tion (bs) radio transmission and reception 2012 Release 1123gpp ts 36104 v112rdquo 2012
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Wireless Communications and Mobile Computing 5
(a) RB allocation (b) Impact of RB allocation on interferer
Figure 1 In Figure 1(a) an example of RB allocation is given eNodeB 0 uses 6 RB (the first one has index 14 and the other ones are indexedfrom 0 to 4) It uses the RB with index 0 to communicate with the typical user1198770 is the index of the last resource used by eNodeB 0 As it usesRB indexed from 14 to 4 1198770 = 4 hereThe number of used RB for the other eNodeB are1198631 = 41198632 = 3 etc An eNodeB interferes if and onlyif it reuses the resource 0 in this example eNodeB 3 and 6 In Figure 1(b) we plot a sample of the point process describing eNodeB locationwith the same RB assignmentThe eNodeB interfering with the typical user is then a dependent thinning of the original point process wherethe thinning involves eNodes 3 6 and 10 (this last interfering eNodeB was not shown in Figure 1(a))Their Voronoı cells are colored in green
resource blocks for a given eNodeB 119894 (119894 gt 0) denoted 119863119894 isthen given by
P (119863119894 = 119896) = 1205870 120588119896119896 (3)
where 1205870 = P(119863119894 = 0) and 119877119861119898119886119909 ge 119896 ge 0These RB are allocated in a cyclic order If the last119877119861 used
by eNodeB 119894 minus 1 has index 119896 eNodeB 119894 uses 119877119861 indexed from(119896 + 1)119898119900119889(119877119861119898119886119909) to (119896 + 119863119894) 119898119900119889(119877119861119898119886119909)For eNodeB 0 we do not consider the total number
of allocated RB (1198630) but instead a random variable 1198770 Itdescribes the index of the last RB used by eNodeB 0 Indeedfor this particular eNodeB the quantity used in practice tocompute the next allocation (RB indexes used by eNodeB 1)is 1198770 rather than 1198630 A formal definition of 1198770 is given inappendix (Appendix A) An example of allocation is given inFigure 1
The distribution of 1198770 is set according to the stationarydistribution of a Markov chain The transition probabilitiesof this Markov chain are
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (5)
where (119897 119898) isin 0 1 119877119861119898119886119909 minus 12 The motivation forthis particular construction is to keep the probability ofusing the resource 0 between eNodeB homogeneous Moreprecisely it is built in order to verify the property given inProposition 1 Details about the Markov chain constructionare given in appendix (Appendix A) It is worth noting thatother distributions for the resource demands (given by (3)in our case) can be considered as well As soon as thedistribution of1198770 verifies Proposition 1 themethod proposedin this paper holds
34 Property of These Assignment Strategies We define moreprecisely the sequence of rv (119908119894)119894isinN It indicates whicheNodeB interferes with the typical user It was already usedin (1)
119908119894 = 1 if eNodeB at 119883119894 uses RB with index 00 otherwise (6)
By convention we set 1199080 = 1 as In the following weshall thus assume that P(119908119894 = 1 | 1199080 = 1) = P(119908119894 = 1)Proposition 1 For the three allocation strategies defined inSections 31 32 and 33 the following property holds
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (7)
The proofs for the first two strategies are straightforwardFor the 119872119872119877119861119898119886119909119877119861119898119886119909 strategy the distribution of 1198770has been set to verify this property (see Appendix A)
35 Heuristics We compare these strategies to heuristics thataim to optimize interference or spatial reuse for a givenconfiguration It allows us to compare our RB assignmentand performs in a cyclic manner around a typical user tostrategies where a controller in charge of a set of eNodeBwill assign RB in order to optimize a certain objective func-tion For the heuristic minimizing the sum of interferencenamed ldquominimize interferencerdquo hereafter the consideredoptimization problem is similar to the one developed in[3] The problem has been shown NP-hard so we use agreedy algorithm to find a solution The number of usersassociated with each eNodeB follows the same distribution asthe119872119872119877119861119898119886119909119877119861119898119886119909 allocation Then we consider usersin a random order and apply Algorithm 1 to associate an RBto a user It chooses the resource block that minimizes thesum of interference Obviously we compute interference only
6 Wireless Communications and Mobile Computing
Data 119877119861119898119886119909 Total number of RBcost sum of interference for the current allocationbestCost sum of interference for the best allocation strategy initialized to -1Result Assign an RB to a user This algorithm is called by the controller for each
new user(1) for each 119903119887 isin 0 1 119877119861119898119886119909 minus 1 do(2) if rb is free then(3) assign rb to this user(4) costlarr sum Interference()(5) if bestCost lt0 or costltbestCost then(6) Save this allocation strategy(7) bestCost = cost(8) end(9) end(10) end(11) Assign the saved allocation strategy
Algorithm 1 Minimizing global interference
for the users already assigned The typical user is consideredin last when the system has reached the targeted load Thisalgorithm mimics an assignment strategy where the RB areassigned at the arrival of the users request without changingthe already assigned RB
The second heuristic ldquomaximize reuse distancerdquo maxi-mizes the distance at which the RB are reused Each user isconsidered in a random order Different RB are assigned tothe 119877119861119898119886119909 first users When assigning an RB to the otherusers the controller chooses the RB for which the reusedistance is maximum The typical user is considered in lastThis second heuristic may correspond to a case where thecontroller does not have information on channel conditionsand interference but knows the distances between eNodeB
4 Interference Characterization
We derive the mean and the variance of interference for thethree assignment strategies defined in the previous section
The point process modeling interferers is a dependentthinning of the original PPP Consequently conditions formean (respectively variance) to be finite with a PPP also holdfor our point process the path loss function 119897(sdot)must belongto 1198711 (respectively 1198712)41 Distribution of Distances between the Typical User andeNodeB (119883119894) As a preamble we give the PDF of thedistance between the typical user at the origin and eNodeBBoth PDF of 119883119894 and joint distribution of (119883119894 119883119895) arederived These PDF are used in the computation of the meanand the variance of interference
In the numerical evaluation we shall condition interfer-ence by the distance 1198830 It allows us to study interferencefor a given distance between the typical user and its attachedeNodeB It is also motivated by the computation of the SIRwhere both interference and the typical user signal strengthdepend on the distance 1198830
For our model the PDF of 119883119894with 119894 gt 0 given 1198830 = 119903is
1198911198941198830=119903 (119906 119903) = (120582119890120587)119894(119894 minus 1)2119906 (1199062 minus 1199032)119894minus1 119890minus120582119890120587(1199062minus1199032)1119906gt119903 (8)
The joint PDF of (119883119894 119883119895)with 119895 gt 119894 gt 0 given 1198830 =119903 is1198911198941198951198830=119903
(119906 V 119903)= (120582119890120587)119895(119894 minus 1) (119895 minus 119894 minus 1)4119906V (V2 minus 1199062)
119895minus119894minus1
times (1199062 minus 1199032)119894minus1 119890minus120582119890120587(V2minus1199032)1Vgt119906gt119903(9)
To obtain the PDF when 1198830 is not set (119891119883119894(sdot) and119891119883119894119883119895(sdot)) it suffices to integrate the two conditional PDFwith regard to the PDF of 1198830 given in (2)
42 Mean of Interference Themean is derived from (1)
E [119868 (100381710038171003817100381711988301003817100381710038171003817)]
= E [ℎ1] +infinsum119894=1
E [119908119894]E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) sdot 119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)] (10)
In this equation 119908119894 has been separated from the expecta-tion as it is independent of the process 119873119890 (according to thedefined strategies) We derive E[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)] andE[119908119894] in the two next sections
421 Computation of E[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)]No Power Control In absence of power control ie when119879119909(sdot) is constant or independent of the process 119873119890 a closedformula may be expressed for (10) E[119879119909(119883119894 minus119880119894) sdot 119897(119883119894)]
Wireless Communications and Mobile Computing 7
is then given by E[119879119909]E[119897(119883119894)] Expectation of 119897(119883119894) isobtained from the distribution of 119883119894 given in Section 41
E [119868 (100381710038171003817100381711988301003817100381710038171003817)]
= E [ℎ1] +infinsum119894=1
E [119908119894]E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817)]E [119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)] (11)
Power Control When the transmission power depends onthe distance between the receiver and its attached eNodeB(119879119909(119883119894 minus 119880119894)) the computations are more complex119864[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)] cannot be approximated byE[119879119909(119883119894minus119880119894)]119864[119897(119883119894)] as the size of the Voronoı cell withnucleus 119883119894 depends on its distance to the origin The jointdistribution of (119883119894 119883119894 minus119880119894) being unknown we proposethe following approximation
The PDF of 119880119894 minus 119883119894 is the same as 1198800 minus 1198830 givenby (2) Its parameter 119888119894(119883119894) depends on 119883119894 119888119894(119883119894) =14radic120582119890(120574119894119883119894 + 119887) with 120574119894 = (12 minus 119887)E[119883119894] and 119887 = 033The motivation and the computation details for this PDF aregiven in appendix (Appendix B) We obtain
E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) sdot 119897 (1003817100381710038171003817119883119894
Often in real systems the transmission power cannot beset arbitrarily and is limited to a set of predetermined valuesThe transmission power function can then be representedas a step function 119879119909(119903) = sum119873119879
119894=1 1199051198941119903isin[120572119894minus1120572119894] where 119873119879
is the number of possible transmission powers 119905119894 the 119894119905ℎtransmission power value and [120572119894minus1 120572119894] the distance intervalbetween a user and its eNodeB at which this transmissionpower is used An example of such setting is given in thenumerical evaluation section In this case (14) becomes
E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) 119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)]= 119873119879sum
119894=1
119905119895 int+infin
0(119890minus120582119890119888119894(119903)1205871205722119895 minus 119890minus120582119890119888119894(119903)1205871205722119895minus1)119891119883119894 (119903) 119889119903
(15)
When the computation is performed for a given distance1198830 the PDF 119891119883119894(sdot) in (12) and (15) must be replaced by1198911198941198830=119903(sdot sdot) (given in Section 41)
422 Computation of E[119908119894] Finally in order to compute(10) we need to expressE[119908119894] First note that E[119908119894] = P(119908119894 =1)Proposition 2 The probability for eNodeB 119894 to interfere withthe typical user is given by
(i) Independent allocation
P (119908119894 = 1) = 1Δ (16)
(ii) Static allocation
P (119908119894 = 1) = 1119894sdot119898119900119889(Δ)=0 (17)
E [119879119909 (1003817100381710038171003817119880119894 minus 1198831198941003817100381710038171003817) 119879119909 (10038171003817100381710038171003817119880119895 minus 119883119895
As for the mean complexity lies in the correlationbetween 119883119894 and 119880119894 minus 119883119894 The term E[119879119909(119880119894 minus 119883119894)2 sdot119897(119883119894)2] is computed with the same method as the firstmoment
Computation of E[119879119909(119880119894minus119883119894)119879119909(119880119895minus119883119895)119897(119883119894)119897(119883119895)]As 119880119894 minus 119883119894 (respectively 119880119895 minus 119883119895) depends on 119883119894(respectively 119883119895) we condition by the distribution of(119883119894 119883119895) given in Section 41 Given 119883119894 and 119883119895 we usethe same PDF as in (B1) assuming that 119880119894 minus 119883119894 and 119880119895 minus119883119895 are independent The considered joint distribution of(119883119894 119883119895 119880119894 minus 119883119894 119880119895 minus 119883119895) becomes
1198910119880119894minus119883119894 (119906 119903) 1198910119880119895minus119883119895 (V 119904) 119891119883119894119883119895 (119903 119904) (21)
When the distance 1198830 is fixed 119891119883119894119883119895(sdot sdot) must bereplaced by the PDF 119891119894119895
1198830=119903(sdot sdot sdot) given in Section 41
Computation of E[119908119894119908119895] It has been shown that the sequence(119908119894)119894gt0 verifies (7) for the three strategies It allows us toexpress E[119908119894119908119895] with 119894 gt 119895 as
E [119908119894119908119895] = P (119908119894minus119895 = 1)P (119908119895 = 1) (22)
8 Wireless Communications and Mobile Computing
Proposition 3 The joint probability for two eNodeB 119894 and 119895(119895 gt 119894) to interfere with the typical user is given by
(i) Independent allocation
E [119908119894119908119895] = 1Δ2 (23)
(ii) Static allocation
E [119908119894119908119895] = 1119894sdot119898119900119889(Δ)=0119895sdot119898119900119889(Δ)=0 (24)
where P(119908 = 1) is given by (18)5 Signal over Interference Ratio (SIR)
In our model there is a strong correlation between theinterfering eNodeB It is generated by the allocation strategiesand cannot be neglected Also a correlation exists betweenthe location of an eNodeB and the size of its Voronoı cellConsequently classical approach based on PPP which uses
Laplacian transform for instance cannot be applied hereand a formal derivation of interference distribution seemsintractable
Nevertheless the different simulations presented in thenext section will show that the PDF of interference can beapproximated by a log-normal distribution The parametersof this distribution mean and variance denoted by 119898119868119889119861
(sdot)and 120590119868119889119861(sdot) are directly derived from the previous analyticalcomputations The classical mapping between log-normaland normal parameters can be applied to derive parametersof the normal distribution when interferences are expressedin decibel In the following a variable is indexed by 119889119861 whenit is expressed in decibel
We get
P (119878119868119877119889119861 le 120573119889119861) = P (10sdot log10 (119875119905 (10038171003817100381710038171198830
1003817100381710038171003817) ℎ0119897 (100381710038171003817100381711988301003817100381710038171003817)) minus 119868119889119861 (10038171003817100381710038171198830
1003817100381710038171003817) le 120573119889119861) (26)
= P (119868119889119861 (100381710038171003817100381711988301003817100381710038171003817) ge minus120573119889119861 + 10
When ℎ0 is not constant the expectation with regard toits distribution must be taken into account in (28) In (29)ℎ0 is assumed to be constant that equals to 1 In these twoequations 119890119903119891(sdot) is the error function6 Numerical Results
We consider an E-UTRA channel with a bandwidth of5MHz with 119877119861119898119886119909 = 15 [33] The path loss function isthe same as [3] It is expressed in dB 119897(119903) = minus1281 minus376 sdot log10(119903) where 119903 is the distance (in km) 119879119909(sdot) isset in such a way to guarantee to each user a minimumreceiving power We set the transmission power function119879119909(sdot) to ensure a signal power greater than or equal tominus724dBm at the reception as specified in [33] For each 50meters (from 50 to 500 meters) we compute the minimumtransmitting power required to reach this threshold (119879119909(119903) sdot119897(119903) ge minus724 dBm for each interval of 50 meters leading to 10possible transmission powers)This step function models thecase where eNodeB has a finite set of predetermined powerThe process intensity modeling eNodeB is equal to 225 perkm2 It corresponds to the intensity of base stations in Paris(httpswwwantennesmobilesfr) Random variables ℎ119894 are
supposed constant equal to 1 This assumption facilitatesinterpretation of the results but any distribution can beconsidered as well It simply adds a factor in terms of variance(cf (20)) We simulate the different strategies through asimulator coded inC available here (httpwwwanthonybus-sonfrindexphppublications) In all simulations and nu-merical results we consider 50 eNodeB The different sumsin the equations (eg (10) and (20)) are then limited to 50For each set of parameters simulations have be run from10000 to several millions times depending on the evaluatedquantities The number of simulationssamples has been setin order to have negligible confidence intervals They areconsequently not shown in the different figures
Mean and Variance of Interference In Figure 2 we plot themean and the standard deviation of interference obtainedfrom simulations and computed from formulas (10) (19) and(20)when the distance 1198830 variesThe theoretical evaluationcloselymatches empirical estimators obtained by simulationsAs expected the highest interference level is observed forthe independent allocation and the lowest level for the staticallocation The static and the 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tions offer equivalent results with a multiplication factor
Figure 2 Interferencemean and standard deviation as a function of the distance between the typical user and its eNodeB (1198830) Simulationresults are given by points and solid lines and theoretical evaluations by the dotted lines
Table 2 Mean and standarddeviation for the119872119872119877119861119898119886119909119877119861m119886119909 allocation and the two heuristics
Assignment strategy Δ = 3 Δ = 6 Δ = 9mean std-dev mean std-dev mean std-dev
119872119872119877119861119898119886119909119877119861119898119886119909 178119890 minus 11 861119890 minus 11 366119890 minus 12 750119890 minus 12 155119890 minus 12 316119890 minus 12Minimize Interference 994119890 minus 12 307119890 minus 11 186119890 minus 12 188119890 minus 12 756119890 minus 13 626119890 minus 13Maximize reuse distance 526119890 minus 11 751119890 minus 10 118119890 minus 11 107119890 minus 10 258119890 minus 12 109119890 minus 11
varying from 11 to 87 for the mean interference Insteadthe independent allocation differs deeply Mean and standarddeviation are multiplied by a factor ranging from 17 to 700for the mean and can reach up to 26 times 104 for the standarddeviation (for 1198830 = 002 km) The use of the thinnedpoint process is thus questionable to approximate realisticassignment strategies This strategy is no more considered inthe following
Extrapolation of Interference Distribution We compareempirical distributions obtained from simulations to knowndistributions for the different strategies and the two heuristicswhen Δ = 3 6 and 9 Three of these empirical distributionsare shown in Figure 3 (in dB) The distribution parametershave been set according to the maximum likelihood Thedistributions that best fit simulations vary according to theallocation strategy and the reuse factorΔ For 4 cases over 12the distribution minimizing error is the normal distributionAs soon as an asymmetry is observed other distributionsare more accurate log-normal Weibull and Inverse GammaWe perform two hypothesis tests T-test and Smirnov-Kolmogorov for all these scenarios The alternative hypoth-esis is systematically ruled out For the 119872119872119877119861119898119886119909119877119861119898119886119909
strategy best fits are given by the normal (Δ = 3 and 6) andWeibull distributions (Δ = 9)
Nevertheless as mentioned in Section 5 when estimatingthe SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy weuse the normal law to model interference Indeed even if thisdistribution is not systematically the most accurate it offers agood approximation as shown inFigure 4 where the empiricaland normalCDF are plottedMoreover simulationswill showthat this assumption does not impact the accuracy of theanalytical model
Heuristics Simulations have shown that allocations resultingfrom the two heuristics correlate the transmission powersthe distance between eNodeB reusing the same resourceand the random variables 119908119894 Whenminimizing interferencecorrelations are caused by resource blocks allocated to usersthat require a low transmission power and that can be reusedat a short distance and inversely The transmission poweris then correlated to 119908119894 and to the distance of eNodeB thatreuse the same resource These complex phenomena impedethe proposal of an analytical model for these two heuristicsFor comparison purposes Table 2 reportsmean and standarddeviation for the two heuristics and the119872119872119877119861119898119886119909119877119861119898119886119909
Figure 3 Interference distribution empirical (histogram) and extrapolation (curve) Among the classical distributions the bestextrapolations are Gamma (m=1003 var=977) normal (m=2823 var=2216) and Weibull (a=2514b=575) for the heuristic MMRBallocation Δ = 6 and static allocation Δ = 9 respectively
Figure 4 CDF of interference for the 119872119872119877119861119898119886119909119877119861119898119886119909 strategy Empirical distributions are represented through the solid lines CDFof the Normal distribution are plotted with dotted lines Their parameters are obtained from the theoretical formulas The 119871+infin errors are338119890minus02 493119890minus02 and 565119890minus02 for Δ = 3 6 and 9 respectively
Figure 5 Mean interference when the number of RB variesThe load is adjusted to keep a constant reuse factor (Δ = 36 or 9) The distance 1198830 is set to 50 meters Dotted linescorrespond to the theoretical evaluation ofmean interference for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation and points to simulations
allocationTheheuristicminimizing interference presents thebest results This difference was expected as the objectivefunction minimizes interference and because the secondheuristic maximizes the reuse distance without taking intoaccount transmission powers 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tion offers interesting results with an intermediary interfer-ence level between the two heuristics Even if interference islower for the heuristic minimizing interference its complex-ity in terms of feedbackmeasures from users and eNodeBis significantly greater than the other strategies Indeed itassumes that the scheduler knows before an assignment theinterference contribution of each eNodeB on each user
Impact of the Number of RB We performed simulations andtheoretical estimations of interference for different values of119877119861119898119886119909 6 15 25 50 75 and 100 corresponding to differentbandwidth of E-UTRA [33] Results are shown in Figure 5 forthe 119872119872119877119861119898119886119909119877119861119898119886119909 allocation and the two heuristicsThe workload is adapted in order to keep the same Δreuse factor It appears that the mean interference is quiteinsensitive to the number of RB except when 119877119861119898119886119909 = 6The impacting factor is thus the reuse factor Δ rather thanthe number of RBAlso the same hierarchy between the threestrategies is observed except for 119877119861119898119886119909 = 6SIR Distribution We evaluate the SIR CDF according to(29) for the 119872119872119877119861119898119886119909119877119861119898119886119909 allocation This CDF iscompared to the one obtained by simulations in Figure 6(a)We clearly observe that the analytical model offers very
tight estimates of the SIR distribution The assumptionabout the normal distribution of interference does notintroduce a noticeable error The SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy and the two heuristics arecompared in Figure 6(b) The three CDF are similar and the119872119872119877119861119898119886119909119877119861119898119886119909 allocation still offers an intermediatedistribution between the two heuristics The 119871+infin normbetween the three CDF is 0192 0266 and 0221 for Δ = 36 and 9 respectivelyModulation and Coding Rate The SIR distribution allowsus to estimate modulations and coding rates that could beoffered to users We consider the thresholds between therequired SIR and coding rate given in [33] Applied to the SIRdistribution it gives the proportion of users that benefits froma certain modulation scheme and coding rate The couplemodulationcoding rate is referred to as transmission rate inthe following This proportion is given in Figure 7(a) for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation when the density of eNodeBincreases from 225 (the default intensity considered in theprevious evaluation) to 20 eNodeB per km2 It appears thatthe network densification even with a factor 10 does notsignificantly improve the transmission rate Only the besttransmission rate (64QAM-CR=45) really benefits from thisdensification with a proportion of users increasing from 35to 116
The spatial reuse has much more impact on the transmis-sion rates as it is shown in Figure 7(b) It clearly increasesthe transmission rates that are shifted from (QPSK CR=12-64QAM CR=23) for Δ = 3 to (16QAM CR=12-64QAMCR=45) for Δ = 9 The mean transmission rate increasesalmost of a factor 2 from 231 to 424 bitsbaud
Finally we compare in Figure 7(c) the transmission ratesfor the three allocation strategies As already observed inthe previous plots the heuristic minimizing interferenceoffers better performance but is comparable to the two otherstrategies For instance for Δ = 6 the mean transmis-sion rates are close 363 323 and 392 bitsbaud for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation the heuristic maximizingthe reuse distance and the one minimizing interferencerespectively
7 Conclusion
In recent LTE standards a server may assign RB to a set ofeNodeB to control resource usage and optimize performancein a global way Even if it exists solutions to address this prob-lem for a given configuration and topology literature lacksmodels that evaluate performance of these RB allocations fora wide range of scenarios and at large scale
To address this problem we propound a spatial stochasticmodel that takes into account RB assignment strategiesrealistic traffic demands and power control We proposeanalytical estimates for the two first moments of interferenceThis computation is based on two approximations the distri-bution of the distance between a point uniformly distributedin a cell that models the user-eNodeB distance and thejoint distribution of the distances eNodeB-origin and user-eNodeB For the latter simulations have shown that these
(a) SIR empirical and theoretical CDF for the119872119872119877119861119898119886119909119877119861119898119886119909allocation
MMRB Delta=3MMRB Delta=6MMRB Delta=9MI Delta=3 MI Delta=6
MI Delta=9 MD Delta=3 MD Delta=6 MD Delta=9
0 10 20 30 40minus10훽 (dB)
00e+00
20eminus01
40eminus01
60eminus01
80eminus01
10e+00
CDF
minus SI
NR
(b) SIR empirical CDF for the three strategies MI maximizeinterference MD maximize distance
Figure 6 (a) We compare the distribution of SIR obtained by simulations (solid lines) and given by (29) (dotted lines) Errors with regardto the 119871+infin norm are 247119890 minus 02 154119890 minus 02 and 158119890 minus 02 for Δ = 3 6 and 9 respectively (b) CDF for the three allocation strategies(119872119872119877119861119898119886119909119877119861119898119886119909 and Heuristics)
(c) 119872119872119877119861119898119886119909119877119861119898119886119909 and the two heuristics Δ = 6 120582119890 = 225
Figure 7 Modulation and coding rate for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation strategy when the density of eNodeB increases (Figure 7(a))for different reuse factor (Figure 7(b)) and comparison with the two heuristics (Figure 7(c))
variables are strongly correlated particularly for the points ofthe process close to the origin This correlation impacts thetransmission power used by the eNodeB and consequentlyinterference and SIR distributions The derivation of the SIRdistribution allows us to express the classical outage but alsothe transmission rates in terms of modulationcoding ratethat provide interesting insights on the throughput offers tothe users Also it appears that eNodeB densification does not
significantly improve the network performance in terms ofthroughput except for a small percentage of users for whichthe transmission rate is increased Spatial reuse expressedthrough the parameter Δ in our study has a much moreimpact on the performance as the transmission rates aresignificantly increased This spatial reuse must be expressedas the ratio between the total number of available resourcesover the number of usersrequests per cell Indeed we have
Wireless Communications and Mobile Computing 13
observed that the system performance is quite insensitive tovariation of these quantities when this ratio stays constant
Beside simulations show that the RB assignment strate-gies developed for the model give results close to the onesgiven by classical optimization problem which minimizesglobal interference or maximizes spatial reuse distances Itempirically shows that our analytical model is able to evaluateperformance of classical optimization approaches but at largescale in terms of number of nodes configurations andtopologies
Appendix
A Construction of the Markov Chain
In this section we introduce the Markov chain used to set thedistribution of 1198770 We consider that the RB used between thetypical user and its eNodeB has index 119903119890119904 (119903119890119904 isin 0 119877119861119898119886119909minus1)We shall prove that this index does not impact calculation(we have chosen 119903119890119904 = 0 in this paper by sack of simplicity)
We set the distribution of the number of used resourceblocks at eNodeB 0 in such a way that it leads to the followingproperty (119895 gt 119894 ge 0)
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (A1)
In order to obtain this property we first introduce somepreliminary notations and results An example of allocationwith the different notations is shown in Figure 1 We define asequence of rv (119877119899)119899ge0 If eNodeB 119894 is the 119899119905ℎ eNodeB using119903119890119904 and if 119864119899119889119899 is the index of the last RB used by this eNodeBthen
119877119899 = 119864119899119889119899 minus 119903119890119904 if 119864119899119889119899 gt 119903119890119904119864119899119889119899 + 119877119861119898119886119909 minus 119903119890119904 otherwise (A2)
If eNodeB 119894 is the 119899119905ℎ interfering eNodeB (using 119903119890119904) theneNodeB 119895 (119895 gt 119894) is the 119899 + 1119905ℎ if and only if
119863119896 ge 119877119861119898119886119909 (A4)
In this case 119877119899+1 is given by
119877119899+1 = 119877119899 +119895sum
119896=119894+1
119863119896 minus 119877119861119898119886119909 (A5)
As (119863119894)119894gt0 are iid the sequence (119877119899)119899ge0 is a homoge-neous Markov chain The transition probabilities for (119897 119898) isin0 1 119877119861119898119886119909 minus 12 are given by 119875119897119898 = P(119877119899+1 = 119898 | 119877119899 = 119897)
The event 119877119899+1 = 119898 given that 119877119899 = 119897 occurs if andonly if it exists 119895 ge 0 such that
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (A10)
We set the distribution of 1198770 with the correspondingstationary distribution Consequently the sequence (119877119899)119899ge0becomes identically distributed where 119877119899 follows the sta-tionary distribution for all 119899 ge 0 Moreover according to(A3) and (A4) the property given in (A1) holds Indeed 119877119899sum119895minus1
119896=119894+1119863119896 and sum119895
119896=119894+1119863119896 have the same distribution as 1198770sum119895minus119894minus1
119896=1119863119896 andsum119895minus119894
119896=1119863119896 respectively In the model evaluation
the stationary distribution is obtained numerically from thetransition probabilities As soon as 1198770 follows the stationarydistribution the index 119903119890119904 can be chosen arbitrarily
B Conditional Distribution of 119880119894minus119883119894Simulations have shown that given 119883119894 the distribution of119883119894 minus 119880119894 is still close to the one given by (2) but with adifferent parameter So we assume that the distribution of119883119894 minus 119880119894 given 119883119894 is equal to (2) but with a parameterfunction of 119883119894 denoted 119888(119883119894)
These simulations have also shown that E[119883119894 minus 119880119894 |119883119894 = 119903]may be approximated as an affine function 119886119894119903 + 119887119894We define 119888119894(sdot) accordingly It leads to
To set 119886119894 and 119887119894 we use the two following propertiesE [E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817 | 10038171003817100381710038171198831198941003817100381710038171003817 = 119903]] = E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817] (B3)
= 12radic120582119890 (B4)
E [E [1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817 | 1003817100381710038171003817119883119894
1003817100381710038171003817 = 119903]] = E [119886119894 10038171003817100381710038171198831198941003817100381710038171003817 + 119887119894] (B5)
Equation (B4) is derived from (2) and (B5) from ourassumption on 119888119894(119903) 119886119894 is then set as a function of 119887119894 E[119883119894]and 120582119890 119887119894 is obtained from our simulations For a givenintensity 119887119894 has been observed almost constant with regardto 119894 The best approximation as a function of the intensity isgiven by 119887119894 = 033radic120582119890 After a fewmanipulations we obtain
First note that by convention we have P(119908119894 = 1) = P(119908119894 =1 | 1199080 = 1) The event 119908119894 = 1 occurs if and only if it exists119895 ge 1 such that1198770 + 119894minus1sum
There is no data used in this paper Nevertheless the code ofour simulator is made available (the link is in the paper)
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] ldquoRequirements for further advancements for e-utra (lte-advanced) 3gpp technical specification ts 36913 release 13rdquo2015 httpwww3gpporg
[2] A S Hamza S S Khalifa H S Hamza and K Elsayed ldquoAsurvey on inter-cell interference coordination techniques inofdma-based cellular networksrdquo IEEE Communications Surveysamp Tutorials vol 15 no 4 pp 1642ndash1670 2013
[3] D Lopez-Perez A Ladanyi A JuttnerH Rivano and J ZhangldquoOptimization method for the joint allocation of modulationschemes coding rates resource blocks and power in self-organizing LTE networksrdquo in Proceedings of the IEEE INFO-COM 2011 pp 111ndash115 April 2011
[4] A Guo andM Haenggi ldquoSpatial stochasticmodels andmetricsfor the structure of base stations in cellular networksrdquo IEEE
Wireless Communications and Mobile Computing 15
Transactions on Wireless Communications vol 12 no 11 pp5800ndash5812 2013
[5] C-H Lee C-Y Shih and Y-S Chen ldquoStochastic geometrybased models for modeling cellular networks in urban areasrdquoWireless Networks vol 19 no 6 pp 1063ndash1072 2013
[6] W Lu and M D Renzo ldquoStochastic geometry modeling ofcellular networks analysis simulation and experimental valida-tionrdquo in Proceedings of the 18th ACM International Conferenceon Modelling Analysis and Simulation of Wireless and MobileSystems (MSWiM rsquo15) pp 179ndash188 New York NY USANovember 2015
[7] Y L Lee T C Chuah J Loo and A Vinel ldquoRecent advancesin radio resource management for heterogeneous LTELTE-Anetworksrdquo IEEE Communications Surveys amp Tutorials vol 16no 4 pp 2142ndash2180 2014
[8] F Mhiri K Sethom and R Bouallegue ldquoA survey on inter-ference management techniques in femtocell self-organizingnetworksrdquo Journal of Network and Computer Applications vol36 no 1 pp 58ndash65 2013
[9] X Chen L Li and X Xiang ldquoAnt colony learning methodfor joint MCS and resource block allocation in LTE Femtocelldownlink for multimedia applications with QoS guaranteesrdquoMultimedia Tools andApplications vol 76 no 3 pp 4035ndash40542017
[10] D Lopez-Perez X Chu A V Vasilakos and H Claussen ldquoOndistributed and coordinated resource allocation for interferencemitigation in self-organizing lte networksrdquo IEEEACMTransac-tions on Networking vol 21 no 4 pp 1145ndash1158 2013
[11] H Zhang H Liu J Cheng and V C M Leung ldquoDownlinkenergy efficiency of power allocation and wireless backhaulbandwidth allocation in heterogeneous small cell networksrdquoIEEE Transactions on Communications no 99 2017
[12] H Zhang W Zheng X Chu et al ldquoJoint subchannel andpower allocation in interference-limited OFDMA femtocellswith heterogeneous QoS guaranteerdquo in Proceedings of theIEEEGlobal Communications Conference (GLOBECOM rsquo12) pp4572ndash4577 December 2012
[13] H Elsawy E Hossain and M Haenggi ldquoStochastic geometryfor modeling analysis and design of multi-tier and cognitivecellular wireless networks a surveyrdquo IEEE CommunicationsSurveys amp Tutorials vol 15 no 3 pp 996ndash1019 2013
[14] H ElSawy A Sultan-Salem M-S Alouini and M Z WinldquoModeling and analysis of cellular networks using stochasticgeometry a tutorialrdquo IEEE Communications Surveys amp Tutori-als vol 19 no 1 pp 167ndash203 2017
[15] R Kwan and C Leung ldquoA survey of scheduling and inter-ference mitigation in lterdquo Journal of Electrical and ComputerEngineeringmdashSpecial Issue on LTELTE-AdvancedCellular Com-munication Networks pp 11ndash110 2010
[16] J G Andrews F Baccelli and R K Ganti ldquoA tractable approachto coverage and rate in cellular networksrdquo IEEE Transactions onCommunications vol 59 no 11 pp 3122ndash3134 2011
[17] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for OFDMA cellu-lar networksrdquo IEEE Transactions on Wireless Communicationsvol 10 no 12 pp 4294ndash4305 2011
[18] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for heterogeneouscellular networksrdquo IEEE Transactions on Communications vol60 no 7 pp 2029ndash2039 2012
[19] H Zhuang and T Ohtsuki ldquoA model based on poisson pointprocess for downlink K tiers fractional frequency reuse hetero-geneous networksrdquo Physical Communication vol 13 pp 3ndash122014 Special Issue on Heterogeneous and Small Cell Networks
[20] A A Gebremariam T Bao D Siracusa T Rasheed F Granelliand L Goratti ldquoDynamic strict fractional frequency reusefor software-defined 5G networksrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo16) pp 1ndash6May 2016
[21] N Lee D Morales-Jimenez A Lozano and R W HeathldquoSpectral efficiency of dynamic coordinated beamforming astochastic geometry approachrdquo IEEE Transactions on WirelessCommunications vol 14 no 1 pp 230ndash241 2015
[22] X Zhang and M Haenggi ldquoA Stochastic geometry analysisof inter-cell interference coordination and intra-cell diversityrdquoIEEE Transactions on Wireless Communications vol 13 no 12pp 6655ndash6669 2014
[23] F Baccelli andAGiovanidis ldquoA stochastic geometry frameworkfor analyzing pairwise-cooperative cellular networksrdquo IEEETransactions on Wireless Communications vol 14 no 2 pp794ndash808 2015
[24] J Yoon and G Hwang ldquoDistance-based inter-cell interferencecoordination in small cell networks stochastic geometry mod-eling and analysisrdquo IEEE Transactions on Wireless Communica-tions 2018
[25] B Błaszczyszyn M Jovanovicy and M K Karray ldquoHow userthroughput depends on the traffic demand in large cellularnetworksrdquo in Proceedings of the 12th International Symposiumon Modeling and Optimization in Mobile Ad Hoc and WirelessNetworks (WiOpt rsquo14) pp 611ndash619 May 2014
[26] H H Yang and T Q S Quek ldquoSIR coverage analysis incellular networks with temporal traffic a stochastic geometryapproachrdquo Corrossion 2018
[27] S M Yu and S-L Kim ldquoDownlink capacity and base stationdensity in cellular networksrdquo in Proceedings of the 11th Interna-tional SymposiumandWorkshops onModeling andOptimizationin Mobile Ad Hoc and Wireless Networks (WiOpt rsquo13) pp 119ndash124 May 2013
[28] YWang M Haenggi and Z Tan ldquoThemeta distribution of thesir for cellular networks with power controlrdquo Corrossion 2017
[29] M S Omar S A Hassan H Pervaiz et al ldquoMulti-objectiveoptimization in 5g hybrid networksrdquo IEEE Internet of ThingsJournal 2018
[30] Y Leo A Busson C Sarraute and E Fleury ldquoCall detail recordsto characterize usages and mobility events of phone usersrdquoComputer Communications vol 95 pp 43ndash53 2016
[31] I L Cherif Spectral and Energy Efficiency in 5G WirelessNetworks [PhD thesis] Univeristy Paris Saclay 2016
[32] S Mukherjee Analytical Modeling of Heterogeneous CellularNetworks Analytical Modeling of Heterogeneous Cellular Net-works Geometry Coverage and Capacity Cambridge Univer-sity Press 2014
[33] ldquoTechnical specification group radio access network base sta-tion (bs) radio transmission and reception 2012 Release 1123gpp ts 36104 v112rdquo 2012
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
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6 Wireless Communications and Mobile Computing
Data 119877119861119898119886119909 Total number of RBcost sum of interference for the current allocationbestCost sum of interference for the best allocation strategy initialized to -1Result Assign an RB to a user This algorithm is called by the controller for each
new user(1) for each 119903119887 isin 0 1 119877119861119898119886119909 minus 1 do(2) if rb is free then(3) assign rb to this user(4) costlarr sum Interference()(5) if bestCost lt0 or costltbestCost then(6) Save this allocation strategy(7) bestCost = cost(8) end(9) end(10) end(11) Assign the saved allocation strategy
Algorithm 1 Minimizing global interference
for the users already assigned The typical user is consideredin last when the system has reached the targeted load Thisalgorithm mimics an assignment strategy where the RB areassigned at the arrival of the users request without changingthe already assigned RB
The second heuristic ldquomaximize reuse distancerdquo maxi-mizes the distance at which the RB are reused Each user isconsidered in a random order Different RB are assigned tothe 119877119861119898119886119909 first users When assigning an RB to the otherusers the controller chooses the RB for which the reusedistance is maximum The typical user is considered in lastThis second heuristic may correspond to a case where thecontroller does not have information on channel conditionsand interference but knows the distances between eNodeB
4 Interference Characterization
We derive the mean and the variance of interference for thethree assignment strategies defined in the previous section
The point process modeling interferers is a dependentthinning of the original PPP Consequently conditions formean (respectively variance) to be finite with a PPP also holdfor our point process the path loss function 119897(sdot)must belongto 1198711 (respectively 1198712)41 Distribution of Distances between the Typical User andeNodeB (119883119894) As a preamble we give the PDF of thedistance between the typical user at the origin and eNodeBBoth PDF of 119883119894 and joint distribution of (119883119894 119883119895) arederived These PDF are used in the computation of the meanand the variance of interference
In the numerical evaluation we shall condition interfer-ence by the distance 1198830 It allows us to study interferencefor a given distance between the typical user and its attachedeNodeB It is also motivated by the computation of the SIRwhere both interference and the typical user signal strengthdepend on the distance 1198830
For our model the PDF of 119883119894with 119894 gt 0 given 1198830 = 119903is
1198911198941198830=119903 (119906 119903) = (120582119890120587)119894(119894 minus 1)2119906 (1199062 minus 1199032)119894minus1 119890minus120582119890120587(1199062minus1199032)1119906gt119903 (8)
The joint PDF of (119883119894 119883119895)with 119895 gt 119894 gt 0 given 1198830 =119903 is1198911198941198951198830=119903
(119906 V 119903)= (120582119890120587)119895(119894 minus 1) (119895 minus 119894 minus 1)4119906V (V2 minus 1199062)
119895minus119894minus1
times (1199062 minus 1199032)119894minus1 119890minus120582119890120587(V2minus1199032)1Vgt119906gt119903(9)
To obtain the PDF when 1198830 is not set (119891119883119894(sdot) and119891119883119894119883119895(sdot)) it suffices to integrate the two conditional PDFwith regard to the PDF of 1198830 given in (2)
42 Mean of Interference Themean is derived from (1)
E [119868 (100381710038171003817100381711988301003817100381710038171003817)]
= E [ℎ1] +infinsum119894=1
E [119908119894]E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) sdot 119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)] (10)
In this equation 119908119894 has been separated from the expecta-tion as it is independent of the process 119873119890 (according to thedefined strategies) We derive E[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)] andE[119908119894] in the two next sections
421 Computation of E[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)]No Power Control In absence of power control ie when119879119909(sdot) is constant or independent of the process 119873119890 a closedformula may be expressed for (10) E[119879119909(119883119894 minus119880119894) sdot 119897(119883119894)]
Wireless Communications and Mobile Computing 7
is then given by E[119879119909]E[119897(119883119894)] Expectation of 119897(119883119894) isobtained from the distribution of 119883119894 given in Section 41
E [119868 (100381710038171003817100381711988301003817100381710038171003817)]
= E [ℎ1] +infinsum119894=1
E [119908119894]E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817)]E [119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)] (11)
Power Control When the transmission power depends onthe distance between the receiver and its attached eNodeB(119879119909(119883119894 minus 119880119894)) the computations are more complex119864[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)] cannot be approximated byE[119879119909(119883119894minus119880119894)]119864[119897(119883119894)] as the size of the Voronoı cell withnucleus 119883119894 depends on its distance to the origin The jointdistribution of (119883119894 119883119894 minus119880119894) being unknown we proposethe following approximation
The PDF of 119880119894 minus 119883119894 is the same as 1198800 minus 1198830 givenby (2) Its parameter 119888119894(119883119894) depends on 119883119894 119888119894(119883119894) =14radic120582119890(120574119894119883119894 + 119887) with 120574119894 = (12 minus 119887)E[119883119894] and 119887 = 033The motivation and the computation details for this PDF aregiven in appendix (Appendix B) We obtain
E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) sdot 119897 (1003817100381710038171003817119883119894
Often in real systems the transmission power cannot beset arbitrarily and is limited to a set of predetermined valuesThe transmission power function can then be representedas a step function 119879119909(119903) = sum119873119879
119894=1 1199051198941119903isin[120572119894minus1120572119894] where 119873119879
is the number of possible transmission powers 119905119894 the 119894119905ℎtransmission power value and [120572119894minus1 120572119894] the distance intervalbetween a user and its eNodeB at which this transmissionpower is used An example of such setting is given in thenumerical evaluation section In this case (14) becomes
E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) 119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)]= 119873119879sum
119894=1
119905119895 int+infin
0(119890minus120582119890119888119894(119903)1205871205722119895 minus 119890minus120582119890119888119894(119903)1205871205722119895minus1)119891119883119894 (119903) 119889119903
(15)
When the computation is performed for a given distance1198830 the PDF 119891119883119894(sdot) in (12) and (15) must be replaced by1198911198941198830=119903(sdot sdot) (given in Section 41)
422 Computation of E[119908119894] Finally in order to compute(10) we need to expressE[119908119894] First note that E[119908119894] = P(119908119894 =1)Proposition 2 The probability for eNodeB 119894 to interfere withthe typical user is given by
(i) Independent allocation
P (119908119894 = 1) = 1Δ (16)
(ii) Static allocation
P (119908119894 = 1) = 1119894sdot119898119900119889(Δ)=0 (17)
E [119879119909 (1003817100381710038171003817119880119894 minus 1198831198941003817100381710038171003817) 119879119909 (10038171003817100381710038171003817119880119895 minus 119883119895
As for the mean complexity lies in the correlationbetween 119883119894 and 119880119894 minus 119883119894 The term E[119879119909(119880119894 minus 119883119894)2 sdot119897(119883119894)2] is computed with the same method as the firstmoment
Computation of E[119879119909(119880119894minus119883119894)119879119909(119880119895minus119883119895)119897(119883119894)119897(119883119895)]As 119880119894 minus 119883119894 (respectively 119880119895 minus 119883119895) depends on 119883119894(respectively 119883119895) we condition by the distribution of(119883119894 119883119895) given in Section 41 Given 119883119894 and 119883119895 we usethe same PDF as in (B1) assuming that 119880119894 minus 119883119894 and 119880119895 minus119883119895 are independent The considered joint distribution of(119883119894 119883119895 119880119894 minus 119883119894 119880119895 minus 119883119895) becomes
1198910119880119894minus119883119894 (119906 119903) 1198910119880119895minus119883119895 (V 119904) 119891119883119894119883119895 (119903 119904) (21)
When the distance 1198830 is fixed 119891119883119894119883119895(sdot sdot) must bereplaced by the PDF 119891119894119895
1198830=119903(sdot sdot sdot) given in Section 41
Computation of E[119908119894119908119895] It has been shown that the sequence(119908119894)119894gt0 verifies (7) for the three strategies It allows us toexpress E[119908119894119908119895] with 119894 gt 119895 as
E [119908119894119908119895] = P (119908119894minus119895 = 1)P (119908119895 = 1) (22)
8 Wireless Communications and Mobile Computing
Proposition 3 The joint probability for two eNodeB 119894 and 119895(119895 gt 119894) to interfere with the typical user is given by
(i) Independent allocation
E [119908119894119908119895] = 1Δ2 (23)
(ii) Static allocation
E [119908119894119908119895] = 1119894sdot119898119900119889(Δ)=0119895sdot119898119900119889(Δ)=0 (24)
where P(119908 = 1) is given by (18)5 Signal over Interference Ratio (SIR)
In our model there is a strong correlation between theinterfering eNodeB It is generated by the allocation strategiesand cannot be neglected Also a correlation exists betweenthe location of an eNodeB and the size of its Voronoı cellConsequently classical approach based on PPP which uses
Laplacian transform for instance cannot be applied hereand a formal derivation of interference distribution seemsintractable
Nevertheless the different simulations presented in thenext section will show that the PDF of interference can beapproximated by a log-normal distribution The parametersof this distribution mean and variance denoted by 119898119868119889119861
(sdot)and 120590119868119889119861(sdot) are directly derived from the previous analyticalcomputations The classical mapping between log-normaland normal parameters can be applied to derive parametersof the normal distribution when interferences are expressedin decibel In the following a variable is indexed by 119889119861 whenit is expressed in decibel
We get
P (119878119868119877119889119861 le 120573119889119861) = P (10sdot log10 (119875119905 (10038171003817100381710038171198830
1003817100381710038171003817) ℎ0119897 (100381710038171003817100381711988301003817100381710038171003817)) minus 119868119889119861 (10038171003817100381710038171198830
1003817100381710038171003817) le 120573119889119861) (26)
= P (119868119889119861 (100381710038171003817100381711988301003817100381710038171003817) ge minus120573119889119861 + 10
When ℎ0 is not constant the expectation with regard toits distribution must be taken into account in (28) In (29)ℎ0 is assumed to be constant that equals to 1 In these twoequations 119890119903119891(sdot) is the error function6 Numerical Results
We consider an E-UTRA channel with a bandwidth of5MHz with 119877119861119898119886119909 = 15 [33] The path loss function isthe same as [3] It is expressed in dB 119897(119903) = minus1281 minus376 sdot log10(119903) where 119903 is the distance (in km) 119879119909(sdot) isset in such a way to guarantee to each user a minimumreceiving power We set the transmission power function119879119909(sdot) to ensure a signal power greater than or equal tominus724dBm at the reception as specified in [33] For each 50meters (from 50 to 500 meters) we compute the minimumtransmitting power required to reach this threshold (119879119909(119903) sdot119897(119903) ge minus724 dBm for each interval of 50 meters leading to 10possible transmission powers)This step function models thecase where eNodeB has a finite set of predetermined powerThe process intensity modeling eNodeB is equal to 225 perkm2 It corresponds to the intensity of base stations in Paris(httpswwwantennesmobilesfr) Random variables ℎ119894 are
supposed constant equal to 1 This assumption facilitatesinterpretation of the results but any distribution can beconsidered as well It simply adds a factor in terms of variance(cf (20)) We simulate the different strategies through asimulator coded inC available here (httpwwwanthonybus-sonfrindexphppublications) In all simulations and nu-merical results we consider 50 eNodeB The different sumsin the equations (eg (10) and (20)) are then limited to 50For each set of parameters simulations have be run from10000 to several millions times depending on the evaluatedquantities The number of simulationssamples has been setin order to have negligible confidence intervals They areconsequently not shown in the different figures
Mean and Variance of Interference In Figure 2 we plot themean and the standard deviation of interference obtainedfrom simulations and computed from formulas (10) (19) and(20)when the distance 1198830 variesThe theoretical evaluationcloselymatches empirical estimators obtained by simulationsAs expected the highest interference level is observed forthe independent allocation and the lowest level for the staticallocation The static and the 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tions offer equivalent results with a multiplication factor
Figure 2 Interferencemean and standard deviation as a function of the distance between the typical user and its eNodeB (1198830) Simulationresults are given by points and solid lines and theoretical evaluations by the dotted lines
Table 2 Mean and standarddeviation for the119872119872119877119861119898119886119909119877119861m119886119909 allocation and the two heuristics
Assignment strategy Δ = 3 Δ = 6 Δ = 9mean std-dev mean std-dev mean std-dev
119872119872119877119861119898119886119909119877119861119898119886119909 178119890 minus 11 861119890 minus 11 366119890 minus 12 750119890 minus 12 155119890 minus 12 316119890 minus 12Minimize Interference 994119890 minus 12 307119890 minus 11 186119890 minus 12 188119890 minus 12 756119890 minus 13 626119890 minus 13Maximize reuse distance 526119890 minus 11 751119890 minus 10 118119890 minus 11 107119890 minus 10 258119890 minus 12 109119890 minus 11
varying from 11 to 87 for the mean interference Insteadthe independent allocation differs deeply Mean and standarddeviation are multiplied by a factor ranging from 17 to 700for the mean and can reach up to 26 times 104 for the standarddeviation (for 1198830 = 002 km) The use of the thinnedpoint process is thus questionable to approximate realisticassignment strategies This strategy is no more considered inthe following
Extrapolation of Interference Distribution We compareempirical distributions obtained from simulations to knowndistributions for the different strategies and the two heuristicswhen Δ = 3 6 and 9 Three of these empirical distributionsare shown in Figure 3 (in dB) The distribution parametershave been set according to the maximum likelihood Thedistributions that best fit simulations vary according to theallocation strategy and the reuse factorΔ For 4 cases over 12the distribution minimizing error is the normal distributionAs soon as an asymmetry is observed other distributionsare more accurate log-normal Weibull and Inverse GammaWe perform two hypothesis tests T-test and Smirnov-Kolmogorov for all these scenarios The alternative hypoth-esis is systematically ruled out For the 119872119872119877119861119898119886119909119877119861119898119886119909
strategy best fits are given by the normal (Δ = 3 and 6) andWeibull distributions (Δ = 9)
Nevertheless as mentioned in Section 5 when estimatingthe SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy weuse the normal law to model interference Indeed even if thisdistribution is not systematically the most accurate it offers agood approximation as shown inFigure 4 where the empiricaland normalCDF are plottedMoreover simulationswill showthat this assumption does not impact the accuracy of theanalytical model
Heuristics Simulations have shown that allocations resultingfrom the two heuristics correlate the transmission powersthe distance between eNodeB reusing the same resourceand the random variables 119908119894 Whenminimizing interferencecorrelations are caused by resource blocks allocated to usersthat require a low transmission power and that can be reusedat a short distance and inversely The transmission poweris then correlated to 119908119894 and to the distance of eNodeB thatreuse the same resource These complex phenomena impedethe proposal of an analytical model for these two heuristicsFor comparison purposes Table 2 reportsmean and standarddeviation for the two heuristics and the119872119872119877119861119898119886119909119877119861119898119886119909
Figure 3 Interference distribution empirical (histogram) and extrapolation (curve) Among the classical distributions the bestextrapolations are Gamma (m=1003 var=977) normal (m=2823 var=2216) and Weibull (a=2514b=575) for the heuristic MMRBallocation Δ = 6 and static allocation Δ = 9 respectively
Figure 4 CDF of interference for the 119872119872119877119861119898119886119909119877119861119898119886119909 strategy Empirical distributions are represented through the solid lines CDFof the Normal distribution are plotted with dotted lines Their parameters are obtained from the theoretical formulas The 119871+infin errors are338119890minus02 493119890minus02 and 565119890minus02 for Δ = 3 6 and 9 respectively
Figure 5 Mean interference when the number of RB variesThe load is adjusted to keep a constant reuse factor (Δ = 36 or 9) The distance 1198830 is set to 50 meters Dotted linescorrespond to the theoretical evaluation ofmean interference for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation and points to simulations
allocationTheheuristicminimizing interference presents thebest results This difference was expected as the objectivefunction minimizes interference and because the secondheuristic maximizes the reuse distance without taking intoaccount transmission powers 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tion offers interesting results with an intermediary interfer-ence level between the two heuristics Even if interference islower for the heuristic minimizing interference its complex-ity in terms of feedbackmeasures from users and eNodeBis significantly greater than the other strategies Indeed itassumes that the scheduler knows before an assignment theinterference contribution of each eNodeB on each user
Impact of the Number of RB We performed simulations andtheoretical estimations of interference for different values of119877119861119898119886119909 6 15 25 50 75 and 100 corresponding to differentbandwidth of E-UTRA [33] Results are shown in Figure 5 forthe 119872119872119877119861119898119886119909119877119861119898119886119909 allocation and the two heuristicsThe workload is adapted in order to keep the same Δreuse factor It appears that the mean interference is quiteinsensitive to the number of RB except when 119877119861119898119886119909 = 6The impacting factor is thus the reuse factor Δ rather thanthe number of RBAlso the same hierarchy between the threestrategies is observed except for 119877119861119898119886119909 = 6SIR Distribution We evaluate the SIR CDF according to(29) for the 119872119872119877119861119898119886119909119877119861119898119886119909 allocation This CDF iscompared to the one obtained by simulations in Figure 6(a)We clearly observe that the analytical model offers very
tight estimates of the SIR distribution The assumptionabout the normal distribution of interference does notintroduce a noticeable error The SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy and the two heuristics arecompared in Figure 6(b) The three CDF are similar and the119872119872119877119861119898119886119909119877119861119898119886119909 allocation still offers an intermediatedistribution between the two heuristics The 119871+infin normbetween the three CDF is 0192 0266 and 0221 for Δ = 36 and 9 respectivelyModulation and Coding Rate The SIR distribution allowsus to estimate modulations and coding rates that could beoffered to users We consider the thresholds between therequired SIR and coding rate given in [33] Applied to the SIRdistribution it gives the proportion of users that benefits froma certain modulation scheme and coding rate The couplemodulationcoding rate is referred to as transmission rate inthe following This proportion is given in Figure 7(a) for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation when the density of eNodeBincreases from 225 (the default intensity considered in theprevious evaluation) to 20 eNodeB per km2 It appears thatthe network densification even with a factor 10 does notsignificantly improve the transmission rate Only the besttransmission rate (64QAM-CR=45) really benefits from thisdensification with a proportion of users increasing from 35to 116
The spatial reuse has much more impact on the transmis-sion rates as it is shown in Figure 7(b) It clearly increasesthe transmission rates that are shifted from (QPSK CR=12-64QAM CR=23) for Δ = 3 to (16QAM CR=12-64QAMCR=45) for Δ = 9 The mean transmission rate increasesalmost of a factor 2 from 231 to 424 bitsbaud
Finally we compare in Figure 7(c) the transmission ratesfor the three allocation strategies As already observed inthe previous plots the heuristic minimizing interferenceoffers better performance but is comparable to the two otherstrategies For instance for Δ = 6 the mean transmis-sion rates are close 363 323 and 392 bitsbaud for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation the heuristic maximizingthe reuse distance and the one minimizing interferencerespectively
7 Conclusion
In recent LTE standards a server may assign RB to a set ofeNodeB to control resource usage and optimize performancein a global way Even if it exists solutions to address this prob-lem for a given configuration and topology literature lacksmodels that evaluate performance of these RB allocations fora wide range of scenarios and at large scale
To address this problem we propound a spatial stochasticmodel that takes into account RB assignment strategiesrealistic traffic demands and power control We proposeanalytical estimates for the two first moments of interferenceThis computation is based on two approximations the distri-bution of the distance between a point uniformly distributedin a cell that models the user-eNodeB distance and thejoint distribution of the distances eNodeB-origin and user-eNodeB For the latter simulations have shown that these
(a) SIR empirical and theoretical CDF for the119872119872119877119861119898119886119909119877119861119898119886119909allocation
MMRB Delta=3MMRB Delta=6MMRB Delta=9MI Delta=3 MI Delta=6
MI Delta=9 MD Delta=3 MD Delta=6 MD Delta=9
0 10 20 30 40minus10훽 (dB)
00e+00
20eminus01
40eminus01
60eminus01
80eminus01
10e+00
CDF
minus SI
NR
(b) SIR empirical CDF for the three strategies MI maximizeinterference MD maximize distance
Figure 6 (a) We compare the distribution of SIR obtained by simulations (solid lines) and given by (29) (dotted lines) Errors with regardto the 119871+infin norm are 247119890 minus 02 154119890 minus 02 and 158119890 minus 02 for Δ = 3 6 and 9 respectively (b) CDF for the three allocation strategies(119872119872119877119861119898119886119909119877119861119898119886119909 and Heuristics)
(c) 119872119872119877119861119898119886119909119877119861119898119886119909 and the two heuristics Δ = 6 120582119890 = 225
Figure 7 Modulation and coding rate for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation strategy when the density of eNodeB increases (Figure 7(a))for different reuse factor (Figure 7(b)) and comparison with the two heuristics (Figure 7(c))
variables are strongly correlated particularly for the points ofthe process close to the origin This correlation impacts thetransmission power used by the eNodeB and consequentlyinterference and SIR distributions The derivation of the SIRdistribution allows us to express the classical outage but alsothe transmission rates in terms of modulationcoding ratethat provide interesting insights on the throughput offers tothe users Also it appears that eNodeB densification does not
significantly improve the network performance in terms ofthroughput except for a small percentage of users for whichthe transmission rate is increased Spatial reuse expressedthrough the parameter Δ in our study has a much moreimpact on the performance as the transmission rates aresignificantly increased This spatial reuse must be expressedas the ratio between the total number of available resourcesover the number of usersrequests per cell Indeed we have
Wireless Communications and Mobile Computing 13
observed that the system performance is quite insensitive tovariation of these quantities when this ratio stays constant
Beside simulations show that the RB assignment strate-gies developed for the model give results close to the onesgiven by classical optimization problem which minimizesglobal interference or maximizes spatial reuse distances Itempirically shows that our analytical model is able to evaluateperformance of classical optimization approaches but at largescale in terms of number of nodes configurations andtopologies
Appendix
A Construction of the Markov Chain
In this section we introduce the Markov chain used to set thedistribution of 1198770 We consider that the RB used between thetypical user and its eNodeB has index 119903119890119904 (119903119890119904 isin 0 119877119861119898119886119909minus1)We shall prove that this index does not impact calculation(we have chosen 119903119890119904 = 0 in this paper by sack of simplicity)
We set the distribution of the number of used resourceblocks at eNodeB 0 in such a way that it leads to the followingproperty (119895 gt 119894 ge 0)
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (A1)
In order to obtain this property we first introduce somepreliminary notations and results An example of allocationwith the different notations is shown in Figure 1 We define asequence of rv (119877119899)119899ge0 If eNodeB 119894 is the 119899119905ℎ eNodeB using119903119890119904 and if 119864119899119889119899 is the index of the last RB used by this eNodeBthen
119877119899 = 119864119899119889119899 minus 119903119890119904 if 119864119899119889119899 gt 119903119890119904119864119899119889119899 + 119877119861119898119886119909 minus 119903119890119904 otherwise (A2)
If eNodeB 119894 is the 119899119905ℎ interfering eNodeB (using 119903119890119904) theneNodeB 119895 (119895 gt 119894) is the 119899 + 1119905ℎ if and only if
119863119896 ge 119877119861119898119886119909 (A4)
In this case 119877119899+1 is given by
119877119899+1 = 119877119899 +119895sum
119896=119894+1
119863119896 minus 119877119861119898119886119909 (A5)
As (119863119894)119894gt0 are iid the sequence (119877119899)119899ge0 is a homoge-neous Markov chain The transition probabilities for (119897 119898) isin0 1 119877119861119898119886119909 minus 12 are given by 119875119897119898 = P(119877119899+1 = 119898 | 119877119899 = 119897)
The event 119877119899+1 = 119898 given that 119877119899 = 119897 occurs if andonly if it exists 119895 ge 0 such that
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (A10)
We set the distribution of 1198770 with the correspondingstationary distribution Consequently the sequence (119877119899)119899ge0becomes identically distributed where 119877119899 follows the sta-tionary distribution for all 119899 ge 0 Moreover according to(A3) and (A4) the property given in (A1) holds Indeed 119877119899sum119895minus1
119896=119894+1119863119896 and sum119895
119896=119894+1119863119896 have the same distribution as 1198770sum119895minus119894minus1
119896=1119863119896 andsum119895minus119894
119896=1119863119896 respectively In the model evaluation
the stationary distribution is obtained numerically from thetransition probabilities As soon as 1198770 follows the stationarydistribution the index 119903119890119904 can be chosen arbitrarily
B Conditional Distribution of 119880119894minus119883119894Simulations have shown that given 119883119894 the distribution of119883119894 minus 119880119894 is still close to the one given by (2) but with adifferent parameter So we assume that the distribution of119883119894 minus 119880119894 given 119883119894 is equal to (2) but with a parameterfunction of 119883119894 denoted 119888(119883119894)
These simulations have also shown that E[119883119894 minus 119880119894 |119883119894 = 119903]may be approximated as an affine function 119886119894119903 + 119887119894We define 119888119894(sdot) accordingly It leads to
To set 119886119894 and 119887119894 we use the two following propertiesE [E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817 | 10038171003817100381710038171198831198941003817100381710038171003817 = 119903]] = E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817] (B3)
= 12radic120582119890 (B4)
E [E [1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817 | 1003817100381710038171003817119883119894
1003817100381710038171003817 = 119903]] = E [119886119894 10038171003817100381710038171198831198941003817100381710038171003817 + 119887119894] (B5)
Equation (B4) is derived from (2) and (B5) from ourassumption on 119888119894(119903) 119886119894 is then set as a function of 119887119894 E[119883119894]and 120582119890 119887119894 is obtained from our simulations For a givenintensity 119887119894 has been observed almost constant with regardto 119894 The best approximation as a function of the intensity isgiven by 119887119894 = 033radic120582119890 After a fewmanipulations we obtain
First note that by convention we have P(119908119894 = 1) = P(119908119894 =1 | 1199080 = 1) The event 119908119894 = 1 occurs if and only if it exists119895 ge 1 such that1198770 + 119894minus1sum
There is no data used in this paper Nevertheless the code ofour simulator is made available (the link is in the paper)
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] ldquoRequirements for further advancements for e-utra (lte-advanced) 3gpp technical specification ts 36913 release 13rdquo2015 httpwww3gpporg
[2] A S Hamza S S Khalifa H S Hamza and K Elsayed ldquoAsurvey on inter-cell interference coordination techniques inofdma-based cellular networksrdquo IEEE Communications Surveysamp Tutorials vol 15 no 4 pp 1642ndash1670 2013
[3] D Lopez-Perez A Ladanyi A JuttnerH Rivano and J ZhangldquoOptimization method for the joint allocation of modulationschemes coding rates resource blocks and power in self-organizing LTE networksrdquo in Proceedings of the IEEE INFO-COM 2011 pp 111ndash115 April 2011
[4] A Guo andM Haenggi ldquoSpatial stochasticmodels andmetricsfor the structure of base stations in cellular networksrdquo IEEE
Wireless Communications and Mobile Computing 15
Transactions on Wireless Communications vol 12 no 11 pp5800ndash5812 2013
[5] C-H Lee C-Y Shih and Y-S Chen ldquoStochastic geometrybased models for modeling cellular networks in urban areasrdquoWireless Networks vol 19 no 6 pp 1063ndash1072 2013
[6] W Lu and M D Renzo ldquoStochastic geometry modeling ofcellular networks analysis simulation and experimental valida-tionrdquo in Proceedings of the 18th ACM International Conferenceon Modelling Analysis and Simulation of Wireless and MobileSystems (MSWiM rsquo15) pp 179ndash188 New York NY USANovember 2015
[7] Y L Lee T C Chuah J Loo and A Vinel ldquoRecent advancesin radio resource management for heterogeneous LTELTE-Anetworksrdquo IEEE Communications Surveys amp Tutorials vol 16no 4 pp 2142ndash2180 2014
[8] F Mhiri K Sethom and R Bouallegue ldquoA survey on inter-ference management techniques in femtocell self-organizingnetworksrdquo Journal of Network and Computer Applications vol36 no 1 pp 58ndash65 2013
[9] X Chen L Li and X Xiang ldquoAnt colony learning methodfor joint MCS and resource block allocation in LTE Femtocelldownlink for multimedia applications with QoS guaranteesrdquoMultimedia Tools andApplications vol 76 no 3 pp 4035ndash40542017
[10] D Lopez-Perez X Chu A V Vasilakos and H Claussen ldquoOndistributed and coordinated resource allocation for interferencemitigation in self-organizing lte networksrdquo IEEEACMTransac-tions on Networking vol 21 no 4 pp 1145ndash1158 2013
[11] H Zhang H Liu J Cheng and V C M Leung ldquoDownlinkenergy efficiency of power allocation and wireless backhaulbandwidth allocation in heterogeneous small cell networksrdquoIEEE Transactions on Communications no 99 2017
[12] H Zhang W Zheng X Chu et al ldquoJoint subchannel andpower allocation in interference-limited OFDMA femtocellswith heterogeneous QoS guaranteerdquo in Proceedings of theIEEEGlobal Communications Conference (GLOBECOM rsquo12) pp4572ndash4577 December 2012
[13] H Elsawy E Hossain and M Haenggi ldquoStochastic geometryfor modeling analysis and design of multi-tier and cognitivecellular wireless networks a surveyrdquo IEEE CommunicationsSurveys amp Tutorials vol 15 no 3 pp 996ndash1019 2013
[14] H ElSawy A Sultan-Salem M-S Alouini and M Z WinldquoModeling and analysis of cellular networks using stochasticgeometry a tutorialrdquo IEEE Communications Surveys amp Tutori-als vol 19 no 1 pp 167ndash203 2017
[15] R Kwan and C Leung ldquoA survey of scheduling and inter-ference mitigation in lterdquo Journal of Electrical and ComputerEngineeringmdashSpecial Issue on LTELTE-AdvancedCellular Com-munication Networks pp 11ndash110 2010
[16] J G Andrews F Baccelli and R K Ganti ldquoA tractable approachto coverage and rate in cellular networksrdquo IEEE Transactions onCommunications vol 59 no 11 pp 3122ndash3134 2011
[17] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for OFDMA cellu-lar networksrdquo IEEE Transactions on Wireless Communicationsvol 10 no 12 pp 4294ndash4305 2011
[18] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for heterogeneouscellular networksrdquo IEEE Transactions on Communications vol60 no 7 pp 2029ndash2039 2012
[19] H Zhuang and T Ohtsuki ldquoA model based on poisson pointprocess for downlink K tiers fractional frequency reuse hetero-geneous networksrdquo Physical Communication vol 13 pp 3ndash122014 Special Issue on Heterogeneous and Small Cell Networks
[20] A A Gebremariam T Bao D Siracusa T Rasheed F Granelliand L Goratti ldquoDynamic strict fractional frequency reusefor software-defined 5G networksrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo16) pp 1ndash6May 2016
[21] N Lee D Morales-Jimenez A Lozano and R W HeathldquoSpectral efficiency of dynamic coordinated beamforming astochastic geometry approachrdquo IEEE Transactions on WirelessCommunications vol 14 no 1 pp 230ndash241 2015
[22] X Zhang and M Haenggi ldquoA Stochastic geometry analysisof inter-cell interference coordination and intra-cell diversityrdquoIEEE Transactions on Wireless Communications vol 13 no 12pp 6655ndash6669 2014
[23] F Baccelli andAGiovanidis ldquoA stochastic geometry frameworkfor analyzing pairwise-cooperative cellular networksrdquo IEEETransactions on Wireless Communications vol 14 no 2 pp794ndash808 2015
[24] J Yoon and G Hwang ldquoDistance-based inter-cell interferencecoordination in small cell networks stochastic geometry mod-eling and analysisrdquo IEEE Transactions on Wireless Communica-tions 2018
[25] B Błaszczyszyn M Jovanovicy and M K Karray ldquoHow userthroughput depends on the traffic demand in large cellularnetworksrdquo in Proceedings of the 12th International Symposiumon Modeling and Optimization in Mobile Ad Hoc and WirelessNetworks (WiOpt rsquo14) pp 611ndash619 May 2014
[26] H H Yang and T Q S Quek ldquoSIR coverage analysis incellular networks with temporal traffic a stochastic geometryapproachrdquo Corrossion 2018
[27] S M Yu and S-L Kim ldquoDownlink capacity and base stationdensity in cellular networksrdquo in Proceedings of the 11th Interna-tional SymposiumandWorkshops onModeling andOptimizationin Mobile Ad Hoc and Wireless Networks (WiOpt rsquo13) pp 119ndash124 May 2013
[28] YWang M Haenggi and Z Tan ldquoThemeta distribution of thesir for cellular networks with power controlrdquo Corrossion 2017
[29] M S Omar S A Hassan H Pervaiz et al ldquoMulti-objectiveoptimization in 5g hybrid networksrdquo IEEE Internet of ThingsJournal 2018
[30] Y Leo A Busson C Sarraute and E Fleury ldquoCall detail recordsto characterize usages and mobility events of phone usersrdquoComputer Communications vol 95 pp 43ndash53 2016
[31] I L Cherif Spectral and Energy Efficiency in 5G WirelessNetworks [PhD thesis] Univeristy Paris Saclay 2016
[32] S Mukherjee Analytical Modeling of Heterogeneous CellularNetworks Analytical Modeling of Heterogeneous Cellular Net-works Geometry Coverage and Capacity Cambridge Univer-sity Press 2014
[33] ldquoTechnical specification group radio access network base sta-tion (bs) radio transmission and reception 2012 Release 1123gpp ts 36104 v112rdquo 2012
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Wireless Communications and Mobile Computing 7
is then given by E[119879119909]E[119897(119883119894)] Expectation of 119897(119883119894) isobtained from the distribution of 119883119894 given in Section 41
E [119868 (100381710038171003817100381711988301003817100381710038171003817)]
= E [ℎ1] +infinsum119894=1
E [119908119894]E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817)]E [119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)] (11)
Power Control When the transmission power depends onthe distance between the receiver and its attached eNodeB(119879119909(119883119894 minus 119880119894)) the computations are more complex119864[119879119909(119883119894 minus 119880119894) sdot 119897(119883119894)] cannot be approximated byE[119879119909(119883119894minus119880119894)]119864[119897(119883119894)] as the size of the Voronoı cell withnucleus 119883119894 depends on its distance to the origin The jointdistribution of (119883119894 119883119894 minus119880119894) being unknown we proposethe following approximation
The PDF of 119880119894 minus 119883119894 is the same as 1198800 minus 1198830 givenby (2) Its parameter 119888119894(119883119894) depends on 119883119894 119888119894(119883119894) =14radic120582119890(120574119894119883119894 + 119887) with 120574119894 = (12 minus 119887)E[119883119894] and 119887 = 033The motivation and the computation details for this PDF aregiven in appendix (Appendix B) We obtain
E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) sdot 119897 (1003817100381710038171003817119883119894
Often in real systems the transmission power cannot beset arbitrarily and is limited to a set of predetermined valuesThe transmission power function can then be representedas a step function 119879119909(119903) = sum119873119879
119894=1 1199051198941119903isin[120572119894minus1120572119894] where 119873119879
is the number of possible transmission powers 119905119894 the 119894119905ℎtransmission power value and [120572119894minus1 120572119894] the distance intervalbetween a user and its eNodeB at which this transmissionpower is used An example of such setting is given in thenumerical evaluation section In this case (14) becomes
E [119879119909 (1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817) 119897 (1003817100381710038171003817119883119894
1003817100381710038171003817)]= 119873119879sum
119894=1
119905119895 int+infin
0(119890minus120582119890119888119894(119903)1205871205722119895 minus 119890minus120582119890119888119894(119903)1205871205722119895minus1)119891119883119894 (119903) 119889119903
(15)
When the computation is performed for a given distance1198830 the PDF 119891119883119894(sdot) in (12) and (15) must be replaced by1198911198941198830=119903(sdot sdot) (given in Section 41)
422 Computation of E[119908119894] Finally in order to compute(10) we need to expressE[119908119894] First note that E[119908119894] = P(119908119894 =1)Proposition 2 The probability for eNodeB 119894 to interfere withthe typical user is given by
(i) Independent allocation
P (119908119894 = 1) = 1Δ (16)
(ii) Static allocation
P (119908119894 = 1) = 1119894sdot119898119900119889(Δ)=0 (17)
E [119879119909 (1003817100381710038171003817119880119894 minus 1198831198941003817100381710038171003817) 119879119909 (10038171003817100381710038171003817119880119895 minus 119883119895
As for the mean complexity lies in the correlationbetween 119883119894 and 119880119894 minus 119883119894 The term E[119879119909(119880119894 minus 119883119894)2 sdot119897(119883119894)2] is computed with the same method as the firstmoment
Computation of E[119879119909(119880119894minus119883119894)119879119909(119880119895minus119883119895)119897(119883119894)119897(119883119895)]As 119880119894 minus 119883119894 (respectively 119880119895 minus 119883119895) depends on 119883119894(respectively 119883119895) we condition by the distribution of(119883119894 119883119895) given in Section 41 Given 119883119894 and 119883119895 we usethe same PDF as in (B1) assuming that 119880119894 minus 119883119894 and 119880119895 minus119883119895 are independent The considered joint distribution of(119883119894 119883119895 119880119894 minus 119883119894 119880119895 minus 119883119895) becomes
1198910119880119894minus119883119894 (119906 119903) 1198910119880119895minus119883119895 (V 119904) 119891119883119894119883119895 (119903 119904) (21)
When the distance 1198830 is fixed 119891119883119894119883119895(sdot sdot) must bereplaced by the PDF 119891119894119895
1198830=119903(sdot sdot sdot) given in Section 41
Computation of E[119908119894119908119895] It has been shown that the sequence(119908119894)119894gt0 verifies (7) for the three strategies It allows us toexpress E[119908119894119908119895] with 119894 gt 119895 as
E [119908119894119908119895] = P (119908119894minus119895 = 1)P (119908119895 = 1) (22)
8 Wireless Communications and Mobile Computing
Proposition 3 The joint probability for two eNodeB 119894 and 119895(119895 gt 119894) to interfere with the typical user is given by
(i) Independent allocation
E [119908119894119908119895] = 1Δ2 (23)
(ii) Static allocation
E [119908119894119908119895] = 1119894sdot119898119900119889(Δ)=0119895sdot119898119900119889(Δ)=0 (24)
where P(119908 = 1) is given by (18)5 Signal over Interference Ratio (SIR)
In our model there is a strong correlation between theinterfering eNodeB It is generated by the allocation strategiesand cannot be neglected Also a correlation exists betweenthe location of an eNodeB and the size of its Voronoı cellConsequently classical approach based on PPP which uses
Laplacian transform for instance cannot be applied hereand a formal derivation of interference distribution seemsintractable
Nevertheless the different simulations presented in thenext section will show that the PDF of interference can beapproximated by a log-normal distribution The parametersof this distribution mean and variance denoted by 119898119868119889119861
(sdot)and 120590119868119889119861(sdot) are directly derived from the previous analyticalcomputations The classical mapping between log-normaland normal parameters can be applied to derive parametersof the normal distribution when interferences are expressedin decibel In the following a variable is indexed by 119889119861 whenit is expressed in decibel
We get
P (119878119868119877119889119861 le 120573119889119861) = P (10sdot log10 (119875119905 (10038171003817100381710038171198830
1003817100381710038171003817) ℎ0119897 (100381710038171003817100381711988301003817100381710038171003817)) minus 119868119889119861 (10038171003817100381710038171198830
1003817100381710038171003817) le 120573119889119861) (26)
= P (119868119889119861 (100381710038171003817100381711988301003817100381710038171003817) ge minus120573119889119861 + 10
When ℎ0 is not constant the expectation with regard toits distribution must be taken into account in (28) In (29)ℎ0 is assumed to be constant that equals to 1 In these twoequations 119890119903119891(sdot) is the error function6 Numerical Results
We consider an E-UTRA channel with a bandwidth of5MHz with 119877119861119898119886119909 = 15 [33] The path loss function isthe same as [3] It is expressed in dB 119897(119903) = minus1281 minus376 sdot log10(119903) where 119903 is the distance (in km) 119879119909(sdot) isset in such a way to guarantee to each user a minimumreceiving power We set the transmission power function119879119909(sdot) to ensure a signal power greater than or equal tominus724dBm at the reception as specified in [33] For each 50meters (from 50 to 500 meters) we compute the minimumtransmitting power required to reach this threshold (119879119909(119903) sdot119897(119903) ge minus724 dBm for each interval of 50 meters leading to 10possible transmission powers)This step function models thecase where eNodeB has a finite set of predetermined powerThe process intensity modeling eNodeB is equal to 225 perkm2 It corresponds to the intensity of base stations in Paris(httpswwwantennesmobilesfr) Random variables ℎ119894 are
supposed constant equal to 1 This assumption facilitatesinterpretation of the results but any distribution can beconsidered as well It simply adds a factor in terms of variance(cf (20)) We simulate the different strategies through asimulator coded inC available here (httpwwwanthonybus-sonfrindexphppublications) In all simulations and nu-merical results we consider 50 eNodeB The different sumsin the equations (eg (10) and (20)) are then limited to 50For each set of parameters simulations have be run from10000 to several millions times depending on the evaluatedquantities The number of simulationssamples has been setin order to have negligible confidence intervals They areconsequently not shown in the different figures
Mean and Variance of Interference In Figure 2 we plot themean and the standard deviation of interference obtainedfrom simulations and computed from formulas (10) (19) and(20)when the distance 1198830 variesThe theoretical evaluationcloselymatches empirical estimators obtained by simulationsAs expected the highest interference level is observed forthe independent allocation and the lowest level for the staticallocation The static and the 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tions offer equivalent results with a multiplication factor
Figure 2 Interferencemean and standard deviation as a function of the distance between the typical user and its eNodeB (1198830) Simulationresults are given by points and solid lines and theoretical evaluations by the dotted lines
Table 2 Mean and standarddeviation for the119872119872119877119861119898119886119909119877119861m119886119909 allocation and the two heuristics
Assignment strategy Δ = 3 Δ = 6 Δ = 9mean std-dev mean std-dev mean std-dev
119872119872119877119861119898119886119909119877119861119898119886119909 178119890 minus 11 861119890 minus 11 366119890 minus 12 750119890 minus 12 155119890 minus 12 316119890 minus 12Minimize Interference 994119890 minus 12 307119890 minus 11 186119890 minus 12 188119890 minus 12 756119890 minus 13 626119890 minus 13Maximize reuse distance 526119890 minus 11 751119890 minus 10 118119890 minus 11 107119890 minus 10 258119890 minus 12 109119890 minus 11
varying from 11 to 87 for the mean interference Insteadthe independent allocation differs deeply Mean and standarddeviation are multiplied by a factor ranging from 17 to 700for the mean and can reach up to 26 times 104 for the standarddeviation (for 1198830 = 002 km) The use of the thinnedpoint process is thus questionable to approximate realisticassignment strategies This strategy is no more considered inthe following
Extrapolation of Interference Distribution We compareempirical distributions obtained from simulations to knowndistributions for the different strategies and the two heuristicswhen Δ = 3 6 and 9 Three of these empirical distributionsare shown in Figure 3 (in dB) The distribution parametershave been set according to the maximum likelihood Thedistributions that best fit simulations vary according to theallocation strategy and the reuse factorΔ For 4 cases over 12the distribution minimizing error is the normal distributionAs soon as an asymmetry is observed other distributionsare more accurate log-normal Weibull and Inverse GammaWe perform two hypothesis tests T-test and Smirnov-Kolmogorov for all these scenarios The alternative hypoth-esis is systematically ruled out For the 119872119872119877119861119898119886119909119877119861119898119886119909
strategy best fits are given by the normal (Δ = 3 and 6) andWeibull distributions (Δ = 9)
Nevertheless as mentioned in Section 5 when estimatingthe SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy weuse the normal law to model interference Indeed even if thisdistribution is not systematically the most accurate it offers agood approximation as shown inFigure 4 where the empiricaland normalCDF are plottedMoreover simulationswill showthat this assumption does not impact the accuracy of theanalytical model
Heuristics Simulations have shown that allocations resultingfrom the two heuristics correlate the transmission powersthe distance between eNodeB reusing the same resourceand the random variables 119908119894 Whenminimizing interferencecorrelations are caused by resource blocks allocated to usersthat require a low transmission power and that can be reusedat a short distance and inversely The transmission poweris then correlated to 119908119894 and to the distance of eNodeB thatreuse the same resource These complex phenomena impedethe proposal of an analytical model for these two heuristicsFor comparison purposes Table 2 reportsmean and standarddeviation for the two heuristics and the119872119872119877119861119898119886119909119877119861119898119886119909
Figure 3 Interference distribution empirical (histogram) and extrapolation (curve) Among the classical distributions the bestextrapolations are Gamma (m=1003 var=977) normal (m=2823 var=2216) and Weibull (a=2514b=575) for the heuristic MMRBallocation Δ = 6 and static allocation Δ = 9 respectively
Figure 4 CDF of interference for the 119872119872119877119861119898119886119909119877119861119898119886119909 strategy Empirical distributions are represented through the solid lines CDFof the Normal distribution are plotted with dotted lines Their parameters are obtained from the theoretical formulas The 119871+infin errors are338119890minus02 493119890minus02 and 565119890minus02 for Δ = 3 6 and 9 respectively
Figure 5 Mean interference when the number of RB variesThe load is adjusted to keep a constant reuse factor (Δ = 36 or 9) The distance 1198830 is set to 50 meters Dotted linescorrespond to the theoretical evaluation ofmean interference for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation and points to simulations
allocationTheheuristicminimizing interference presents thebest results This difference was expected as the objectivefunction minimizes interference and because the secondheuristic maximizes the reuse distance without taking intoaccount transmission powers 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tion offers interesting results with an intermediary interfer-ence level between the two heuristics Even if interference islower for the heuristic minimizing interference its complex-ity in terms of feedbackmeasures from users and eNodeBis significantly greater than the other strategies Indeed itassumes that the scheduler knows before an assignment theinterference contribution of each eNodeB on each user
Impact of the Number of RB We performed simulations andtheoretical estimations of interference for different values of119877119861119898119886119909 6 15 25 50 75 and 100 corresponding to differentbandwidth of E-UTRA [33] Results are shown in Figure 5 forthe 119872119872119877119861119898119886119909119877119861119898119886119909 allocation and the two heuristicsThe workload is adapted in order to keep the same Δreuse factor It appears that the mean interference is quiteinsensitive to the number of RB except when 119877119861119898119886119909 = 6The impacting factor is thus the reuse factor Δ rather thanthe number of RBAlso the same hierarchy between the threestrategies is observed except for 119877119861119898119886119909 = 6SIR Distribution We evaluate the SIR CDF according to(29) for the 119872119872119877119861119898119886119909119877119861119898119886119909 allocation This CDF iscompared to the one obtained by simulations in Figure 6(a)We clearly observe that the analytical model offers very
tight estimates of the SIR distribution The assumptionabout the normal distribution of interference does notintroduce a noticeable error The SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy and the two heuristics arecompared in Figure 6(b) The three CDF are similar and the119872119872119877119861119898119886119909119877119861119898119886119909 allocation still offers an intermediatedistribution between the two heuristics The 119871+infin normbetween the three CDF is 0192 0266 and 0221 for Δ = 36 and 9 respectivelyModulation and Coding Rate The SIR distribution allowsus to estimate modulations and coding rates that could beoffered to users We consider the thresholds between therequired SIR and coding rate given in [33] Applied to the SIRdistribution it gives the proportion of users that benefits froma certain modulation scheme and coding rate The couplemodulationcoding rate is referred to as transmission rate inthe following This proportion is given in Figure 7(a) for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation when the density of eNodeBincreases from 225 (the default intensity considered in theprevious evaluation) to 20 eNodeB per km2 It appears thatthe network densification even with a factor 10 does notsignificantly improve the transmission rate Only the besttransmission rate (64QAM-CR=45) really benefits from thisdensification with a proportion of users increasing from 35to 116
The spatial reuse has much more impact on the transmis-sion rates as it is shown in Figure 7(b) It clearly increasesthe transmission rates that are shifted from (QPSK CR=12-64QAM CR=23) for Δ = 3 to (16QAM CR=12-64QAMCR=45) for Δ = 9 The mean transmission rate increasesalmost of a factor 2 from 231 to 424 bitsbaud
Finally we compare in Figure 7(c) the transmission ratesfor the three allocation strategies As already observed inthe previous plots the heuristic minimizing interferenceoffers better performance but is comparable to the two otherstrategies For instance for Δ = 6 the mean transmis-sion rates are close 363 323 and 392 bitsbaud for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation the heuristic maximizingthe reuse distance and the one minimizing interferencerespectively
7 Conclusion
In recent LTE standards a server may assign RB to a set ofeNodeB to control resource usage and optimize performancein a global way Even if it exists solutions to address this prob-lem for a given configuration and topology literature lacksmodels that evaluate performance of these RB allocations fora wide range of scenarios and at large scale
To address this problem we propound a spatial stochasticmodel that takes into account RB assignment strategiesrealistic traffic demands and power control We proposeanalytical estimates for the two first moments of interferenceThis computation is based on two approximations the distri-bution of the distance between a point uniformly distributedin a cell that models the user-eNodeB distance and thejoint distribution of the distances eNodeB-origin and user-eNodeB For the latter simulations have shown that these
(a) SIR empirical and theoretical CDF for the119872119872119877119861119898119886119909119877119861119898119886119909allocation
MMRB Delta=3MMRB Delta=6MMRB Delta=9MI Delta=3 MI Delta=6
MI Delta=9 MD Delta=3 MD Delta=6 MD Delta=9
0 10 20 30 40minus10훽 (dB)
00e+00
20eminus01
40eminus01
60eminus01
80eminus01
10e+00
CDF
minus SI
NR
(b) SIR empirical CDF for the three strategies MI maximizeinterference MD maximize distance
Figure 6 (a) We compare the distribution of SIR obtained by simulations (solid lines) and given by (29) (dotted lines) Errors with regardto the 119871+infin norm are 247119890 minus 02 154119890 minus 02 and 158119890 minus 02 for Δ = 3 6 and 9 respectively (b) CDF for the three allocation strategies(119872119872119877119861119898119886119909119877119861119898119886119909 and Heuristics)
(c) 119872119872119877119861119898119886119909119877119861119898119886119909 and the two heuristics Δ = 6 120582119890 = 225
Figure 7 Modulation and coding rate for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation strategy when the density of eNodeB increases (Figure 7(a))for different reuse factor (Figure 7(b)) and comparison with the two heuristics (Figure 7(c))
variables are strongly correlated particularly for the points ofthe process close to the origin This correlation impacts thetransmission power used by the eNodeB and consequentlyinterference and SIR distributions The derivation of the SIRdistribution allows us to express the classical outage but alsothe transmission rates in terms of modulationcoding ratethat provide interesting insights on the throughput offers tothe users Also it appears that eNodeB densification does not
significantly improve the network performance in terms ofthroughput except for a small percentage of users for whichthe transmission rate is increased Spatial reuse expressedthrough the parameter Δ in our study has a much moreimpact on the performance as the transmission rates aresignificantly increased This spatial reuse must be expressedas the ratio between the total number of available resourcesover the number of usersrequests per cell Indeed we have
Wireless Communications and Mobile Computing 13
observed that the system performance is quite insensitive tovariation of these quantities when this ratio stays constant
Beside simulations show that the RB assignment strate-gies developed for the model give results close to the onesgiven by classical optimization problem which minimizesglobal interference or maximizes spatial reuse distances Itempirically shows that our analytical model is able to evaluateperformance of classical optimization approaches but at largescale in terms of number of nodes configurations andtopologies
Appendix
A Construction of the Markov Chain
In this section we introduce the Markov chain used to set thedistribution of 1198770 We consider that the RB used between thetypical user and its eNodeB has index 119903119890119904 (119903119890119904 isin 0 119877119861119898119886119909minus1)We shall prove that this index does not impact calculation(we have chosen 119903119890119904 = 0 in this paper by sack of simplicity)
We set the distribution of the number of used resourceblocks at eNodeB 0 in such a way that it leads to the followingproperty (119895 gt 119894 ge 0)
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (A1)
In order to obtain this property we first introduce somepreliminary notations and results An example of allocationwith the different notations is shown in Figure 1 We define asequence of rv (119877119899)119899ge0 If eNodeB 119894 is the 119899119905ℎ eNodeB using119903119890119904 and if 119864119899119889119899 is the index of the last RB used by this eNodeBthen
119877119899 = 119864119899119889119899 minus 119903119890119904 if 119864119899119889119899 gt 119903119890119904119864119899119889119899 + 119877119861119898119886119909 minus 119903119890119904 otherwise (A2)
If eNodeB 119894 is the 119899119905ℎ interfering eNodeB (using 119903119890119904) theneNodeB 119895 (119895 gt 119894) is the 119899 + 1119905ℎ if and only if
119863119896 ge 119877119861119898119886119909 (A4)
In this case 119877119899+1 is given by
119877119899+1 = 119877119899 +119895sum
119896=119894+1
119863119896 minus 119877119861119898119886119909 (A5)
As (119863119894)119894gt0 are iid the sequence (119877119899)119899ge0 is a homoge-neous Markov chain The transition probabilities for (119897 119898) isin0 1 119877119861119898119886119909 minus 12 are given by 119875119897119898 = P(119877119899+1 = 119898 | 119877119899 = 119897)
The event 119877119899+1 = 119898 given that 119877119899 = 119897 occurs if andonly if it exists 119895 ge 0 such that
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (A10)
We set the distribution of 1198770 with the correspondingstationary distribution Consequently the sequence (119877119899)119899ge0becomes identically distributed where 119877119899 follows the sta-tionary distribution for all 119899 ge 0 Moreover according to(A3) and (A4) the property given in (A1) holds Indeed 119877119899sum119895minus1
119896=119894+1119863119896 and sum119895
119896=119894+1119863119896 have the same distribution as 1198770sum119895minus119894minus1
119896=1119863119896 andsum119895minus119894
119896=1119863119896 respectively In the model evaluation
the stationary distribution is obtained numerically from thetransition probabilities As soon as 1198770 follows the stationarydistribution the index 119903119890119904 can be chosen arbitrarily
B Conditional Distribution of 119880119894minus119883119894Simulations have shown that given 119883119894 the distribution of119883119894 minus 119880119894 is still close to the one given by (2) but with adifferent parameter So we assume that the distribution of119883119894 minus 119880119894 given 119883119894 is equal to (2) but with a parameterfunction of 119883119894 denoted 119888(119883119894)
These simulations have also shown that E[119883119894 minus 119880119894 |119883119894 = 119903]may be approximated as an affine function 119886119894119903 + 119887119894We define 119888119894(sdot) accordingly It leads to
To set 119886119894 and 119887119894 we use the two following propertiesE [E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817 | 10038171003817100381710038171198831198941003817100381710038171003817 = 119903]] = E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817] (B3)
= 12radic120582119890 (B4)
E [E [1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817 | 1003817100381710038171003817119883119894
1003817100381710038171003817 = 119903]] = E [119886119894 10038171003817100381710038171198831198941003817100381710038171003817 + 119887119894] (B5)
Equation (B4) is derived from (2) and (B5) from ourassumption on 119888119894(119903) 119886119894 is then set as a function of 119887119894 E[119883119894]and 120582119890 119887119894 is obtained from our simulations For a givenintensity 119887119894 has been observed almost constant with regardto 119894 The best approximation as a function of the intensity isgiven by 119887119894 = 033radic120582119890 After a fewmanipulations we obtain
First note that by convention we have P(119908119894 = 1) = P(119908119894 =1 | 1199080 = 1) The event 119908119894 = 1 occurs if and only if it exists119895 ge 1 such that1198770 + 119894minus1sum
There is no data used in this paper Nevertheless the code ofour simulator is made available (the link is in the paper)
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] ldquoRequirements for further advancements for e-utra (lte-advanced) 3gpp technical specification ts 36913 release 13rdquo2015 httpwww3gpporg
[2] A S Hamza S S Khalifa H S Hamza and K Elsayed ldquoAsurvey on inter-cell interference coordination techniques inofdma-based cellular networksrdquo IEEE Communications Surveysamp Tutorials vol 15 no 4 pp 1642ndash1670 2013
[3] D Lopez-Perez A Ladanyi A JuttnerH Rivano and J ZhangldquoOptimization method for the joint allocation of modulationschemes coding rates resource blocks and power in self-organizing LTE networksrdquo in Proceedings of the IEEE INFO-COM 2011 pp 111ndash115 April 2011
[4] A Guo andM Haenggi ldquoSpatial stochasticmodels andmetricsfor the structure of base stations in cellular networksrdquo IEEE
Wireless Communications and Mobile Computing 15
Transactions on Wireless Communications vol 12 no 11 pp5800ndash5812 2013
[5] C-H Lee C-Y Shih and Y-S Chen ldquoStochastic geometrybased models for modeling cellular networks in urban areasrdquoWireless Networks vol 19 no 6 pp 1063ndash1072 2013
[6] W Lu and M D Renzo ldquoStochastic geometry modeling ofcellular networks analysis simulation and experimental valida-tionrdquo in Proceedings of the 18th ACM International Conferenceon Modelling Analysis and Simulation of Wireless and MobileSystems (MSWiM rsquo15) pp 179ndash188 New York NY USANovember 2015
[7] Y L Lee T C Chuah J Loo and A Vinel ldquoRecent advancesin radio resource management for heterogeneous LTELTE-Anetworksrdquo IEEE Communications Surveys amp Tutorials vol 16no 4 pp 2142ndash2180 2014
[8] F Mhiri K Sethom and R Bouallegue ldquoA survey on inter-ference management techniques in femtocell self-organizingnetworksrdquo Journal of Network and Computer Applications vol36 no 1 pp 58ndash65 2013
[9] X Chen L Li and X Xiang ldquoAnt colony learning methodfor joint MCS and resource block allocation in LTE Femtocelldownlink for multimedia applications with QoS guaranteesrdquoMultimedia Tools andApplications vol 76 no 3 pp 4035ndash40542017
[10] D Lopez-Perez X Chu A V Vasilakos and H Claussen ldquoOndistributed and coordinated resource allocation for interferencemitigation in self-organizing lte networksrdquo IEEEACMTransac-tions on Networking vol 21 no 4 pp 1145ndash1158 2013
[11] H Zhang H Liu J Cheng and V C M Leung ldquoDownlinkenergy efficiency of power allocation and wireless backhaulbandwidth allocation in heterogeneous small cell networksrdquoIEEE Transactions on Communications no 99 2017
[12] H Zhang W Zheng X Chu et al ldquoJoint subchannel andpower allocation in interference-limited OFDMA femtocellswith heterogeneous QoS guaranteerdquo in Proceedings of theIEEEGlobal Communications Conference (GLOBECOM rsquo12) pp4572ndash4577 December 2012
[13] H Elsawy E Hossain and M Haenggi ldquoStochastic geometryfor modeling analysis and design of multi-tier and cognitivecellular wireless networks a surveyrdquo IEEE CommunicationsSurveys amp Tutorials vol 15 no 3 pp 996ndash1019 2013
[14] H ElSawy A Sultan-Salem M-S Alouini and M Z WinldquoModeling and analysis of cellular networks using stochasticgeometry a tutorialrdquo IEEE Communications Surveys amp Tutori-als vol 19 no 1 pp 167ndash203 2017
[15] R Kwan and C Leung ldquoA survey of scheduling and inter-ference mitigation in lterdquo Journal of Electrical and ComputerEngineeringmdashSpecial Issue on LTELTE-AdvancedCellular Com-munication Networks pp 11ndash110 2010
[16] J G Andrews F Baccelli and R K Ganti ldquoA tractable approachto coverage and rate in cellular networksrdquo IEEE Transactions onCommunications vol 59 no 11 pp 3122ndash3134 2011
[17] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for OFDMA cellu-lar networksrdquo IEEE Transactions on Wireless Communicationsvol 10 no 12 pp 4294ndash4305 2011
[18] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for heterogeneouscellular networksrdquo IEEE Transactions on Communications vol60 no 7 pp 2029ndash2039 2012
[19] H Zhuang and T Ohtsuki ldquoA model based on poisson pointprocess for downlink K tiers fractional frequency reuse hetero-geneous networksrdquo Physical Communication vol 13 pp 3ndash122014 Special Issue on Heterogeneous and Small Cell Networks
[20] A A Gebremariam T Bao D Siracusa T Rasheed F Granelliand L Goratti ldquoDynamic strict fractional frequency reusefor software-defined 5G networksrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo16) pp 1ndash6May 2016
[21] N Lee D Morales-Jimenez A Lozano and R W HeathldquoSpectral efficiency of dynamic coordinated beamforming astochastic geometry approachrdquo IEEE Transactions on WirelessCommunications vol 14 no 1 pp 230ndash241 2015
[22] X Zhang and M Haenggi ldquoA Stochastic geometry analysisof inter-cell interference coordination and intra-cell diversityrdquoIEEE Transactions on Wireless Communications vol 13 no 12pp 6655ndash6669 2014
[23] F Baccelli andAGiovanidis ldquoA stochastic geometry frameworkfor analyzing pairwise-cooperative cellular networksrdquo IEEETransactions on Wireless Communications vol 14 no 2 pp794ndash808 2015
[24] J Yoon and G Hwang ldquoDistance-based inter-cell interferencecoordination in small cell networks stochastic geometry mod-eling and analysisrdquo IEEE Transactions on Wireless Communica-tions 2018
[25] B Błaszczyszyn M Jovanovicy and M K Karray ldquoHow userthroughput depends on the traffic demand in large cellularnetworksrdquo in Proceedings of the 12th International Symposiumon Modeling and Optimization in Mobile Ad Hoc and WirelessNetworks (WiOpt rsquo14) pp 611ndash619 May 2014
[26] H H Yang and T Q S Quek ldquoSIR coverage analysis incellular networks with temporal traffic a stochastic geometryapproachrdquo Corrossion 2018
[27] S M Yu and S-L Kim ldquoDownlink capacity and base stationdensity in cellular networksrdquo in Proceedings of the 11th Interna-tional SymposiumandWorkshops onModeling andOptimizationin Mobile Ad Hoc and Wireless Networks (WiOpt rsquo13) pp 119ndash124 May 2013
[28] YWang M Haenggi and Z Tan ldquoThemeta distribution of thesir for cellular networks with power controlrdquo Corrossion 2017
[29] M S Omar S A Hassan H Pervaiz et al ldquoMulti-objectiveoptimization in 5g hybrid networksrdquo IEEE Internet of ThingsJournal 2018
[30] Y Leo A Busson C Sarraute and E Fleury ldquoCall detail recordsto characterize usages and mobility events of phone usersrdquoComputer Communications vol 95 pp 43ndash53 2016
[31] I L Cherif Spectral and Energy Efficiency in 5G WirelessNetworks [PhD thesis] Univeristy Paris Saclay 2016
[32] S Mukherjee Analytical Modeling of Heterogeneous CellularNetworks Analytical Modeling of Heterogeneous Cellular Net-works Geometry Coverage and Capacity Cambridge Univer-sity Press 2014
[33] ldquoTechnical specification group radio access network base sta-tion (bs) radio transmission and reception 2012 Release 1123gpp ts 36104 v112rdquo 2012
where P(119908 = 1) is given by (18)5 Signal over Interference Ratio (SIR)
In our model there is a strong correlation between theinterfering eNodeB It is generated by the allocation strategiesand cannot be neglected Also a correlation exists betweenthe location of an eNodeB and the size of its Voronoı cellConsequently classical approach based on PPP which uses
Laplacian transform for instance cannot be applied hereand a formal derivation of interference distribution seemsintractable
Nevertheless the different simulations presented in thenext section will show that the PDF of interference can beapproximated by a log-normal distribution The parametersof this distribution mean and variance denoted by 119898119868119889119861
(sdot)and 120590119868119889119861(sdot) are directly derived from the previous analyticalcomputations The classical mapping between log-normaland normal parameters can be applied to derive parametersof the normal distribution when interferences are expressedin decibel In the following a variable is indexed by 119889119861 whenit is expressed in decibel
We get
P (119878119868119877119889119861 le 120573119889119861) = P (10sdot log10 (119875119905 (10038171003817100381710038171198830
1003817100381710038171003817) ℎ0119897 (100381710038171003817100381711988301003817100381710038171003817)) minus 119868119889119861 (10038171003817100381710038171198830
1003817100381710038171003817) le 120573119889119861) (26)
= P (119868119889119861 (100381710038171003817100381711988301003817100381710038171003817) ge minus120573119889119861 + 10
When ℎ0 is not constant the expectation with regard toits distribution must be taken into account in (28) In (29)ℎ0 is assumed to be constant that equals to 1 In these twoequations 119890119903119891(sdot) is the error function6 Numerical Results
We consider an E-UTRA channel with a bandwidth of5MHz with 119877119861119898119886119909 = 15 [33] The path loss function isthe same as [3] It is expressed in dB 119897(119903) = minus1281 minus376 sdot log10(119903) where 119903 is the distance (in km) 119879119909(sdot) isset in such a way to guarantee to each user a minimumreceiving power We set the transmission power function119879119909(sdot) to ensure a signal power greater than or equal tominus724dBm at the reception as specified in [33] For each 50meters (from 50 to 500 meters) we compute the minimumtransmitting power required to reach this threshold (119879119909(119903) sdot119897(119903) ge minus724 dBm for each interval of 50 meters leading to 10possible transmission powers)This step function models thecase where eNodeB has a finite set of predetermined powerThe process intensity modeling eNodeB is equal to 225 perkm2 It corresponds to the intensity of base stations in Paris(httpswwwantennesmobilesfr) Random variables ℎ119894 are
supposed constant equal to 1 This assumption facilitatesinterpretation of the results but any distribution can beconsidered as well It simply adds a factor in terms of variance(cf (20)) We simulate the different strategies through asimulator coded inC available here (httpwwwanthonybus-sonfrindexphppublications) In all simulations and nu-merical results we consider 50 eNodeB The different sumsin the equations (eg (10) and (20)) are then limited to 50For each set of parameters simulations have be run from10000 to several millions times depending on the evaluatedquantities The number of simulationssamples has been setin order to have negligible confidence intervals They areconsequently not shown in the different figures
Mean and Variance of Interference In Figure 2 we plot themean and the standard deviation of interference obtainedfrom simulations and computed from formulas (10) (19) and(20)when the distance 1198830 variesThe theoretical evaluationcloselymatches empirical estimators obtained by simulationsAs expected the highest interference level is observed forthe independent allocation and the lowest level for the staticallocation The static and the 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tions offer equivalent results with a multiplication factor
Figure 2 Interferencemean and standard deviation as a function of the distance between the typical user and its eNodeB (1198830) Simulationresults are given by points and solid lines and theoretical evaluations by the dotted lines
Table 2 Mean and standarddeviation for the119872119872119877119861119898119886119909119877119861m119886119909 allocation and the two heuristics
Assignment strategy Δ = 3 Δ = 6 Δ = 9mean std-dev mean std-dev mean std-dev
119872119872119877119861119898119886119909119877119861119898119886119909 178119890 minus 11 861119890 minus 11 366119890 minus 12 750119890 minus 12 155119890 minus 12 316119890 minus 12Minimize Interference 994119890 minus 12 307119890 minus 11 186119890 minus 12 188119890 minus 12 756119890 minus 13 626119890 minus 13Maximize reuse distance 526119890 minus 11 751119890 minus 10 118119890 minus 11 107119890 minus 10 258119890 minus 12 109119890 minus 11
varying from 11 to 87 for the mean interference Insteadthe independent allocation differs deeply Mean and standarddeviation are multiplied by a factor ranging from 17 to 700for the mean and can reach up to 26 times 104 for the standarddeviation (for 1198830 = 002 km) The use of the thinnedpoint process is thus questionable to approximate realisticassignment strategies This strategy is no more considered inthe following
Extrapolation of Interference Distribution We compareempirical distributions obtained from simulations to knowndistributions for the different strategies and the two heuristicswhen Δ = 3 6 and 9 Three of these empirical distributionsare shown in Figure 3 (in dB) The distribution parametershave been set according to the maximum likelihood Thedistributions that best fit simulations vary according to theallocation strategy and the reuse factorΔ For 4 cases over 12the distribution minimizing error is the normal distributionAs soon as an asymmetry is observed other distributionsare more accurate log-normal Weibull and Inverse GammaWe perform two hypothesis tests T-test and Smirnov-Kolmogorov for all these scenarios The alternative hypoth-esis is systematically ruled out For the 119872119872119877119861119898119886119909119877119861119898119886119909
strategy best fits are given by the normal (Δ = 3 and 6) andWeibull distributions (Δ = 9)
Nevertheless as mentioned in Section 5 when estimatingthe SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy weuse the normal law to model interference Indeed even if thisdistribution is not systematically the most accurate it offers agood approximation as shown inFigure 4 where the empiricaland normalCDF are plottedMoreover simulationswill showthat this assumption does not impact the accuracy of theanalytical model
Heuristics Simulations have shown that allocations resultingfrom the two heuristics correlate the transmission powersthe distance between eNodeB reusing the same resourceand the random variables 119908119894 Whenminimizing interferencecorrelations are caused by resource blocks allocated to usersthat require a low transmission power and that can be reusedat a short distance and inversely The transmission poweris then correlated to 119908119894 and to the distance of eNodeB thatreuse the same resource These complex phenomena impedethe proposal of an analytical model for these two heuristicsFor comparison purposes Table 2 reportsmean and standarddeviation for the two heuristics and the119872119872119877119861119898119886119909119877119861119898119886119909
Figure 3 Interference distribution empirical (histogram) and extrapolation (curve) Among the classical distributions the bestextrapolations are Gamma (m=1003 var=977) normal (m=2823 var=2216) and Weibull (a=2514b=575) for the heuristic MMRBallocation Δ = 6 and static allocation Δ = 9 respectively
Figure 4 CDF of interference for the 119872119872119877119861119898119886119909119877119861119898119886119909 strategy Empirical distributions are represented through the solid lines CDFof the Normal distribution are plotted with dotted lines Their parameters are obtained from the theoretical formulas The 119871+infin errors are338119890minus02 493119890minus02 and 565119890minus02 for Δ = 3 6 and 9 respectively
Figure 5 Mean interference when the number of RB variesThe load is adjusted to keep a constant reuse factor (Δ = 36 or 9) The distance 1198830 is set to 50 meters Dotted linescorrespond to the theoretical evaluation ofmean interference for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation and points to simulations
allocationTheheuristicminimizing interference presents thebest results This difference was expected as the objectivefunction minimizes interference and because the secondheuristic maximizes the reuse distance without taking intoaccount transmission powers 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tion offers interesting results with an intermediary interfer-ence level between the two heuristics Even if interference islower for the heuristic minimizing interference its complex-ity in terms of feedbackmeasures from users and eNodeBis significantly greater than the other strategies Indeed itassumes that the scheduler knows before an assignment theinterference contribution of each eNodeB on each user
Impact of the Number of RB We performed simulations andtheoretical estimations of interference for different values of119877119861119898119886119909 6 15 25 50 75 and 100 corresponding to differentbandwidth of E-UTRA [33] Results are shown in Figure 5 forthe 119872119872119877119861119898119886119909119877119861119898119886119909 allocation and the two heuristicsThe workload is adapted in order to keep the same Δreuse factor It appears that the mean interference is quiteinsensitive to the number of RB except when 119877119861119898119886119909 = 6The impacting factor is thus the reuse factor Δ rather thanthe number of RBAlso the same hierarchy between the threestrategies is observed except for 119877119861119898119886119909 = 6SIR Distribution We evaluate the SIR CDF according to(29) for the 119872119872119877119861119898119886119909119877119861119898119886119909 allocation This CDF iscompared to the one obtained by simulations in Figure 6(a)We clearly observe that the analytical model offers very
tight estimates of the SIR distribution The assumptionabout the normal distribution of interference does notintroduce a noticeable error The SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy and the two heuristics arecompared in Figure 6(b) The three CDF are similar and the119872119872119877119861119898119886119909119877119861119898119886119909 allocation still offers an intermediatedistribution between the two heuristics The 119871+infin normbetween the three CDF is 0192 0266 and 0221 for Δ = 36 and 9 respectivelyModulation and Coding Rate The SIR distribution allowsus to estimate modulations and coding rates that could beoffered to users We consider the thresholds between therequired SIR and coding rate given in [33] Applied to the SIRdistribution it gives the proportion of users that benefits froma certain modulation scheme and coding rate The couplemodulationcoding rate is referred to as transmission rate inthe following This proportion is given in Figure 7(a) for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation when the density of eNodeBincreases from 225 (the default intensity considered in theprevious evaluation) to 20 eNodeB per km2 It appears thatthe network densification even with a factor 10 does notsignificantly improve the transmission rate Only the besttransmission rate (64QAM-CR=45) really benefits from thisdensification with a proportion of users increasing from 35to 116
The spatial reuse has much more impact on the transmis-sion rates as it is shown in Figure 7(b) It clearly increasesthe transmission rates that are shifted from (QPSK CR=12-64QAM CR=23) for Δ = 3 to (16QAM CR=12-64QAMCR=45) for Δ = 9 The mean transmission rate increasesalmost of a factor 2 from 231 to 424 bitsbaud
Finally we compare in Figure 7(c) the transmission ratesfor the three allocation strategies As already observed inthe previous plots the heuristic minimizing interferenceoffers better performance but is comparable to the two otherstrategies For instance for Δ = 6 the mean transmis-sion rates are close 363 323 and 392 bitsbaud for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation the heuristic maximizingthe reuse distance and the one minimizing interferencerespectively
7 Conclusion
In recent LTE standards a server may assign RB to a set ofeNodeB to control resource usage and optimize performancein a global way Even if it exists solutions to address this prob-lem for a given configuration and topology literature lacksmodels that evaluate performance of these RB allocations fora wide range of scenarios and at large scale
To address this problem we propound a spatial stochasticmodel that takes into account RB assignment strategiesrealistic traffic demands and power control We proposeanalytical estimates for the two first moments of interferenceThis computation is based on two approximations the distri-bution of the distance between a point uniformly distributedin a cell that models the user-eNodeB distance and thejoint distribution of the distances eNodeB-origin and user-eNodeB For the latter simulations have shown that these
(a) SIR empirical and theoretical CDF for the119872119872119877119861119898119886119909119877119861119898119886119909allocation
MMRB Delta=3MMRB Delta=6MMRB Delta=9MI Delta=3 MI Delta=6
MI Delta=9 MD Delta=3 MD Delta=6 MD Delta=9
0 10 20 30 40minus10훽 (dB)
00e+00
20eminus01
40eminus01
60eminus01
80eminus01
10e+00
CDF
minus SI
NR
(b) SIR empirical CDF for the three strategies MI maximizeinterference MD maximize distance
Figure 6 (a) We compare the distribution of SIR obtained by simulations (solid lines) and given by (29) (dotted lines) Errors with regardto the 119871+infin norm are 247119890 minus 02 154119890 minus 02 and 158119890 minus 02 for Δ = 3 6 and 9 respectively (b) CDF for the three allocation strategies(119872119872119877119861119898119886119909119877119861119898119886119909 and Heuristics)
(c) 119872119872119877119861119898119886119909119877119861119898119886119909 and the two heuristics Δ = 6 120582119890 = 225
Figure 7 Modulation and coding rate for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation strategy when the density of eNodeB increases (Figure 7(a))for different reuse factor (Figure 7(b)) and comparison with the two heuristics (Figure 7(c))
variables are strongly correlated particularly for the points ofthe process close to the origin This correlation impacts thetransmission power used by the eNodeB and consequentlyinterference and SIR distributions The derivation of the SIRdistribution allows us to express the classical outage but alsothe transmission rates in terms of modulationcoding ratethat provide interesting insights on the throughput offers tothe users Also it appears that eNodeB densification does not
significantly improve the network performance in terms ofthroughput except for a small percentage of users for whichthe transmission rate is increased Spatial reuse expressedthrough the parameter Δ in our study has a much moreimpact on the performance as the transmission rates aresignificantly increased This spatial reuse must be expressedas the ratio between the total number of available resourcesover the number of usersrequests per cell Indeed we have
Wireless Communications and Mobile Computing 13
observed that the system performance is quite insensitive tovariation of these quantities when this ratio stays constant
Beside simulations show that the RB assignment strate-gies developed for the model give results close to the onesgiven by classical optimization problem which minimizesglobal interference or maximizes spatial reuse distances Itempirically shows that our analytical model is able to evaluateperformance of classical optimization approaches but at largescale in terms of number of nodes configurations andtopologies
Appendix
A Construction of the Markov Chain
In this section we introduce the Markov chain used to set thedistribution of 1198770 We consider that the RB used between thetypical user and its eNodeB has index 119903119890119904 (119903119890119904 isin 0 119877119861119898119886119909minus1)We shall prove that this index does not impact calculation(we have chosen 119903119890119904 = 0 in this paper by sack of simplicity)
We set the distribution of the number of used resourceblocks at eNodeB 0 in such a way that it leads to the followingproperty (119895 gt 119894 ge 0)
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (A1)
In order to obtain this property we first introduce somepreliminary notations and results An example of allocationwith the different notations is shown in Figure 1 We define asequence of rv (119877119899)119899ge0 If eNodeB 119894 is the 119899119905ℎ eNodeB using119903119890119904 and if 119864119899119889119899 is the index of the last RB used by this eNodeBthen
119877119899 = 119864119899119889119899 minus 119903119890119904 if 119864119899119889119899 gt 119903119890119904119864119899119889119899 + 119877119861119898119886119909 minus 119903119890119904 otherwise (A2)
If eNodeB 119894 is the 119899119905ℎ interfering eNodeB (using 119903119890119904) theneNodeB 119895 (119895 gt 119894) is the 119899 + 1119905ℎ if and only if
119863119896 ge 119877119861119898119886119909 (A4)
In this case 119877119899+1 is given by
119877119899+1 = 119877119899 +119895sum
119896=119894+1
119863119896 minus 119877119861119898119886119909 (A5)
As (119863119894)119894gt0 are iid the sequence (119877119899)119899ge0 is a homoge-neous Markov chain The transition probabilities for (119897 119898) isin0 1 119877119861119898119886119909 minus 12 are given by 119875119897119898 = P(119877119899+1 = 119898 | 119877119899 = 119897)
The event 119877119899+1 = 119898 given that 119877119899 = 119897 occurs if andonly if it exists 119895 ge 0 such that
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (A10)
We set the distribution of 1198770 with the correspondingstationary distribution Consequently the sequence (119877119899)119899ge0becomes identically distributed where 119877119899 follows the sta-tionary distribution for all 119899 ge 0 Moreover according to(A3) and (A4) the property given in (A1) holds Indeed 119877119899sum119895minus1
119896=119894+1119863119896 and sum119895
119896=119894+1119863119896 have the same distribution as 1198770sum119895minus119894minus1
119896=1119863119896 andsum119895minus119894
119896=1119863119896 respectively In the model evaluation
the stationary distribution is obtained numerically from thetransition probabilities As soon as 1198770 follows the stationarydistribution the index 119903119890119904 can be chosen arbitrarily
B Conditional Distribution of 119880119894minus119883119894Simulations have shown that given 119883119894 the distribution of119883119894 minus 119880119894 is still close to the one given by (2) but with adifferent parameter So we assume that the distribution of119883119894 minus 119880119894 given 119883119894 is equal to (2) but with a parameterfunction of 119883119894 denoted 119888(119883119894)
These simulations have also shown that E[119883119894 minus 119880119894 |119883119894 = 119903]may be approximated as an affine function 119886119894119903 + 119887119894We define 119888119894(sdot) accordingly It leads to
To set 119886119894 and 119887119894 we use the two following propertiesE [E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817 | 10038171003817100381710038171198831198941003817100381710038171003817 = 119903]] = E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817] (B3)
= 12radic120582119890 (B4)
E [E [1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817 | 1003817100381710038171003817119883119894
1003817100381710038171003817 = 119903]] = E [119886119894 10038171003817100381710038171198831198941003817100381710038171003817 + 119887119894] (B5)
Equation (B4) is derived from (2) and (B5) from ourassumption on 119888119894(119903) 119886119894 is then set as a function of 119887119894 E[119883119894]and 120582119890 119887119894 is obtained from our simulations For a givenintensity 119887119894 has been observed almost constant with regardto 119894 The best approximation as a function of the intensity isgiven by 119887119894 = 033radic120582119890 After a fewmanipulations we obtain
First note that by convention we have P(119908119894 = 1) = P(119908119894 =1 | 1199080 = 1) The event 119908119894 = 1 occurs if and only if it exists119895 ge 1 such that1198770 + 119894minus1sum
There is no data used in this paper Nevertheless the code ofour simulator is made available (the link is in the paper)
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] ldquoRequirements for further advancements for e-utra (lte-advanced) 3gpp technical specification ts 36913 release 13rdquo2015 httpwww3gpporg
[2] A S Hamza S S Khalifa H S Hamza and K Elsayed ldquoAsurvey on inter-cell interference coordination techniques inofdma-based cellular networksrdquo IEEE Communications Surveysamp Tutorials vol 15 no 4 pp 1642ndash1670 2013
[3] D Lopez-Perez A Ladanyi A JuttnerH Rivano and J ZhangldquoOptimization method for the joint allocation of modulationschemes coding rates resource blocks and power in self-organizing LTE networksrdquo in Proceedings of the IEEE INFO-COM 2011 pp 111ndash115 April 2011
[4] A Guo andM Haenggi ldquoSpatial stochasticmodels andmetricsfor the structure of base stations in cellular networksrdquo IEEE
Wireless Communications and Mobile Computing 15
Transactions on Wireless Communications vol 12 no 11 pp5800ndash5812 2013
[5] C-H Lee C-Y Shih and Y-S Chen ldquoStochastic geometrybased models for modeling cellular networks in urban areasrdquoWireless Networks vol 19 no 6 pp 1063ndash1072 2013
[6] W Lu and M D Renzo ldquoStochastic geometry modeling ofcellular networks analysis simulation and experimental valida-tionrdquo in Proceedings of the 18th ACM International Conferenceon Modelling Analysis and Simulation of Wireless and MobileSystems (MSWiM rsquo15) pp 179ndash188 New York NY USANovember 2015
[7] Y L Lee T C Chuah J Loo and A Vinel ldquoRecent advancesin radio resource management for heterogeneous LTELTE-Anetworksrdquo IEEE Communications Surveys amp Tutorials vol 16no 4 pp 2142ndash2180 2014
[8] F Mhiri K Sethom and R Bouallegue ldquoA survey on inter-ference management techniques in femtocell self-organizingnetworksrdquo Journal of Network and Computer Applications vol36 no 1 pp 58ndash65 2013
[9] X Chen L Li and X Xiang ldquoAnt colony learning methodfor joint MCS and resource block allocation in LTE Femtocelldownlink for multimedia applications with QoS guaranteesrdquoMultimedia Tools andApplications vol 76 no 3 pp 4035ndash40542017
[10] D Lopez-Perez X Chu A V Vasilakos and H Claussen ldquoOndistributed and coordinated resource allocation for interferencemitigation in self-organizing lte networksrdquo IEEEACMTransac-tions on Networking vol 21 no 4 pp 1145ndash1158 2013
[11] H Zhang H Liu J Cheng and V C M Leung ldquoDownlinkenergy efficiency of power allocation and wireless backhaulbandwidth allocation in heterogeneous small cell networksrdquoIEEE Transactions on Communications no 99 2017
[12] H Zhang W Zheng X Chu et al ldquoJoint subchannel andpower allocation in interference-limited OFDMA femtocellswith heterogeneous QoS guaranteerdquo in Proceedings of theIEEEGlobal Communications Conference (GLOBECOM rsquo12) pp4572ndash4577 December 2012
[13] H Elsawy E Hossain and M Haenggi ldquoStochastic geometryfor modeling analysis and design of multi-tier and cognitivecellular wireless networks a surveyrdquo IEEE CommunicationsSurveys amp Tutorials vol 15 no 3 pp 996ndash1019 2013
[14] H ElSawy A Sultan-Salem M-S Alouini and M Z WinldquoModeling and analysis of cellular networks using stochasticgeometry a tutorialrdquo IEEE Communications Surveys amp Tutori-als vol 19 no 1 pp 167ndash203 2017
[15] R Kwan and C Leung ldquoA survey of scheduling and inter-ference mitigation in lterdquo Journal of Electrical and ComputerEngineeringmdashSpecial Issue on LTELTE-AdvancedCellular Com-munication Networks pp 11ndash110 2010
[16] J G Andrews F Baccelli and R K Ganti ldquoA tractable approachto coverage and rate in cellular networksrdquo IEEE Transactions onCommunications vol 59 no 11 pp 3122ndash3134 2011
[17] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for OFDMA cellu-lar networksrdquo IEEE Transactions on Wireless Communicationsvol 10 no 12 pp 4294ndash4305 2011
[18] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for heterogeneouscellular networksrdquo IEEE Transactions on Communications vol60 no 7 pp 2029ndash2039 2012
[19] H Zhuang and T Ohtsuki ldquoA model based on poisson pointprocess for downlink K tiers fractional frequency reuse hetero-geneous networksrdquo Physical Communication vol 13 pp 3ndash122014 Special Issue on Heterogeneous and Small Cell Networks
[20] A A Gebremariam T Bao D Siracusa T Rasheed F Granelliand L Goratti ldquoDynamic strict fractional frequency reusefor software-defined 5G networksrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo16) pp 1ndash6May 2016
[21] N Lee D Morales-Jimenez A Lozano and R W HeathldquoSpectral efficiency of dynamic coordinated beamforming astochastic geometry approachrdquo IEEE Transactions on WirelessCommunications vol 14 no 1 pp 230ndash241 2015
[22] X Zhang and M Haenggi ldquoA Stochastic geometry analysisof inter-cell interference coordination and intra-cell diversityrdquoIEEE Transactions on Wireless Communications vol 13 no 12pp 6655ndash6669 2014
[23] F Baccelli andAGiovanidis ldquoA stochastic geometry frameworkfor analyzing pairwise-cooperative cellular networksrdquo IEEETransactions on Wireless Communications vol 14 no 2 pp794ndash808 2015
[24] J Yoon and G Hwang ldquoDistance-based inter-cell interferencecoordination in small cell networks stochastic geometry mod-eling and analysisrdquo IEEE Transactions on Wireless Communica-tions 2018
[25] B Błaszczyszyn M Jovanovicy and M K Karray ldquoHow userthroughput depends on the traffic demand in large cellularnetworksrdquo in Proceedings of the 12th International Symposiumon Modeling and Optimization in Mobile Ad Hoc and WirelessNetworks (WiOpt rsquo14) pp 611ndash619 May 2014
[26] H H Yang and T Q S Quek ldquoSIR coverage analysis incellular networks with temporal traffic a stochastic geometryapproachrdquo Corrossion 2018
[27] S M Yu and S-L Kim ldquoDownlink capacity and base stationdensity in cellular networksrdquo in Proceedings of the 11th Interna-tional SymposiumandWorkshops onModeling andOptimizationin Mobile Ad Hoc and Wireless Networks (WiOpt rsquo13) pp 119ndash124 May 2013
[28] YWang M Haenggi and Z Tan ldquoThemeta distribution of thesir for cellular networks with power controlrdquo Corrossion 2017
[29] M S Omar S A Hassan H Pervaiz et al ldquoMulti-objectiveoptimization in 5g hybrid networksrdquo IEEE Internet of ThingsJournal 2018
[30] Y Leo A Busson C Sarraute and E Fleury ldquoCall detail recordsto characterize usages and mobility events of phone usersrdquoComputer Communications vol 95 pp 43ndash53 2016
[31] I L Cherif Spectral and Energy Efficiency in 5G WirelessNetworks [PhD thesis] Univeristy Paris Saclay 2016
[32] S Mukherjee Analytical Modeling of Heterogeneous CellularNetworks Analytical Modeling of Heterogeneous Cellular Net-works Geometry Coverage and Capacity Cambridge Univer-sity Press 2014
[33] ldquoTechnical specification group radio access network base sta-tion (bs) radio transmission and reception 2012 Release 1123gpp ts 36104 v112rdquo 2012
Figure 2 Interferencemean and standard deviation as a function of the distance between the typical user and its eNodeB (1198830) Simulationresults are given by points and solid lines and theoretical evaluations by the dotted lines
Table 2 Mean and standarddeviation for the119872119872119877119861119898119886119909119877119861m119886119909 allocation and the two heuristics
Assignment strategy Δ = 3 Δ = 6 Δ = 9mean std-dev mean std-dev mean std-dev
119872119872119877119861119898119886119909119877119861119898119886119909 178119890 minus 11 861119890 minus 11 366119890 minus 12 750119890 minus 12 155119890 minus 12 316119890 minus 12Minimize Interference 994119890 minus 12 307119890 minus 11 186119890 minus 12 188119890 minus 12 756119890 minus 13 626119890 minus 13Maximize reuse distance 526119890 minus 11 751119890 minus 10 118119890 minus 11 107119890 minus 10 258119890 minus 12 109119890 minus 11
varying from 11 to 87 for the mean interference Insteadthe independent allocation differs deeply Mean and standarddeviation are multiplied by a factor ranging from 17 to 700for the mean and can reach up to 26 times 104 for the standarddeviation (for 1198830 = 002 km) The use of the thinnedpoint process is thus questionable to approximate realisticassignment strategies This strategy is no more considered inthe following
Extrapolation of Interference Distribution We compareempirical distributions obtained from simulations to knowndistributions for the different strategies and the two heuristicswhen Δ = 3 6 and 9 Three of these empirical distributionsare shown in Figure 3 (in dB) The distribution parametershave been set according to the maximum likelihood Thedistributions that best fit simulations vary according to theallocation strategy and the reuse factorΔ For 4 cases over 12the distribution minimizing error is the normal distributionAs soon as an asymmetry is observed other distributionsare more accurate log-normal Weibull and Inverse GammaWe perform two hypothesis tests T-test and Smirnov-Kolmogorov for all these scenarios The alternative hypoth-esis is systematically ruled out For the 119872119872119877119861119898119886119909119877119861119898119886119909
strategy best fits are given by the normal (Δ = 3 and 6) andWeibull distributions (Δ = 9)
Nevertheless as mentioned in Section 5 when estimatingthe SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy weuse the normal law to model interference Indeed even if thisdistribution is not systematically the most accurate it offers agood approximation as shown inFigure 4 where the empiricaland normalCDF are plottedMoreover simulationswill showthat this assumption does not impact the accuracy of theanalytical model
Heuristics Simulations have shown that allocations resultingfrom the two heuristics correlate the transmission powersthe distance between eNodeB reusing the same resourceand the random variables 119908119894 Whenminimizing interferencecorrelations are caused by resource blocks allocated to usersthat require a low transmission power and that can be reusedat a short distance and inversely The transmission poweris then correlated to 119908119894 and to the distance of eNodeB thatreuse the same resource These complex phenomena impedethe proposal of an analytical model for these two heuristicsFor comparison purposes Table 2 reportsmean and standarddeviation for the two heuristics and the119872119872119877119861119898119886119909119877119861119898119886119909
Figure 3 Interference distribution empirical (histogram) and extrapolation (curve) Among the classical distributions the bestextrapolations are Gamma (m=1003 var=977) normal (m=2823 var=2216) and Weibull (a=2514b=575) for the heuristic MMRBallocation Δ = 6 and static allocation Δ = 9 respectively
Figure 4 CDF of interference for the 119872119872119877119861119898119886119909119877119861119898119886119909 strategy Empirical distributions are represented through the solid lines CDFof the Normal distribution are plotted with dotted lines Their parameters are obtained from the theoretical formulas The 119871+infin errors are338119890minus02 493119890minus02 and 565119890minus02 for Δ = 3 6 and 9 respectively
Figure 5 Mean interference when the number of RB variesThe load is adjusted to keep a constant reuse factor (Δ = 36 or 9) The distance 1198830 is set to 50 meters Dotted linescorrespond to the theoretical evaluation ofmean interference for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation and points to simulations
allocationTheheuristicminimizing interference presents thebest results This difference was expected as the objectivefunction minimizes interference and because the secondheuristic maximizes the reuse distance without taking intoaccount transmission powers 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tion offers interesting results with an intermediary interfer-ence level between the two heuristics Even if interference islower for the heuristic minimizing interference its complex-ity in terms of feedbackmeasures from users and eNodeBis significantly greater than the other strategies Indeed itassumes that the scheduler knows before an assignment theinterference contribution of each eNodeB on each user
Impact of the Number of RB We performed simulations andtheoretical estimations of interference for different values of119877119861119898119886119909 6 15 25 50 75 and 100 corresponding to differentbandwidth of E-UTRA [33] Results are shown in Figure 5 forthe 119872119872119877119861119898119886119909119877119861119898119886119909 allocation and the two heuristicsThe workload is adapted in order to keep the same Δreuse factor It appears that the mean interference is quiteinsensitive to the number of RB except when 119877119861119898119886119909 = 6The impacting factor is thus the reuse factor Δ rather thanthe number of RBAlso the same hierarchy between the threestrategies is observed except for 119877119861119898119886119909 = 6SIR Distribution We evaluate the SIR CDF according to(29) for the 119872119872119877119861119898119886119909119877119861119898119886119909 allocation This CDF iscompared to the one obtained by simulations in Figure 6(a)We clearly observe that the analytical model offers very
tight estimates of the SIR distribution The assumptionabout the normal distribution of interference does notintroduce a noticeable error The SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy and the two heuristics arecompared in Figure 6(b) The three CDF are similar and the119872119872119877119861119898119886119909119877119861119898119886119909 allocation still offers an intermediatedistribution between the two heuristics The 119871+infin normbetween the three CDF is 0192 0266 and 0221 for Δ = 36 and 9 respectivelyModulation and Coding Rate The SIR distribution allowsus to estimate modulations and coding rates that could beoffered to users We consider the thresholds between therequired SIR and coding rate given in [33] Applied to the SIRdistribution it gives the proportion of users that benefits froma certain modulation scheme and coding rate The couplemodulationcoding rate is referred to as transmission rate inthe following This proportion is given in Figure 7(a) for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation when the density of eNodeBincreases from 225 (the default intensity considered in theprevious evaluation) to 20 eNodeB per km2 It appears thatthe network densification even with a factor 10 does notsignificantly improve the transmission rate Only the besttransmission rate (64QAM-CR=45) really benefits from thisdensification with a proportion of users increasing from 35to 116
The spatial reuse has much more impact on the transmis-sion rates as it is shown in Figure 7(b) It clearly increasesthe transmission rates that are shifted from (QPSK CR=12-64QAM CR=23) for Δ = 3 to (16QAM CR=12-64QAMCR=45) for Δ = 9 The mean transmission rate increasesalmost of a factor 2 from 231 to 424 bitsbaud
Finally we compare in Figure 7(c) the transmission ratesfor the three allocation strategies As already observed inthe previous plots the heuristic minimizing interferenceoffers better performance but is comparable to the two otherstrategies For instance for Δ = 6 the mean transmis-sion rates are close 363 323 and 392 bitsbaud for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation the heuristic maximizingthe reuse distance and the one minimizing interferencerespectively
7 Conclusion
In recent LTE standards a server may assign RB to a set ofeNodeB to control resource usage and optimize performancein a global way Even if it exists solutions to address this prob-lem for a given configuration and topology literature lacksmodels that evaluate performance of these RB allocations fora wide range of scenarios and at large scale
To address this problem we propound a spatial stochasticmodel that takes into account RB assignment strategiesrealistic traffic demands and power control We proposeanalytical estimates for the two first moments of interferenceThis computation is based on two approximations the distri-bution of the distance between a point uniformly distributedin a cell that models the user-eNodeB distance and thejoint distribution of the distances eNodeB-origin and user-eNodeB For the latter simulations have shown that these
(a) SIR empirical and theoretical CDF for the119872119872119877119861119898119886119909119877119861119898119886119909allocation
MMRB Delta=3MMRB Delta=6MMRB Delta=9MI Delta=3 MI Delta=6
MI Delta=9 MD Delta=3 MD Delta=6 MD Delta=9
0 10 20 30 40minus10훽 (dB)
00e+00
20eminus01
40eminus01
60eminus01
80eminus01
10e+00
CDF
minus SI
NR
(b) SIR empirical CDF for the three strategies MI maximizeinterference MD maximize distance
Figure 6 (a) We compare the distribution of SIR obtained by simulations (solid lines) and given by (29) (dotted lines) Errors with regardto the 119871+infin norm are 247119890 minus 02 154119890 minus 02 and 158119890 minus 02 for Δ = 3 6 and 9 respectively (b) CDF for the three allocation strategies(119872119872119877119861119898119886119909119877119861119898119886119909 and Heuristics)
(c) 119872119872119877119861119898119886119909119877119861119898119886119909 and the two heuristics Δ = 6 120582119890 = 225
Figure 7 Modulation and coding rate for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation strategy when the density of eNodeB increases (Figure 7(a))for different reuse factor (Figure 7(b)) and comparison with the two heuristics (Figure 7(c))
variables are strongly correlated particularly for the points ofthe process close to the origin This correlation impacts thetransmission power used by the eNodeB and consequentlyinterference and SIR distributions The derivation of the SIRdistribution allows us to express the classical outage but alsothe transmission rates in terms of modulationcoding ratethat provide interesting insights on the throughput offers tothe users Also it appears that eNodeB densification does not
significantly improve the network performance in terms ofthroughput except for a small percentage of users for whichthe transmission rate is increased Spatial reuse expressedthrough the parameter Δ in our study has a much moreimpact on the performance as the transmission rates aresignificantly increased This spatial reuse must be expressedas the ratio between the total number of available resourcesover the number of usersrequests per cell Indeed we have
Wireless Communications and Mobile Computing 13
observed that the system performance is quite insensitive tovariation of these quantities when this ratio stays constant
Beside simulations show that the RB assignment strate-gies developed for the model give results close to the onesgiven by classical optimization problem which minimizesglobal interference or maximizes spatial reuse distances Itempirically shows that our analytical model is able to evaluateperformance of classical optimization approaches but at largescale in terms of number of nodes configurations andtopologies
Appendix
A Construction of the Markov Chain
In this section we introduce the Markov chain used to set thedistribution of 1198770 We consider that the RB used between thetypical user and its eNodeB has index 119903119890119904 (119903119890119904 isin 0 119877119861119898119886119909minus1)We shall prove that this index does not impact calculation(we have chosen 119903119890119904 = 0 in this paper by sack of simplicity)
We set the distribution of the number of used resourceblocks at eNodeB 0 in such a way that it leads to the followingproperty (119895 gt 119894 ge 0)
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (A1)
In order to obtain this property we first introduce somepreliminary notations and results An example of allocationwith the different notations is shown in Figure 1 We define asequence of rv (119877119899)119899ge0 If eNodeB 119894 is the 119899119905ℎ eNodeB using119903119890119904 and if 119864119899119889119899 is the index of the last RB used by this eNodeBthen
119877119899 = 119864119899119889119899 minus 119903119890119904 if 119864119899119889119899 gt 119903119890119904119864119899119889119899 + 119877119861119898119886119909 minus 119903119890119904 otherwise (A2)
If eNodeB 119894 is the 119899119905ℎ interfering eNodeB (using 119903119890119904) theneNodeB 119895 (119895 gt 119894) is the 119899 + 1119905ℎ if and only if
119863119896 ge 119877119861119898119886119909 (A4)
In this case 119877119899+1 is given by
119877119899+1 = 119877119899 +119895sum
119896=119894+1
119863119896 minus 119877119861119898119886119909 (A5)
As (119863119894)119894gt0 are iid the sequence (119877119899)119899ge0 is a homoge-neous Markov chain The transition probabilities for (119897 119898) isin0 1 119877119861119898119886119909 minus 12 are given by 119875119897119898 = P(119877119899+1 = 119898 | 119877119899 = 119897)
The event 119877119899+1 = 119898 given that 119877119899 = 119897 occurs if andonly if it exists 119895 ge 0 such that
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (A10)
We set the distribution of 1198770 with the correspondingstationary distribution Consequently the sequence (119877119899)119899ge0becomes identically distributed where 119877119899 follows the sta-tionary distribution for all 119899 ge 0 Moreover according to(A3) and (A4) the property given in (A1) holds Indeed 119877119899sum119895minus1
119896=119894+1119863119896 and sum119895
119896=119894+1119863119896 have the same distribution as 1198770sum119895minus119894minus1
119896=1119863119896 andsum119895minus119894
119896=1119863119896 respectively In the model evaluation
the stationary distribution is obtained numerically from thetransition probabilities As soon as 1198770 follows the stationarydistribution the index 119903119890119904 can be chosen arbitrarily
B Conditional Distribution of 119880119894minus119883119894Simulations have shown that given 119883119894 the distribution of119883119894 minus 119880119894 is still close to the one given by (2) but with adifferent parameter So we assume that the distribution of119883119894 minus 119880119894 given 119883119894 is equal to (2) but with a parameterfunction of 119883119894 denoted 119888(119883119894)
These simulations have also shown that E[119883119894 minus 119880119894 |119883119894 = 119903]may be approximated as an affine function 119886119894119903 + 119887119894We define 119888119894(sdot) accordingly It leads to
To set 119886119894 and 119887119894 we use the two following propertiesE [E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817 | 10038171003817100381710038171198831198941003817100381710038171003817 = 119903]] = E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817] (B3)
= 12radic120582119890 (B4)
E [E [1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817 | 1003817100381710038171003817119883119894
1003817100381710038171003817 = 119903]] = E [119886119894 10038171003817100381710038171198831198941003817100381710038171003817 + 119887119894] (B5)
Equation (B4) is derived from (2) and (B5) from ourassumption on 119888119894(119903) 119886119894 is then set as a function of 119887119894 E[119883119894]and 120582119890 119887119894 is obtained from our simulations For a givenintensity 119887119894 has been observed almost constant with regardto 119894 The best approximation as a function of the intensity isgiven by 119887119894 = 033radic120582119890 After a fewmanipulations we obtain
First note that by convention we have P(119908119894 = 1) = P(119908119894 =1 | 1199080 = 1) The event 119908119894 = 1 occurs if and only if it exists119895 ge 1 such that1198770 + 119894minus1sum
There is no data used in this paper Nevertheless the code ofour simulator is made available (the link is in the paper)
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] ldquoRequirements for further advancements for e-utra (lte-advanced) 3gpp technical specification ts 36913 release 13rdquo2015 httpwww3gpporg
[2] A S Hamza S S Khalifa H S Hamza and K Elsayed ldquoAsurvey on inter-cell interference coordination techniques inofdma-based cellular networksrdquo IEEE Communications Surveysamp Tutorials vol 15 no 4 pp 1642ndash1670 2013
[3] D Lopez-Perez A Ladanyi A JuttnerH Rivano and J ZhangldquoOptimization method for the joint allocation of modulationschemes coding rates resource blocks and power in self-organizing LTE networksrdquo in Proceedings of the IEEE INFO-COM 2011 pp 111ndash115 April 2011
[4] A Guo andM Haenggi ldquoSpatial stochasticmodels andmetricsfor the structure of base stations in cellular networksrdquo IEEE
Wireless Communications and Mobile Computing 15
Transactions on Wireless Communications vol 12 no 11 pp5800ndash5812 2013
[5] C-H Lee C-Y Shih and Y-S Chen ldquoStochastic geometrybased models for modeling cellular networks in urban areasrdquoWireless Networks vol 19 no 6 pp 1063ndash1072 2013
[6] W Lu and M D Renzo ldquoStochastic geometry modeling ofcellular networks analysis simulation and experimental valida-tionrdquo in Proceedings of the 18th ACM International Conferenceon Modelling Analysis and Simulation of Wireless and MobileSystems (MSWiM rsquo15) pp 179ndash188 New York NY USANovember 2015
[7] Y L Lee T C Chuah J Loo and A Vinel ldquoRecent advancesin radio resource management for heterogeneous LTELTE-Anetworksrdquo IEEE Communications Surveys amp Tutorials vol 16no 4 pp 2142ndash2180 2014
[8] F Mhiri K Sethom and R Bouallegue ldquoA survey on inter-ference management techniques in femtocell self-organizingnetworksrdquo Journal of Network and Computer Applications vol36 no 1 pp 58ndash65 2013
[9] X Chen L Li and X Xiang ldquoAnt colony learning methodfor joint MCS and resource block allocation in LTE Femtocelldownlink for multimedia applications with QoS guaranteesrdquoMultimedia Tools andApplications vol 76 no 3 pp 4035ndash40542017
[10] D Lopez-Perez X Chu A V Vasilakos and H Claussen ldquoOndistributed and coordinated resource allocation for interferencemitigation in self-organizing lte networksrdquo IEEEACMTransac-tions on Networking vol 21 no 4 pp 1145ndash1158 2013
[11] H Zhang H Liu J Cheng and V C M Leung ldquoDownlinkenergy efficiency of power allocation and wireless backhaulbandwidth allocation in heterogeneous small cell networksrdquoIEEE Transactions on Communications no 99 2017
[12] H Zhang W Zheng X Chu et al ldquoJoint subchannel andpower allocation in interference-limited OFDMA femtocellswith heterogeneous QoS guaranteerdquo in Proceedings of theIEEEGlobal Communications Conference (GLOBECOM rsquo12) pp4572ndash4577 December 2012
[13] H Elsawy E Hossain and M Haenggi ldquoStochastic geometryfor modeling analysis and design of multi-tier and cognitivecellular wireless networks a surveyrdquo IEEE CommunicationsSurveys amp Tutorials vol 15 no 3 pp 996ndash1019 2013
[14] H ElSawy A Sultan-Salem M-S Alouini and M Z WinldquoModeling and analysis of cellular networks using stochasticgeometry a tutorialrdquo IEEE Communications Surveys amp Tutori-als vol 19 no 1 pp 167ndash203 2017
[15] R Kwan and C Leung ldquoA survey of scheduling and inter-ference mitigation in lterdquo Journal of Electrical and ComputerEngineeringmdashSpecial Issue on LTELTE-AdvancedCellular Com-munication Networks pp 11ndash110 2010
[16] J G Andrews F Baccelli and R K Ganti ldquoA tractable approachto coverage and rate in cellular networksrdquo IEEE Transactions onCommunications vol 59 no 11 pp 3122ndash3134 2011
[17] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for OFDMA cellu-lar networksrdquo IEEE Transactions on Wireless Communicationsvol 10 no 12 pp 4294ndash4305 2011
[18] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for heterogeneouscellular networksrdquo IEEE Transactions on Communications vol60 no 7 pp 2029ndash2039 2012
[19] H Zhuang and T Ohtsuki ldquoA model based on poisson pointprocess for downlink K tiers fractional frequency reuse hetero-geneous networksrdquo Physical Communication vol 13 pp 3ndash122014 Special Issue on Heterogeneous and Small Cell Networks
[20] A A Gebremariam T Bao D Siracusa T Rasheed F Granelliand L Goratti ldquoDynamic strict fractional frequency reusefor software-defined 5G networksrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo16) pp 1ndash6May 2016
[21] N Lee D Morales-Jimenez A Lozano and R W HeathldquoSpectral efficiency of dynamic coordinated beamforming astochastic geometry approachrdquo IEEE Transactions on WirelessCommunications vol 14 no 1 pp 230ndash241 2015
[22] X Zhang and M Haenggi ldquoA Stochastic geometry analysisof inter-cell interference coordination and intra-cell diversityrdquoIEEE Transactions on Wireless Communications vol 13 no 12pp 6655ndash6669 2014
[23] F Baccelli andAGiovanidis ldquoA stochastic geometry frameworkfor analyzing pairwise-cooperative cellular networksrdquo IEEETransactions on Wireless Communications vol 14 no 2 pp794ndash808 2015
[24] J Yoon and G Hwang ldquoDistance-based inter-cell interferencecoordination in small cell networks stochastic geometry mod-eling and analysisrdquo IEEE Transactions on Wireless Communica-tions 2018
[25] B Błaszczyszyn M Jovanovicy and M K Karray ldquoHow userthroughput depends on the traffic demand in large cellularnetworksrdquo in Proceedings of the 12th International Symposiumon Modeling and Optimization in Mobile Ad Hoc and WirelessNetworks (WiOpt rsquo14) pp 611ndash619 May 2014
[26] H H Yang and T Q S Quek ldquoSIR coverage analysis incellular networks with temporal traffic a stochastic geometryapproachrdquo Corrossion 2018
[27] S M Yu and S-L Kim ldquoDownlink capacity and base stationdensity in cellular networksrdquo in Proceedings of the 11th Interna-tional SymposiumandWorkshops onModeling andOptimizationin Mobile Ad Hoc and Wireless Networks (WiOpt rsquo13) pp 119ndash124 May 2013
[28] YWang M Haenggi and Z Tan ldquoThemeta distribution of thesir for cellular networks with power controlrdquo Corrossion 2017
[29] M S Omar S A Hassan H Pervaiz et al ldquoMulti-objectiveoptimization in 5g hybrid networksrdquo IEEE Internet of ThingsJournal 2018
[30] Y Leo A Busson C Sarraute and E Fleury ldquoCall detail recordsto characterize usages and mobility events of phone usersrdquoComputer Communications vol 95 pp 43ndash53 2016
[31] I L Cherif Spectral and Energy Efficiency in 5G WirelessNetworks [PhD thesis] Univeristy Paris Saclay 2016
[32] S Mukherjee Analytical Modeling of Heterogeneous CellularNetworks Analytical Modeling of Heterogeneous Cellular Net-works Geometry Coverage and Capacity Cambridge Univer-sity Press 2014
[33] ldquoTechnical specification group radio access network base sta-tion (bs) radio transmission and reception 2012 Release 1123gpp ts 36104 v112rdquo 2012
Figure 3 Interference distribution empirical (histogram) and extrapolation (curve) Among the classical distributions the bestextrapolations are Gamma (m=1003 var=977) normal (m=2823 var=2216) and Weibull (a=2514b=575) for the heuristic MMRBallocation Δ = 6 and static allocation Δ = 9 respectively
Figure 4 CDF of interference for the 119872119872119877119861119898119886119909119877119861119898119886119909 strategy Empirical distributions are represented through the solid lines CDFof the Normal distribution are plotted with dotted lines Their parameters are obtained from the theoretical formulas The 119871+infin errors are338119890minus02 493119890minus02 and 565119890minus02 for Δ = 3 6 and 9 respectively
Figure 5 Mean interference when the number of RB variesThe load is adjusted to keep a constant reuse factor (Δ = 36 or 9) The distance 1198830 is set to 50 meters Dotted linescorrespond to the theoretical evaluation ofmean interference for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation and points to simulations
allocationTheheuristicminimizing interference presents thebest results This difference was expected as the objectivefunction minimizes interference and because the secondheuristic maximizes the reuse distance without taking intoaccount transmission powers 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tion offers interesting results with an intermediary interfer-ence level between the two heuristics Even if interference islower for the heuristic minimizing interference its complex-ity in terms of feedbackmeasures from users and eNodeBis significantly greater than the other strategies Indeed itassumes that the scheduler knows before an assignment theinterference contribution of each eNodeB on each user
Impact of the Number of RB We performed simulations andtheoretical estimations of interference for different values of119877119861119898119886119909 6 15 25 50 75 and 100 corresponding to differentbandwidth of E-UTRA [33] Results are shown in Figure 5 forthe 119872119872119877119861119898119886119909119877119861119898119886119909 allocation and the two heuristicsThe workload is adapted in order to keep the same Δreuse factor It appears that the mean interference is quiteinsensitive to the number of RB except when 119877119861119898119886119909 = 6The impacting factor is thus the reuse factor Δ rather thanthe number of RBAlso the same hierarchy between the threestrategies is observed except for 119877119861119898119886119909 = 6SIR Distribution We evaluate the SIR CDF according to(29) for the 119872119872119877119861119898119886119909119877119861119898119886119909 allocation This CDF iscompared to the one obtained by simulations in Figure 6(a)We clearly observe that the analytical model offers very
tight estimates of the SIR distribution The assumptionabout the normal distribution of interference does notintroduce a noticeable error The SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy and the two heuristics arecompared in Figure 6(b) The three CDF are similar and the119872119872119877119861119898119886119909119877119861119898119886119909 allocation still offers an intermediatedistribution between the two heuristics The 119871+infin normbetween the three CDF is 0192 0266 and 0221 for Δ = 36 and 9 respectivelyModulation and Coding Rate The SIR distribution allowsus to estimate modulations and coding rates that could beoffered to users We consider the thresholds between therequired SIR and coding rate given in [33] Applied to the SIRdistribution it gives the proportion of users that benefits froma certain modulation scheme and coding rate The couplemodulationcoding rate is referred to as transmission rate inthe following This proportion is given in Figure 7(a) for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation when the density of eNodeBincreases from 225 (the default intensity considered in theprevious evaluation) to 20 eNodeB per km2 It appears thatthe network densification even with a factor 10 does notsignificantly improve the transmission rate Only the besttransmission rate (64QAM-CR=45) really benefits from thisdensification with a proportion of users increasing from 35to 116
The spatial reuse has much more impact on the transmis-sion rates as it is shown in Figure 7(b) It clearly increasesthe transmission rates that are shifted from (QPSK CR=12-64QAM CR=23) for Δ = 3 to (16QAM CR=12-64QAMCR=45) for Δ = 9 The mean transmission rate increasesalmost of a factor 2 from 231 to 424 bitsbaud
Finally we compare in Figure 7(c) the transmission ratesfor the three allocation strategies As already observed inthe previous plots the heuristic minimizing interferenceoffers better performance but is comparable to the two otherstrategies For instance for Δ = 6 the mean transmis-sion rates are close 363 323 and 392 bitsbaud for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation the heuristic maximizingthe reuse distance and the one minimizing interferencerespectively
7 Conclusion
In recent LTE standards a server may assign RB to a set ofeNodeB to control resource usage and optimize performancein a global way Even if it exists solutions to address this prob-lem for a given configuration and topology literature lacksmodels that evaluate performance of these RB allocations fora wide range of scenarios and at large scale
To address this problem we propound a spatial stochasticmodel that takes into account RB assignment strategiesrealistic traffic demands and power control We proposeanalytical estimates for the two first moments of interferenceThis computation is based on two approximations the distri-bution of the distance between a point uniformly distributedin a cell that models the user-eNodeB distance and thejoint distribution of the distances eNodeB-origin and user-eNodeB For the latter simulations have shown that these
(a) SIR empirical and theoretical CDF for the119872119872119877119861119898119886119909119877119861119898119886119909allocation
MMRB Delta=3MMRB Delta=6MMRB Delta=9MI Delta=3 MI Delta=6
MI Delta=9 MD Delta=3 MD Delta=6 MD Delta=9
0 10 20 30 40minus10훽 (dB)
00e+00
20eminus01
40eminus01
60eminus01
80eminus01
10e+00
CDF
minus SI
NR
(b) SIR empirical CDF for the three strategies MI maximizeinterference MD maximize distance
Figure 6 (a) We compare the distribution of SIR obtained by simulations (solid lines) and given by (29) (dotted lines) Errors with regardto the 119871+infin norm are 247119890 minus 02 154119890 minus 02 and 158119890 minus 02 for Δ = 3 6 and 9 respectively (b) CDF for the three allocation strategies(119872119872119877119861119898119886119909119877119861119898119886119909 and Heuristics)
(c) 119872119872119877119861119898119886119909119877119861119898119886119909 and the two heuristics Δ = 6 120582119890 = 225
Figure 7 Modulation and coding rate for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation strategy when the density of eNodeB increases (Figure 7(a))for different reuse factor (Figure 7(b)) and comparison with the two heuristics (Figure 7(c))
variables are strongly correlated particularly for the points ofthe process close to the origin This correlation impacts thetransmission power used by the eNodeB and consequentlyinterference and SIR distributions The derivation of the SIRdistribution allows us to express the classical outage but alsothe transmission rates in terms of modulationcoding ratethat provide interesting insights on the throughput offers tothe users Also it appears that eNodeB densification does not
significantly improve the network performance in terms ofthroughput except for a small percentage of users for whichthe transmission rate is increased Spatial reuse expressedthrough the parameter Δ in our study has a much moreimpact on the performance as the transmission rates aresignificantly increased This spatial reuse must be expressedas the ratio between the total number of available resourcesover the number of usersrequests per cell Indeed we have
Wireless Communications and Mobile Computing 13
observed that the system performance is quite insensitive tovariation of these quantities when this ratio stays constant
Beside simulations show that the RB assignment strate-gies developed for the model give results close to the onesgiven by classical optimization problem which minimizesglobal interference or maximizes spatial reuse distances Itempirically shows that our analytical model is able to evaluateperformance of classical optimization approaches but at largescale in terms of number of nodes configurations andtopologies
Appendix
A Construction of the Markov Chain
In this section we introduce the Markov chain used to set thedistribution of 1198770 We consider that the RB used between thetypical user and its eNodeB has index 119903119890119904 (119903119890119904 isin 0 119877119861119898119886119909minus1)We shall prove that this index does not impact calculation(we have chosen 119903119890119904 = 0 in this paper by sack of simplicity)
We set the distribution of the number of used resourceblocks at eNodeB 0 in such a way that it leads to the followingproperty (119895 gt 119894 ge 0)
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (A1)
In order to obtain this property we first introduce somepreliminary notations and results An example of allocationwith the different notations is shown in Figure 1 We define asequence of rv (119877119899)119899ge0 If eNodeB 119894 is the 119899119905ℎ eNodeB using119903119890119904 and if 119864119899119889119899 is the index of the last RB used by this eNodeBthen
119877119899 = 119864119899119889119899 minus 119903119890119904 if 119864119899119889119899 gt 119903119890119904119864119899119889119899 + 119877119861119898119886119909 minus 119903119890119904 otherwise (A2)
If eNodeB 119894 is the 119899119905ℎ interfering eNodeB (using 119903119890119904) theneNodeB 119895 (119895 gt 119894) is the 119899 + 1119905ℎ if and only if
119863119896 ge 119877119861119898119886119909 (A4)
In this case 119877119899+1 is given by
119877119899+1 = 119877119899 +119895sum
119896=119894+1
119863119896 minus 119877119861119898119886119909 (A5)
As (119863119894)119894gt0 are iid the sequence (119877119899)119899ge0 is a homoge-neous Markov chain The transition probabilities for (119897 119898) isin0 1 119877119861119898119886119909 minus 12 are given by 119875119897119898 = P(119877119899+1 = 119898 | 119877119899 = 119897)
The event 119877119899+1 = 119898 given that 119877119899 = 119897 occurs if andonly if it exists 119895 ge 0 such that
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (A10)
We set the distribution of 1198770 with the correspondingstationary distribution Consequently the sequence (119877119899)119899ge0becomes identically distributed where 119877119899 follows the sta-tionary distribution for all 119899 ge 0 Moreover according to(A3) and (A4) the property given in (A1) holds Indeed 119877119899sum119895minus1
119896=119894+1119863119896 and sum119895
119896=119894+1119863119896 have the same distribution as 1198770sum119895minus119894minus1
119896=1119863119896 andsum119895minus119894
119896=1119863119896 respectively In the model evaluation
the stationary distribution is obtained numerically from thetransition probabilities As soon as 1198770 follows the stationarydistribution the index 119903119890119904 can be chosen arbitrarily
B Conditional Distribution of 119880119894minus119883119894Simulations have shown that given 119883119894 the distribution of119883119894 minus 119880119894 is still close to the one given by (2) but with adifferent parameter So we assume that the distribution of119883119894 minus 119880119894 given 119883119894 is equal to (2) but with a parameterfunction of 119883119894 denoted 119888(119883119894)
These simulations have also shown that E[119883119894 minus 119880119894 |119883119894 = 119903]may be approximated as an affine function 119886119894119903 + 119887119894We define 119888119894(sdot) accordingly It leads to
To set 119886119894 and 119887119894 we use the two following propertiesE [E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817 | 10038171003817100381710038171198831198941003817100381710038171003817 = 119903]] = E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817] (B3)
= 12radic120582119890 (B4)
E [E [1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817 | 1003817100381710038171003817119883119894
1003817100381710038171003817 = 119903]] = E [119886119894 10038171003817100381710038171198831198941003817100381710038171003817 + 119887119894] (B5)
Equation (B4) is derived from (2) and (B5) from ourassumption on 119888119894(119903) 119886119894 is then set as a function of 119887119894 E[119883119894]and 120582119890 119887119894 is obtained from our simulations For a givenintensity 119887119894 has been observed almost constant with regardto 119894 The best approximation as a function of the intensity isgiven by 119887119894 = 033radic120582119890 After a fewmanipulations we obtain
First note that by convention we have P(119908119894 = 1) = P(119908119894 =1 | 1199080 = 1) The event 119908119894 = 1 occurs if and only if it exists119895 ge 1 such that1198770 + 119894minus1sum
There is no data used in this paper Nevertheless the code ofour simulator is made available (the link is in the paper)
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] ldquoRequirements for further advancements for e-utra (lte-advanced) 3gpp technical specification ts 36913 release 13rdquo2015 httpwww3gpporg
[2] A S Hamza S S Khalifa H S Hamza and K Elsayed ldquoAsurvey on inter-cell interference coordination techniques inofdma-based cellular networksrdquo IEEE Communications Surveysamp Tutorials vol 15 no 4 pp 1642ndash1670 2013
[3] D Lopez-Perez A Ladanyi A JuttnerH Rivano and J ZhangldquoOptimization method for the joint allocation of modulationschemes coding rates resource blocks and power in self-organizing LTE networksrdquo in Proceedings of the IEEE INFO-COM 2011 pp 111ndash115 April 2011
[4] A Guo andM Haenggi ldquoSpatial stochasticmodels andmetricsfor the structure of base stations in cellular networksrdquo IEEE
Wireless Communications and Mobile Computing 15
Transactions on Wireless Communications vol 12 no 11 pp5800ndash5812 2013
[5] C-H Lee C-Y Shih and Y-S Chen ldquoStochastic geometrybased models for modeling cellular networks in urban areasrdquoWireless Networks vol 19 no 6 pp 1063ndash1072 2013
[6] W Lu and M D Renzo ldquoStochastic geometry modeling ofcellular networks analysis simulation and experimental valida-tionrdquo in Proceedings of the 18th ACM International Conferenceon Modelling Analysis and Simulation of Wireless and MobileSystems (MSWiM rsquo15) pp 179ndash188 New York NY USANovember 2015
[7] Y L Lee T C Chuah J Loo and A Vinel ldquoRecent advancesin radio resource management for heterogeneous LTELTE-Anetworksrdquo IEEE Communications Surveys amp Tutorials vol 16no 4 pp 2142ndash2180 2014
[8] F Mhiri K Sethom and R Bouallegue ldquoA survey on inter-ference management techniques in femtocell self-organizingnetworksrdquo Journal of Network and Computer Applications vol36 no 1 pp 58ndash65 2013
[9] X Chen L Li and X Xiang ldquoAnt colony learning methodfor joint MCS and resource block allocation in LTE Femtocelldownlink for multimedia applications with QoS guaranteesrdquoMultimedia Tools andApplications vol 76 no 3 pp 4035ndash40542017
[10] D Lopez-Perez X Chu A V Vasilakos and H Claussen ldquoOndistributed and coordinated resource allocation for interferencemitigation in self-organizing lte networksrdquo IEEEACMTransac-tions on Networking vol 21 no 4 pp 1145ndash1158 2013
[11] H Zhang H Liu J Cheng and V C M Leung ldquoDownlinkenergy efficiency of power allocation and wireless backhaulbandwidth allocation in heterogeneous small cell networksrdquoIEEE Transactions on Communications no 99 2017
[12] H Zhang W Zheng X Chu et al ldquoJoint subchannel andpower allocation in interference-limited OFDMA femtocellswith heterogeneous QoS guaranteerdquo in Proceedings of theIEEEGlobal Communications Conference (GLOBECOM rsquo12) pp4572ndash4577 December 2012
[13] H Elsawy E Hossain and M Haenggi ldquoStochastic geometryfor modeling analysis and design of multi-tier and cognitivecellular wireless networks a surveyrdquo IEEE CommunicationsSurveys amp Tutorials vol 15 no 3 pp 996ndash1019 2013
[14] H ElSawy A Sultan-Salem M-S Alouini and M Z WinldquoModeling and analysis of cellular networks using stochasticgeometry a tutorialrdquo IEEE Communications Surveys amp Tutori-als vol 19 no 1 pp 167ndash203 2017
[15] R Kwan and C Leung ldquoA survey of scheduling and inter-ference mitigation in lterdquo Journal of Electrical and ComputerEngineeringmdashSpecial Issue on LTELTE-AdvancedCellular Com-munication Networks pp 11ndash110 2010
[16] J G Andrews F Baccelli and R K Ganti ldquoA tractable approachto coverage and rate in cellular networksrdquo IEEE Transactions onCommunications vol 59 no 11 pp 3122ndash3134 2011
[17] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for OFDMA cellu-lar networksrdquo IEEE Transactions on Wireless Communicationsvol 10 no 12 pp 4294ndash4305 2011
[18] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for heterogeneouscellular networksrdquo IEEE Transactions on Communications vol60 no 7 pp 2029ndash2039 2012
[19] H Zhuang and T Ohtsuki ldquoA model based on poisson pointprocess for downlink K tiers fractional frequency reuse hetero-geneous networksrdquo Physical Communication vol 13 pp 3ndash122014 Special Issue on Heterogeneous and Small Cell Networks
[20] A A Gebremariam T Bao D Siracusa T Rasheed F Granelliand L Goratti ldquoDynamic strict fractional frequency reusefor software-defined 5G networksrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo16) pp 1ndash6May 2016
[21] N Lee D Morales-Jimenez A Lozano and R W HeathldquoSpectral efficiency of dynamic coordinated beamforming astochastic geometry approachrdquo IEEE Transactions on WirelessCommunications vol 14 no 1 pp 230ndash241 2015
[22] X Zhang and M Haenggi ldquoA Stochastic geometry analysisof inter-cell interference coordination and intra-cell diversityrdquoIEEE Transactions on Wireless Communications vol 13 no 12pp 6655ndash6669 2014
[23] F Baccelli andAGiovanidis ldquoA stochastic geometry frameworkfor analyzing pairwise-cooperative cellular networksrdquo IEEETransactions on Wireless Communications vol 14 no 2 pp794ndash808 2015
[24] J Yoon and G Hwang ldquoDistance-based inter-cell interferencecoordination in small cell networks stochastic geometry mod-eling and analysisrdquo IEEE Transactions on Wireless Communica-tions 2018
[25] B Błaszczyszyn M Jovanovicy and M K Karray ldquoHow userthroughput depends on the traffic demand in large cellularnetworksrdquo in Proceedings of the 12th International Symposiumon Modeling and Optimization in Mobile Ad Hoc and WirelessNetworks (WiOpt rsquo14) pp 611ndash619 May 2014
[26] H H Yang and T Q S Quek ldquoSIR coverage analysis incellular networks with temporal traffic a stochastic geometryapproachrdquo Corrossion 2018
[27] S M Yu and S-L Kim ldquoDownlink capacity and base stationdensity in cellular networksrdquo in Proceedings of the 11th Interna-tional SymposiumandWorkshops onModeling andOptimizationin Mobile Ad Hoc and Wireless Networks (WiOpt rsquo13) pp 119ndash124 May 2013
[28] YWang M Haenggi and Z Tan ldquoThemeta distribution of thesir for cellular networks with power controlrdquo Corrossion 2017
[29] M S Omar S A Hassan H Pervaiz et al ldquoMulti-objectiveoptimization in 5g hybrid networksrdquo IEEE Internet of ThingsJournal 2018
[30] Y Leo A Busson C Sarraute and E Fleury ldquoCall detail recordsto characterize usages and mobility events of phone usersrdquoComputer Communications vol 95 pp 43ndash53 2016
[31] I L Cherif Spectral and Energy Efficiency in 5G WirelessNetworks [PhD thesis] Univeristy Paris Saclay 2016
[32] S Mukherjee Analytical Modeling of Heterogeneous CellularNetworks Analytical Modeling of Heterogeneous Cellular Net-works Geometry Coverage and Capacity Cambridge Univer-sity Press 2014
[33] ldquoTechnical specification group radio access network base sta-tion (bs) radio transmission and reception 2012 Release 1123gpp ts 36104 v112rdquo 2012
Figure 5 Mean interference when the number of RB variesThe load is adjusted to keep a constant reuse factor (Δ = 36 or 9) The distance 1198830 is set to 50 meters Dotted linescorrespond to the theoretical evaluation ofmean interference for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation and points to simulations
allocationTheheuristicminimizing interference presents thebest results This difference was expected as the objectivefunction minimizes interference and because the secondheuristic maximizes the reuse distance without taking intoaccount transmission powers 119872119872119877119861119898119886119909119877119861119898119886119909 alloca-tion offers interesting results with an intermediary interfer-ence level between the two heuristics Even if interference islower for the heuristic minimizing interference its complex-ity in terms of feedbackmeasures from users and eNodeBis significantly greater than the other strategies Indeed itassumes that the scheduler knows before an assignment theinterference contribution of each eNodeB on each user
Impact of the Number of RB We performed simulations andtheoretical estimations of interference for different values of119877119861119898119886119909 6 15 25 50 75 and 100 corresponding to differentbandwidth of E-UTRA [33] Results are shown in Figure 5 forthe 119872119872119877119861119898119886119909119877119861119898119886119909 allocation and the two heuristicsThe workload is adapted in order to keep the same Δreuse factor It appears that the mean interference is quiteinsensitive to the number of RB except when 119877119861119898119886119909 = 6The impacting factor is thus the reuse factor Δ rather thanthe number of RBAlso the same hierarchy between the threestrategies is observed except for 119877119861119898119886119909 = 6SIR Distribution We evaluate the SIR CDF according to(29) for the 119872119872119877119861119898119886119909119877119861119898119886119909 allocation This CDF iscompared to the one obtained by simulations in Figure 6(a)We clearly observe that the analytical model offers very
tight estimates of the SIR distribution The assumptionabout the normal distribution of interference does notintroduce a noticeable error The SIR distribution for the119872119872119877119861119898119886119909119877119861119898119886119909 strategy and the two heuristics arecompared in Figure 6(b) The three CDF are similar and the119872119872119877119861119898119886119909119877119861119898119886119909 allocation still offers an intermediatedistribution between the two heuristics The 119871+infin normbetween the three CDF is 0192 0266 and 0221 for Δ = 36 and 9 respectivelyModulation and Coding Rate The SIR distribution allowsus to estimate modulations and coding rates that could beoffered to users We consider the thresholds between therequired SIR and coding rate given in [33] Applied to the SIRdistribution it gives the proportion of users that benefits froma certain modulation scheme and coding rate The couplemodulationcoding rate is referred to as transmission rate inthe following This proportion is given in Figure 7(a) for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation when the density of eNodeBincreases from 225 (the default intensity considered in theprevious evaluation) to 20 eNodeB per km2 It appears thatthe network densification even with a factor 10 does notsignificantly improve the transmission rate Only the besttransmission rate (64QAM-CR=45) really benefits from thisdensification with a proportion of users increasing from 35to 116
The spatial reuse has much more impact on the transmis-sion rates as it is shown in Figure 7(b) It clearly increasesthe transmission rates that are shifted from (QPSK CR=12-64QAM CR=23) for Δ = 3 to (16QAM CR=12-64QAMCR=45) for Δ = 9 The mean transmission rate increasesalmost of a factor 2 from 231 to 424 bitsbaud
Finally we compare in Figure 7(c) the transmission ratesfor the three allocation strategies As already observed inthe previous plots the heuristic minimizing interferenceoffers better performance but is comparable to the two otherstrategies For instance for Δ = 6 the mean transmis-sion rates are close 363 323 and 392 bitsbaud for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation the heuristic maximizingthe reuse distance and the one minimizing interferencerespectively
7 Conclusion
In recent LTE standards a server may assign RB to a set ofeNodeB to control resource usage and optimize performancein a global way Even if it exists solutions to address this prob-lem for a given configuration and topology literature lacksmodels that evaluate performance of these RB allocations fora wide range of scenarios and at large scale
To address this problem we propound a spatial stochasticmodel that takes into account RB assignment strategiesrealistic traffic demands and power control We proposeanalytical estimates for the two first moments of interferenceThis computation is based on two approximations the distri-bution of the distance between a point uniformly distributedin a cell that models the user-eNodeB distance and thejoint distribution of the distances eNodeB-origin and user-eNodeB For the latter simulations have shown that these
(a) SIR empirical and theoretical CDF for the119872119872119877119861119898119886119909119877119861119898119886119909allocation
MMRB Delta=3MMRB Delta=6MMRB Delta=9MI Delta=3 MI Delta=6
MI Delta=9 MD Delta=3 MD Delta=6 MD Delta=9
0 10 20 30 40minus10훽 (dB)
00e+00
20eminus01
40eminus01
60eminus01
80eminus01
10e+00
CDF
minus SI
NR
(b) SIR empirical CDF for the three strategies MI maximizeinterference MD maximize distance
Figure 6 (a) We compare the distribution of SIR obtained by simulations (solid lines) and given by (29) (dotted lines) Errors with regardto the 119871+infin norm are 247119890 minus 02 154119890 minus 02 and 158119890 minus 02 for Δ = 3 6 and 9 respectively (b) CDF for the three allocation strategies(119872119872119877119861119898119886119909119877119861119898119886119909 and Heuristics)
(c) 119872119872119877119861119898119886119909119877119861119898119886119909 and the two heuristics Δ = 6 120582119890 = 225
Figure 7 Modulation and coding rate for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation strategy when the density of eNodeB increases (Figure 7(a))for different reuse factor (Figure 7(b)) and comparison with the two heuristics (Figure 7(c))
variables are strongly correlated particularly for the points ofthe process close to the origin This correlation impacts thetransmission power used by the eNodeB and consequentlyinterference and SIR distributions The derivation of the SIRdistribution allows us to express the classical outage but alsothe transmission rates in terms of modulationcoding ratethat provide interesting insights on the throughput offers tothe users Also it appears that eNodeB densification does not
significantly improve the network performance in terms ofthroughput except for a small percentage of users for whichthe transmission rate is increased Spatial reuse expressedthrough the parameter Δ in our study has a much moreimpact on the performance as the transmission rates aresignificantly increased This spatial reuse must be expressedas the ratio between the total number of available resourcesover the number of usersrequests per cell Indeed we have
Wireless Communications and Mobile Computing 13
observed that the system performance is quite insensitive tovariation of these quantities when this ratio stays constant
Beside simulations show that the RB assignment strate-gies developed for the model give results close to the onesgiven by classical optimization problem which minimizesglobal interference or maximizes spatial reuse distances Itempirically shows that our analytical model is able to evaluateperformance of classical optimization approaches but at largescale in terms of number of nodes configurations andtopologies
Appendix
A Construction of the Markov Chain
In this section we introduce the Markov chain used to set thedistribution of 1198770 We consider that the RB used between thetypical user and its eNodeB has index 119903119890119904 (119903119890119904 isin 0 119877119861119898119886119909minus1)We shall prove that this index does not impact calculation(we have chosen 119903119890119904 = 0 in this paper by sack of simplicity)
We set the distribution of the number of used resourceblocks at eNodeB 0 in such a way that it leads to the followingproperty (119895 gt 119894 ge 0)
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (A1)
In order to obtain this property we first introduce somepreliminary notations and results An example of allocationwith the different notations is shown in Figure 1 We define asequence of rv (119877119899)119899ge0 If eNodeB 119894 is the 119899119905ℎ eNodeB using119903119890119904 and if 119864119899119889119899 is the index of the last RB used by this eNodeBthen
119877119899 = 119864119899119889119899 minus 119903119890119904 if 119864119899119889119899 gt 119903119890119904119864119899119889119899 + 119877119861119898119886119909 minus 119903119890119904 otherwise (A2)
If eNodeB 119894 is the 119899119905ℎ interfering eNodeB (using 119903119890119904) theneNodeB 119895 (119895 gt 119894) is the 119899 + 1119905ℎ if and only if
119863119896 ge 119877119861119898119886119909 (A4)
In this case 119877119899+1 is given by
119877119899+1 = 119877119899 +119895sum
119896=119894+1
119863119896 minus 119877119861119898119886119909 (A5)
As (119863119894)119894gt0 are iid the sequence (119877119899)119899ge0 is a homoge-neous Markov chain The transition probabilities for (119897 119898) isin0 1 119877119861119898119886119909 minus 12 are given by 119875119897119898 = P(119877119899+1 = 119898 | 119877119899 = 119897)
The event 119877119899+1 = 119898 given that 119877119899 = 119897 occurs if andonly if it exists 119895 ge 0 such that
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (A10)
We set the distribution of 1198770 with the correspondingstationary distribution Consequently the sequence (119877119899)119899ge0becomes identically distributed where 119877119899 follows the sta-tionary distribution for all 119899 ge 0 Moreover according to(A3) and (A4) the property given in (A1) holds Indeed 119877119899sum119895minus1
119896=119894+1119863119896 and sum119895
119896=119894+1119863119896 have the same distribution as 1198770sum119895minus119894minus1
119896=1119863119896 andsum119895minus119894
119896=1119863119896 respectively In the model evaluation
the stationary distribution is obtained numerically from thetransition probabilities As soon as 1198770 follows the stationarydistribution the index 119903119890119904 can be chosen arbitrarily
B Conditional Distribution of 119880119894minus119883119894Simulations have shown that given 119883119894 the distribution of119883119894 minus 119880119894 is still close to the one given by (2) but with adifferent parameter So we assume that the distribution of119883119894 minus 119880119894 given 119883119894 is equal to (2) but with a parameterfunction of 119883119894 denoted 119888(119883119894)
These simulations have also shown that E[119883119894 minus 119880119894 |119883119894 = 119903]may be approximated as an affine function 119886119894119903 + 119887119894We define 119888119894(sdot) accordingly It leads to
To set 119886119894 and 119887119894 we use the two following propertiesE [E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817 | 10038171003817100381710038171198831198941003817100381710038171003817 = 119903]] = E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817] (B3)
= 12radic120582119890 (B4)
E [E [1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817 | 1003817100381710038171003817119883119894
1003817100381710038171003817 = 119903]] = E [119886119894 10038171003817100381710038171198831198941003817100381710038171003817 + 119887119894] (B5)
Equation (B4) is derived from (2) and (B5) from ourassumption on 119888119894(119903) 119886119894 is then set as a function of 119887119894 E[119883119894]and 120582119890 119887119894 is obtained from our simulations For a givenintensity 119887119894 has been observed almost constant with regardto 119894 The best approximation as a function of the intensity isgiven by 119887119894 = 033radic120582119890 After a fewmanipulations we obtain
First note that by convention we have P(119908119894 = 1) = P(119908119894 =1 | 1199080 = 1) The event 119908119894 = 1 occurs if and only if it exists119895 ge 1 such that1198770 + 119894minus1sum
There is no data used in this paper Nevertheless the code ofour simulator is made available (the link is in the paper)
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] ldquoRequirements for further advancements for e-utra (lte-advanced) 3gpp technical specification ts 36913 release 13rdquo2015 httpwww3gpporg
[2] A S Hamza S S Khalifa H S Hamza and K Elsayed ldquoAsurvey on inter-cell interference coordination techniques inofdma-based cellular networksrdquo IEEE Communications Surveysamp Tutorials vol 15 no 4 pp 1642ndash1670 2013
[3] D Lopez-Perez A Ladanyi A JuttnerH Rivano and J ZhangldquoOptimization method for the joint allocation of modulationschemes coding rates resource blocks and power in self-organizing LTE networksrdquo in Proceedings of the IEEE INFO-COM 2011 pp 111ndash115 April 2011
[4] A Guo andM Haenggi ldquoSpatial stochasticmodels andmetricsfor the structure of base stations in cellular networksrdquo IEEE
Wireless Communications and Mobile Computing 15
Transactions on Wireless Communications vol 12 no 11 pp5800ndash5812 2013
[5] C-H Lee C-Y Shih and Y-S Chen ldquoStochastic geometrybased models for modeling cellular networks in urban areasrdquoWireless Networks vol 19 no 6 pp 1063ndash1072 2013
[6] W Lu and M D Renzo ldquoStochastic geometry modeling ofcellular networks analysis simulation and experimental valida-tionrdquo in Proceedings of the 18th ACM International Conferenceon Modelling Analysis and Simulation of Wireless and MobileSystems (MSWiM rsquo15) pp 179ndash188 New York NY USANovember 2015
[7] Y L Lee T C Chuah J Loo and A Vinel ldquoRecent advancesin radio resource management for heterogeneous LTELTE-Anetworksrdquo IEEE Communications Surveys amp Tutorials vol 16no 4 pp 2142ndash2180 2014
[8] F Mhiri K Sethom and R Bouallegue ldquoA survey on inter-ference management techniques in femtocell self-organizingnetworksrdquo Journal of Network and Computer Applications vol36 no 1 pp 58ndash65 2013
[9] X Chen L Li and X Xiang ldquoAnt colony learning methodfor joint MCS and resource block allocation in LTE Femtocelldownlink for multimedia applications with QoS guaranteesrdquoMultimedia Tools andApplications vol 76 no 3 pp 4035ndash40542017
[10] D Lopez-Perez X Chu A V Vasilakos and H Claussen ldquoOndistributed and coordinated resource allocation for interferencemitigation in self-organizing lte networksrdquo IEEEACMTransac-tions on Networking vol 21 no 4 pp 1145ndash1158 2013
[11] H Zhang H Liu J Cheng and V C M Leung ldquoDownlinkenergy efficiency of power allocation and wireless backhaulbandwidth allocation in heterogeneous small cell networksrdquoIEEE Transactions on Communications no 99 2017
[12] H Zhang W Zheng X Chu et al ldquoJoint subchannel andpower allocation in interference-limited OFDMA femtocellswith heterogeneous QoS guaranteerdquo in Proceedings of theIEEEGlobal Communications Conference (GLOBECOM rsquo12) pp4572ndash4577 December 2012
[13] H Elsawy E Hossain and M Haenggi ldquoStochastic geometryfor modeling analysis and design of multi-tier and cognitivecellular wireless networks a surveyrdquo IEEE CommunicationsSurveys amp Tutorials vol 15 no 3 pp 996ndash1019 2013
[14] H ElSawy A Sultan-Salem M-S Alouini and M Z WinldquoModeling and analysis of cellular networks using stochasticgeometry a tutorialrdquo IEEE Communications Surveys amp Tutori-als vol 19 no 1 pp 167ndash203 2017
[15] R Kwan and C Leung ldquoA survey of scheduling and inter-ference mitigation in lterdquo Journal of Electrical and ComputerEngineeringmdashSpecial Issue on LTELTE-AdvancedCellular Com-munication Networks pp 11ndash110 2010
[16] J G Andrews F Baccelli and R K Ganti ldquoA tractable approachto coverage and rate in cellular networksrdquo IEEE Transactions onCommunications vol 59 no 11 pp 3122ndash3134 2011
[17] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for OFDMA cellu-lar networksrdquo IEEE Transactions on Wireless Communicationsvol 10 no 12 pp 4294ndash4305 2011
[18] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for heterogeneouscellular networksrdquo IEEE Transactions on Communications vol60 no 7 pp 2029ndash2039 2012
[19] H Zhuang and T Ohtsuki ldquoA model based on poisson pointprocess for downlink K tiers fractional frequency reuse hetero-geneous networksrdquo Physical Communication vol 13 pp 3ndash122014 Special Issue on Heterogeneous and Small Cell Networks
[20] A A Gebremariam T Bao D Siracusa T Rasheed F Granelliand L Goratti ldquoDynamic strict fractional frequency reusefor software-defined 5G networksrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo16) pp 1ndash6May 2016
[21] N Lee D Morales-Jimenez A Lozano and R W HeathldquoSpectral efficiency of dynamic coordinated beamforming astochastic geometry approachrdquo IEEE Transactions on WirelessCommunications vol 14 no 1 pp 230ndash241 2015
[22] X Zhang and M Haenggi ldquoA Stochastic geometry analysisof inter-cell interference coordination and intra-cell diversityrdquoIEEE Transactions on Wireless Communications vol 13 no 12pp 6655ndash6669 2014
[23] F Baccelli andAGiovanidis ldquoA stochastic geometry frameworkfor analyzing pairwise-cooperative cellular networksrdquo IEEETransactions on Wireless Communications vol 14 no 2 pp794ndash808 2015
[24] J Yoon and G Hwang ldquoDistance-based inter-cell interferencecoordination in small cell networks stochastic geometry mod-eling and analysisrdquo IEEE Transactions on Wireless Communica-tions 2018
[25] B Błaszczyszyn M Jovanovicy and M K Karray ldquoHow userthroughput depends on the traffic demand in large cellularnetworksrdquo in Proceedings of the 12th International Symposiumon Modeling and Optimization in Mobile Ad Hoc and WirelessNetworks (WiOpt rsquo14) pp 611ndash619 May 2014
[26] H H Yang and T Q S Quek ldquoSIR coverage analysis incellular networks with temporal traffic a stochastic geometryapproachrdquo Corrossion 2018
[27] S M Yu and S-L Kim ldquoDownlink capacity and base stationdensity in cellular networksrdquo in Proceedings of the 11th Interna-tional SymposiumandWorkshops onModeling andOptimizationin Mobile Ad Hoc and Wireless Networks (WiOpt rsquo13) pp 119ndash124 May 2013
[28] YWang M Haenggi and Z Tan ldquoThemeta distribution of thesir for cellular networks with power controlrdquo Corrossion 2017
[29] M S Omar S A Hassan H Pervaiz et al ldquoMulti-objectiveoptimization in 5g hybrid networksrdquo IEEE Internet of ThingsJournal 2018
[30] Y Leo A Busson C Sarraute and E Fleury ldquoCall detail recordsto characterize usages and mobility events of phone usersrdquoComputer Communications vol 95 pp 43ndash53 2016
[31] I L Cherif Spectral and Energy Efficiency in 5G WirelessNetworks [PhD thesis] Univeristy Paris Saclay 2016
[32] S Mukherjee Analytical Modeling of Heterogeneous CellularNetworks Analytical Modeling of Heterogeneous Cellular Net-works Geometry Coverage and Capacity Cambridge Univer-sity Press 2014
[33] ldquoTechnical specification group radio access network base sta-tion (bs) radio transmission and reception 2012 Release 1123gpp ts 36104 v112rdquo 2012
(a) SIR empirical and theoretical CDF for the119872119872119877119861119898119886119909119877119861119898119886119909allocation
MMRB Delta=3MMRB Delta=6MMRB Delta=9MI Delta=3 MI Delta=6
MI Delta=9 MD Delta=3 MD Delta=6 MD Delta=9
0 10 20 30 40minus10훽 (dB)
00e+00
20eminus01
40eminus01
60eminus01
80eminus01
10e+00
CDF
minus SI
NR
(b) SIR empirical CDF for the three strategies MI maximizeinterference MD maximize distance
Figure 6 (a) We compare the distribution of SIR obtained by simulations (solid lines) and given by (29) (dotted lines) Errors with regardto the 119871+infin norm are 247119890 minus 02 154119890 minus 02 and 158119890 minus 02 for Δ = 3 6 and 9 respectively (b) CDF for the three allocation strategies(119872119872119877119861119898119886119909119877119861119898119886119909 and Heuristics)
(c) 119872119872119877119861119898119886119909119877119861119898119886119909 and the two heuristics Δ = 6 120582119890 = 225
Figure 7 Modulation and coding rate for the119872119872119877119861119898119886119909119877119861119898119886119909 allocation strategy when the density of eNodeB increases (Figure 7(a))for different reuse factor (Figure 7(b)) and comparison with the two heuristics (Figure 7(c))
variables are strongly correlated particularly for the points ofthe process close to the origin This correlation impacts thetransmission power used by the eNodeB and consequentlyinterference and SIR distributions The derivation of the SIRdistribution allows us to express the classical outage but alsothe transmission rates in terms of modulationcoding ratethat provide interesting insights on the throughput offers tothe users Also it appears that eNodeB densification does not
significantly improve the network performance in terms ofthroughput except for a small percentage of users for whichthe transmission rate is increased Spatial reuse expressedthrough the parameter Δ in our study has a much moreimpact on the performance as the transmission rates aresignificantly increased This spatial reuse must be expressedas the ratio between the total number of available resourcesover the number of usersrequests per cell Indeed we have
Wireless Communications and Mobile Computing 13
observed that the system performance is quite insensitive tovariation of these quantities when this ratio stays constant
Beside simulations show that the RB assignment strate-gies developed for the model give results close to the onesgiven by classical optimization problem which minimizesglobal interference or maximizes spatial reuse distances Itempirically shows that our analytical model is able to evaluateperformance of classical optimization approaches but at largescale in terms of number of nodes configurations andtopologies
Appendix
A Construction of the Markov Chain
In this section we introduce the Markov chain used to set thedistribution of 1198770 We consider that the RB used between thetypical user and its eNodeB has index 119903119890119904 (119903119890119904 isin 0 119877119861119898119886119909minus1)We shall prove that this index does not impact calculation(we have chosen 119903119890119904 = 0 in this paper by sack of simplicity)
We set the distribution of the number of used resourceblocks at eNodeB 0 in such a way that it leads to the followingproperty (119895 gt 119894 ge 0)
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (A1)
In order to obtain this property we first introduce somepreliminary notations and results An example of allocationwith the different notations is shown in Figure 1 We define asequence of rv (119877119899)119899ge0 If eNodeB 119894 is the 119899119905ℎ eNodeB using119903119890119904 and if 119864119899119889119899 is the index of the last RB used by this eNodeBthen
119877119899 = 119864119899119889119899 minus 119903119890119904 if 119864119899119889119899 gt 119903119890119904119864119899119889119899 + 119877119861119898119886119909 minus 119903119890119904 otherwise (A2)
If eNodeB 119894 is the 119899119905ℎ interfering eNodeB (using 119903119890119904) theneNodeB 119895 (119895 gt 119894) is the 119899 + 1119905ℎ if and only if
119863119896 ge 119877119861119898119886119909 (A4)
In this case 119877119899+1 is given by
119877119899+1 = 119877119899 +119895sum
119896=119894+1
119863119896 minus 119877119861119898119886119909 (A5)
As (119863119894)119894gt0 are iid the sequence (119877119899)119899ge0 is a homoge-neous Markov chain The transition probabilities for (119897 119898) isin0 1 119877119861119898119886119909 minus 12 are given by 119875119897119898 = P(119877119899+1 = 119898 | 119877119899 = 119897)
The event 119877119899+1 = 119898 given that 119877119899 = 119897 occurs if andonly if it exists 119895 ge 0 such that
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (A10)
We set the distribution of 1198770 with the correspondingstationary distribution Consequently the sequence (119877119899)119899ge0becomes identically distributed where 119877119899 follows the sta-tionary distribution for all 119899 ge 0 Moreover according to(A3) and (A4) the property given in (A1) holds Indeed 119877119899sum119895minus1
119896=119894+1119863119896 and sum119895
119896=119894+1119863119896 have the same distribution as 1198770sum119895minus119894minus1
119896=1119863119896 andsum119895minus119894
119896=1119863119896 respectively In the model evaluation
the stationary distribution is obtained numerically from thetransition probabilities As soon as 1198770 follows the stationarydistribution the index 119903119890119904 can be chosen arbitrarily
B Conditional Distribution of 119880119894minus119883119894Simulations have shown that given 119883119894 the distribution of119883119894 minus 119880119894 is still close to the one given by (2) but with adifferent parameter So we assume that the distribution of119883119894 minus 119880119894 given 119883119894 is equal to (2) but with a parameterfunction of 119883119894 denoted 119888(119883119894)
These simulations have also shown that E[119883119894 minus 119880119894 |119883119894 = 119903]may be approximated as an affine function 119886119894119903 + 119887119894We define 119888119894(sdot) accordingly It leads to
To set 119886119894 and 119887119894 we use the two following propertiesE [E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817 | 10038171003817100381710038171198831198941003817100381710038171003817 = 119903]] = E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817] (B3)
= 12radic120582119890 (B4)
E [E [1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817 | 1003817100381710038171003817119883119894
1003817100381710038171003817 = 119903]] = E [119886119894 10038171003817100381710038171198831198941003817100381710038171003817 + 119887119894] (B5)
Equation (B4) is derived from (2) and (B5) from ourassumption on 119888119894(119903) 119886119894 is then set as a function of 119887119894 E[119883119894]and 120582119890 119887119894 is obtained from our simulations For a givenintensity 119887119894 has been observed almost constant with regardto 119894 The best approximation as a function of the intensity isgiven by 119887119894 = 033radic120582119890 After a fewmanipulations we obtain
First note that by convention we have P(119908119894 = 1) = P(119908119894 =1 | 1199080 = 1) The event 119908119894 = 1 occurs if and only if it exists119895 ge 1 such that1198770 + 119894minus1sum
There is no data used in this paper Nevertheless the code ofour simulator is made available (the link is in the paper)
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] ldquoRequirements for further advancements for e-utra (lte-advanced) 3gpp technical specification ts 36913 release 13rdquo2015 httpwww3gpporg
[2] A S Hamza S S Khalifa H S Hamza and K Elsayed ldquoAsurvey on inter-cell interference coordination techniques inofdma-based cellular networksrdquo IEEE Communications Surveysamp Tutorials vol 15 no 4 pp 1642ndash1670 2013
[3] D Lopez-Perez A Ladanyi A JuttnerH Rivano and J ZhangldquoOptimization method for the joint allocation of modulationschemes coding rates resource blocks and power in self-organizing LTE networksrdquo in Proceedings of the IEEE INFO-COM 2011 pp 111ndash115 April 2011
[4] A Guo andM Haenggi ldquoSpatial stochasticmodels andmetricsfor the structure of base stations in cellular networksrdquo IEEE
Wireless Communications and Mobile Computing 15
Transactions on Wireless Communications vol 12 no 11 pp5800ndash5812 2013
[5] C-H Lee C-Y Shih and Y-S Chen ldquoStochastic geometrybased models for modeling cellular networks in urban areasrdquoWireless Networks vol 19 no 6 pp 1063ndash1072 2013
[6] W Lu and M D Renzo ldquoStochastic geometry modeling ofcellular networks analysis simulation and experimental valida-tionrdquo in Proceedings of the 18th ACM International Conferenceon Modelling Analysis and Simulation of Wireless and MobileSystems (MSWiM rsquo15) pp 179ndash188 New York NY USANovember 2015
[7] Y L Lee T C Chuah J Loo and A Vinel ldquoRecent advancesin radio resource management for heterogeneous LTELTE-Anetworksrdquo IEEE Communications Surveys amp Tutorials vol 16no 4 pp 2142ndash2180 2014
[8] F Mhiri K Sethom and R Bouallegue ldquoA survey on inter-ference management techniques in femtocell self-organizingnetworksrdquo Journal of Network and Computer Applications vol36 no 1 pp 58ndash65 2013
[9] X Chen L Li and X Xiang ldquoAnt colony learning methodfor joint MCS and resource block allocation in LTE Femtocelldownlink for multimedia applications with QoS guaranteesrdquoMultimedia Tools andApplications vol 76 no 3 pp 4035ndash40542017
[10] D Lopez-Perez X Chu A V Vasilakos and H Claussen ldquoOndistributed and coordinated resource allocation for interferencemitigation in self-organizing lte networksrdquo IEEEACMTransac-tions on Networking vol 21 no 4 pp 1145ndash1158 2013
[11] H Zhang H Liu J Cheng and V C M Leung ldquoDownlinkenergy efficiency of power allocation and wireless backhaulbandwidth allocation in heterogeneous small cell networksrdquoIEEE Transactions on Communications no 99 2017
[12] H Zhang W Zheng X Chu et al ldquoJoint subchannel andpower allocation in interference-limited OFDMA femtocellswith heterogeneous QoS guaranteerdquo in Proceedings of theIEEEGlobal Communications Conference (GLOBECOM rsquo12) pp4572ndash4577 December 2012
[13] H Elsawy E Hossain and M Haenggi ldquoStochastic geometryfor modeling analysis and design of multi-tier and cognitivecellular wireless networks a surveyrdquo IEEE CommunicationsSurveys amp Tutorials vol 15 no 3 pp 996ndash1019 2013
[14] H ElSawy A Sultan-Salem M-S Alouini and M Z WinldquoModeling and analysis of cellular networks using stochasticgeometry a tutorialrdquo IEEE Communications Surveys amp Tutori-als vol 19 no 1 pp 167ndash203 2017
[15] R Kwan and C Leung ldquoA survey of scheduling and inter-ference mitigation in lterdquo Journal of Electrical and ComputerEngineeringmdashSpecial Issue on LTELTE-AdvancedCellular Com-munication Networks pp 11ndash110 2010
[16] J G Andrews F Baccelli and R K Ganti ldquoA tractable approachto coverage and rate in cellular networksrdquo IEEE Transactions onCommunications vol 59 no 11 pp 3122ndash3134 2011
[17] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for OFDMA cellu-lar networksrdquo IEEE Transactions on Wireless Communicationsvol 10 no 12 pp 4294ndash4305 2011
[18] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for heterogeneouscellular networksrdquo IEEE Transactions on Communications vol60 no 7 pp 2029ndash2039 2012
[19] H Zhuang and T Ohtsuki ldquoA model based on poisson pointprocess for downlink K tiers fractional frequency reuse hetero-geneous networksrdquo Physical Communication vol 13 pp 3ndash122014 Special Issue on Heterogeneous and Small Cell Networks
[20] A A Gebremariam T Bao D Siracusa T Rasheed F Granelliand L Goratti ldquoDynamic strict fractional frequency reusefor software-defined 5G networksrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo16) pp 1ndash6May 2016
[21] N Lee D Morales-Jimenez A Lozano and R W HeathldquoSpectral efficiency of dynamic coordinated beamforming astochastic geometry approachrdquo IEEE Transactions on WirelessCommunications vol 14 no 1 pp 230ndash241 2015
[22] X Zhang and M Haenggi ldquoA Stochastic geometry analysisof inter-cell interference coordination and intra-cell diversityrdquoIEEE Transactions on Wireless Communications vol 13 no 12pp 6655ndash6669 2014
[23] F Baccelli andAGiovanidis ldquoA stochastic geometry frameworkfor analyzing pairwise-cooperative cellular networksrdquo IEEETransactions on Wireless Communications vol 14 no 2 pp794ndash808 2015
[24] J Yoon and G Hwang ldquoDistance-based inter-cell interferencecoordination in small cell networks stochastic geometry mod-eling and analysisrdquo IEEE Transactions on Wireless Communica-tions 2018
[25] B Błaszczyszyn M Jovanovicy and M K Karray ldquoHow userthroughput depends on the traffic demand in large cellularnetworksrdquo in Proceedings of the 12th International Symposiumon Modeling and Optimization in Mobile Ad Hoc and WirelessNetworks (WiOpt rsquo14) pp 611ndash619 May 2014
[26] H H Yang and T Q S Quek ldquoSIR coverage analysis incellular networks with temporal traffic a stochastic geometryapproachrdquo Corrossion 2018
[27] S M Yu and S-L Kim ldquoDownlink capacity and base stationdensity in cellular networksrdquo in Proceedings of the 11th Interna-tional SymposiumandWorkshops onModeling andOptimizationin Mobile Ad Hoc and Wireless Networks (WiOpt rsquo13) pp 119ndash124 May 2013
[28] YWang M Haenggi and Z Tan ldquoThemeta distribution of thesir for cellular networks with power controlrdquo Corrossion 2017
[29] M S Omar S A Hassan H Pervaiz et al ldquoMulti-objectiveoptimization in 5g hybrid networksrdquo IEEE Internet of ThingsJournal 2018
[30] Y Leo A Busson C Sarraute and E Fleury ldquoCall detail recordsto characterize usages and mobility events of phone usersrdquoComputer Communications vol 95 pp 43ndash53 2016
[31] I L Cherif Spectral and Energy Efficiency in 5G WirelessNetworks [PhD thesis] Univeristy Paris Saclay 2016
[32] S Mukherjee Analytical Modeling of Heterogeneous CellularNetworks Analytical Modeling of Heterogeneous Cellular Net-works Geometry Coverage and Capacity Cambridge Univer-sity Press 2014
[33] ldquoTechnical specification group radio access network base sta-tion (bs) radio transmission and reception 2012 Release 1123gpp ts 36104 v112rdquo 2012
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Wireless Communications and Mobile Computing 13
observed that the system performance is quite insensitive tovariation of these quantities when this ratio stays constant
Beside simulations show that the RB assignment strate-gies developed for the model give results close to the onesgiven by classical optimization problem which minimizesglobal interference or maximizes spatial reuse distances Itempirically shows that our analytical model is able to evaluateperformance of classical optimization approaches but at largescale in terms of number of nodes configurations andtopologies
Appendix
A Construction of the Markov Chain
In this section we introduce the Markov chain used to set thedistribution of 1198770 We consider that the RB used between thetypical user and its eNodeB has index 119903119890119904 (119903119890119904 isin 0 119877119861119898119886119909minus1)We shall prove that this index does not impact calculation(we have chosen 119903119890119904 = 0 in this paper by sack of simplicity)
We set the distribution of the number of used resourceblocks at eNodeB 0 in such a way that it leads to the followingproperty (119895 gt 119894 ge 0)
P (119908119895 = 1 | 119908119894 = 1) = P (119908119895minus119894 = 1 | 1199080 = 1) (A1)
In order to obtain this property we first introduce somepreliminary notations and results An example of allocationwith the different notations is shown in Figure 1 We define asequence of rv (119877119899)119899ge0 If eNodeB 119894 is the 119899119905ℎ eNodeB using119903119890119904 and if 119864119899119889119899 is the index of the last RB used by this eNodeBthen
119877119899 = 119864119899119889119899 minus 119903119890119904 if 119864119899119889119899 gt 119903119890119904119864119899119889119899 + 119877119861119898119886119909 minus 119903119890119904 otherwise (A2)
If eNodeB 119894 is the 119899119905ℎ interfering eNodeB (using 119903119890119904) theneNodeB 119895 (119895 gt 119894) is the 119899 + 1119905ℎ if and only if
119863119896 ge 119877119861119898119886119909 (A4)
In this case 119877119899+1 is given by
119877119899+1 = 119877119899 +119895sum
119896=119894+1
119863119896 minus 119877119861119898119886119909 (A5)
As (119863119894)119894gt0 are iid the sequence (119877119899)119899ge0 is a homoge-neous Markov chain The transition probabilities for (119897 119898) isin0 1 119877119861119898119886119909 minus 12 are given by 119875119897119898 = P(119877119899+1 = 119898 | 119877119899 = 119897)
The event 119877119899+1 = 119898 given that 119877119899 = 119897 occurs if andonly if it exists 119895 ge 0 such that
1205870 120588119877119861119898119886119909+119898minus119906minus119897(119877119861119898119886119909 + 119898 minus 119906 minus 119897)1205871198950
(119895120588)119906119906 (A10)
We set the distribution of 1198770 with the correspondingstationary distribution Consequently the sequence (119877119899)119899ge0becomes identically distributed where 119877119899 follows the sta-tionary distribution for all 119899 ge 0 Moreover according to(A3) and (A4) the property given in (A1) holds Indeed 119877119899sum119895minus1
119896=119894+1119863119896 and sum119895
119896=119894+1119863119896 have the same distribution as 1198770sum119895minus119894minus1
119896=1119863119896 andsum119895minus119894
119896=1119863119896 respectively In the model evaluation
the stationary distribution is obtained numerically from thetransition probabilities As soon as 1198770 follows the stationarydistribution the index 119903119890119904 can be chosen arbitrarily
B Conditional Distribution of 119880119894minus119883119894Simulations have shown that given 119883119894 the distribution of119883119894 minus 119880119894 is still close to the one given by (2) but with adifferent parameter So we assume that the distribution of119883119894 minus 119880119894 given 119883119894 is equal to (2) but with a parameterfunction of 119883119894 denoted 119888(119883119894)
These simulations have also shown that E[119883119894 minus 119880119894 |119883119894 = 119903]may be approximated as an affine function 119886119894119903 + 119887119894We define 119888119894(sdot) accordingly It leads to
To set 119886119894 and 119887119894 we use the two following propertiesE [E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817 | 10038171003817100381710038171198831198941003817100381710038171003817 = 119903]] = E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817] (B3)
= 12radic120582119890 (B4)
E [E [1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817 | 1003817100381710038171003817119883119894
1003817100381710038171003817 = 119903]] = E [119886119894 10038171003817100381710038171198831198941003817100381710038171003817 + 119887119894] (B5)
Equation (B4) is derived from (2) and (B5) from ourassumption on 119888119894(119903) 119886119894 is then set as a function of 119887119894 E[119883119894]and 120582119890 119887119894 is obtained from our simulations For a givenintensity 119887119894 has been observed almost constant with regardto 119894 The best approximation as a function of the intensity isgiven by 119887119894 = 033radic120582119890 After a fewmanipulations we obtain
First note that by convention we have P(119908119894 = 1) = P(119908119894 =1 | 1199080 = 1) The event 119908119894 = 1 occurs if and only if it exists119895 ge 1 such that1198770 + 119894minus1sum
There is no data used in this paper Nevertheless the code ofour simulator is made available (the link is in the paper)
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] ldquoRequirements for further advancements for e-utra (lte-advanced) 3gpp technical specification ts 36913 release 13rdquo2015 httpwww3gpporg
[2] A S Hamza S S Khalifa H S Hamza and K Elsayed ldquoAsurvey on inter-cell interference coordination techniques inofdma-based cellular networksrdquo IEEE Communications Surveysamp Tutorials vol 15 no 4 pp 1642ndash1670 2013
[3] D Lopez-Perez A Ladanyi A JuttnerH Rivano and J ZhangldquoOptimization method for the joint allocation of modulationschemes coding rates resource blocks and power in self-organizing LTE networksrdquo in Proceedings of the IEEE INFO-COM 2011 pp 111ndash115 April 2011
[4] A Guo andM Haenggi ldquoSpatial stochasticmodels andmetricsfor the structure of base stations in cellular networksrdquo IEEE
Wireless Communications and Mobile Computing 15
Transactions on Wireless Communications vol 12 no 11 pp5800ndash5812 2013
[5] C-H Lee C-Y Shih and Y-S Chen ldquoStochastic geometrybased models for modeling cellular networks in urban areasrdquoWireless Networks vol 19 no 6 pp 1063ndash1072 2013
[6] W Lu and M D Renzo ldquoStochastic geometry modeling ofcellular networks analysis simulation and experimental valida-tionrdquo in Proceedings of the 18th ACM International Conferenceon Modelling Analysis and Simulation of Wireless and MobileSystems (MSWiM rsquo15) pp 179ndash188 New York NY USANovember 2015
[7] Y L Lee T C Chuah J Loo and A Vinel ldquoRecent advancesin radio resource management for heterogeneous LTELTE-Anetworksrdquo IEEE Communications Surveys amp Tutorials vol 16no 4 pp 2142ndash2180 2014
[8] F Mhiri K Sethom and R Bouallegue ldquoA survey on inter-ference management techniques in femtocell self-organizingnetworksrdquo Journal of Network and Computer Applications vol36 no 1 pp 58ndash65 2013
[9] X Chen L Li and X Xiang ldquoAnt colony learning methodfor joint MCS and resource block allocation in LTE Femtocelldownlink for multimedia applications with QoS guaranteesrdquoMultimedia Tools andApplications vol 76 no 3 pp 4035ndash40542017
[10] D Lopez-Perez X Chu A V Vasilakos and H Claussen ldquoOndistributed and coordinated resource allocation for interferencemitigation in self-organizing lte networksrdquo IEEEACMTransac-tions on Networking vol 21 no 4 pp 1145ndash1158 2013
[11] H Zhang H Liu J Cheng and V C M Leung ldquoDownlinkenergy efficiency of power allocation and wireless backhaulbandwidth allocation in heterogeneous small cell networksrdquoIEEE Transactions on Communications no 99 2017
[12] H Zhang W Zheng X Chu et al ldquoJoint subchannel andpower allocation in interference-limited OFDMA femtocellswith heterogeneous QoS guaranteerdquo in Proceedings of theIEEEGlobal Communications Conference (GLOBECOM rsquo12) pp4572ndash4577 December 2012
[13] H Elsawy E Hossain and M Haenggi ldquoStochastic geometryfor modeling analysis and design of multi-tier and cognitivecellular wireless networks a surveyrdquo IEEE CommunicationsSurveys amp Tutorials vol 15 no 3 pp 996ndash1019 2013
[14] H ElSawy A Sultan-Salem M-S Alouini and M Z WinldquoModeling and analysis of cellular networks using stochasticgeometry a tutorialrdquo IEEE Communications Surveys amp Tutori-als vol 19 no 1 pp 167ndash203 2017
[15] R Kwan and C Leung ldquoA survey of scheduling and inter-ference mitigation in lterdquo Journal of Electrical and ComputerEngineeringmdashSpecial Issue on LTELTE-AdvancedCellular Com-munication Networks pp 11ndash110 2010
[16] J G Andrews F Baccelli and R K Ganti ldquoA tractable approachto coverage and rate in cellular networksrdquo IEEE Transactions onCommunications vol 59 no 11 pp 3122ndash3134 2011
[17] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for OFDMA cellu-lar networksrdquo IEEE Transactions on Wireless Communicationsvol 10 no 12 pp 4294ndash4305 2011
[18] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for heterogeneouscellular networksrdquo IEEE Transactions on Communications vol60 no 7 pp 2029ndash2039 2012
[19] H Zhuang and T Ohtsuki ldquoA model based on poisson pointprocess for downlink K tiers fractional frequency reuse hetero-geneous networksrdquo Physical Communication vol 13 pp 3ndash122014 Special Issue on Heterogeneous and Small Cell Networks
[20] A A Gebremariam T Bao D Siracusa T Rasheed F Granelliand L Goratti ldquoDynamic strict fractional frequency reusefor software-defined 5G networksrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo16) pp 1ndash6May 2016
[21] N Lee D Morales-Jimenez A Lozano and R W HeathldquoSpectral efficiency of dynamic coordinated beamforming astochastic geometry approachrdquo IEEE Transactions on WirelessCommunications vol 14 no 1 pp 230ndash241 2015
[22] X Zhang and M Haenggi ldquoA Stochastic geometry analysisof inter-cell interference coordination and intra-cell diversityrdquoIEEE Transactions on Wireless Communications vol 13 no 12pp 6655ndash6669 2014
[23] F Baccelli andAGiovanidis ldquoA stochastic geometry frameworkfor analyzing pairwise-cooperative cellular networksrdquo IEEETransactions on Wireless Communications vol 14 no 2 pp794ndash808 2015
[24] J Yoon and G Hwang ldquoDistance-based inter-cell interferencecoordination in small cell networks stochastic geometry mod-eling and analysisrdquo IEEE Transactions on Wireless Communica-tions 2018
[25] B Błaszczyszyn M Jovanovicy and M K Karray ldquoHow userthroughput depends on the traffic demand in large cellularnetworksrdquo in Proceedings of the 12th International Symposiumon Modeling and Optimization in Mobile Ad Hoc and WirelessNetworks (WiOpt rsquo14) pp 611ndash619 May 2014
[26] H H Yang and T Q S Quek ldquoSIR coverage analysis incellular networks with temporal traffic a stochastic geometryapproachrdquo Corrossion 2018
[27] S M Yu and S-L Kim ldquoDownlink capacity and base stationdensity in cellular networksrdquo in Proceedings of the 11th Interna-tional SymposiumandWorkshops onModeling andOptimizationin Mobile Ad Hoc and Wireless Networks (WiOpt rsquo13) pp 119ndash124 May 2013
[28] YWang M Haenggi and Z Tan ldquoThemeta distribution of thesir for cellular networks with power controlrdquo Corrossion 2017
[29] M S Omar S A Hassan H Pervaiz et al ldquoMulti-objectiveoptimization in 5g hybrid networksrdquo IEEE Internet of ThingsJournal 2018
[30] Y Leo A Busson C Sarraute and E Fleury ldquoCall detail recordsto characterize usages and mobility events of phone usersrdquoComputer Communications vol 95 pp 43ndash53 2016
[31] I L Cherif Spectral and Energy Efficiency in 5G WirelessNetworks [PhD thesis] Univeristy Paris Saclay 2016
[32] S Mukherjee Analytical Modeling of Heterogeneous CellularNetworks Analytical Modeling of Heterogeneous Cellular Net-works Geometry Coverage and Capacity Cambridge Univer-sity Press 2014
[33] ldquoTechnical specification group radio access network base sta-tion (bs) radio transmission and reception 2012 Release 1123gpp ts 36104 v112rdquo 2012
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
14 Wireless Communications and Mobile Computing
These simulations have also shown that E[119883119894 minus 119880119894 |119883119894 = 119903]may be approximated as an affine function 119886119894119903 + 119887119894We define 119888119894(sdot) accordingly It leads to
To set 119886119894 and 119887119894 we use the two following propertiesE [E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817 | 10038171003817100381710038171198831198941003817100381710038171003817 = 119903]] = E [1003817100381710038171003817119883119894 minus 119880119894
1003817100381710038171003817] (B3)
= 12radic120582119890 (B4)
E [E [1003817100381710038171003817119883119894 minus 1198801198941003817100381710038171003817 | 1003817100381710038171003817119883119894
1003817100381710038171003817 = 119903]] = E [119886119894 10038171003817100381710038171198831198941003817100381710038171003817 + 119887119894] (B5)
Equation (B4) is derived from (2) and (B5) from ourassumption on 119888119894(119903) 119886119894 is then set as a function of 119887119894 E[119883119894]and 120582119890 119887119894 is obtained from our simulations For a givenintensity 119887119894 has been observed almost constant with regardto 119894 The best approximation as a function of the intensity isgiven by 119887119894 = 033radic120582119890 After a fewmanipulations we obtain
First note that by convention we have P(119908119894 = 1) = P(119908119894 =1 | 1199080 = 1) The event 119908119894 = 1 occurs if and only if it exists119895 ge 1 such that1198770 + 119894minus1sum
There is no data used in this paper Nevertheless the code ofour simulator is made available (the link is in the paper)
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] ldquoRequirements for further advancements for e-utra (lte-advanced) 3gpp technical specification ts 36913 release 13rdquo2015 httpwww3gpporg
[2] A S Hamza S S Khalifa H S Hamza and K Elsayed ldquoAsurvey on inter-cell interference coordination techniques inofdma-based cellular networksrdquo IEEE Communications Surveysamp Tutorials vol 15 no 4 pp 1642ndash1670 2013
[3] D Lopez-Perez A Ladanyi A JuttnerH Rivano and J ZhangldquoOptimization method for the joint allocation of modulationschemes coding rates resource blocks and power in self-organizing LTE networksrdquo in Proceedings of the IEEE INFO-COM 2011 pp 111ndash115 April 2011
[4] A Guo andM Haenggi ldquoSpatial stochasticmodels andmetricsfor the structure of base stations in cellular networksrdquo IEEE
Wireless Communications and Mobile Computing 15
Transactions on Wireless Communications vol 12 no 11 pp5800ndash5812 2013
[5] C-H Lee C-Y Shih and Y-S Chen ldquoStochastic geometrybased models for modeling cellular networks in urban areasrdquoWireless Networks vol 19 no 6 pp 1063ndash1072 2013
[6] W Lu and M D Renzo ldquoStochastic geometry modeling ofcellular networks analysis simulation and experimental valida-tionrdquo in Proceedings of the 18th ACM International Conferenceon Modelling Analysis and Simulation of Wireless and MobileSystems (MSWiM rsquo15) pp 179ndash188 New York NY USANovember 2015
[7] Y L Lee T C Chuah J Loo and A Vinel ldquoRecent advancesin radio resource management for heterogeneous LTELTE-Anetworksrdquo IEEE Communications Surveys amp Tutorials vol 16no 4 pp 2142ndash2180 2014
[8] F Mhiri K Sethom and R Bouallegue ldquoA survey on inter-ference management techniques in femtocell self-organizingnetworksrdquo Journal of Network and Computer Applications vol36 no 1 pp 58ndash65 2013
[9] X Chen L Li and X Xiang ldquoAnt colony learning methodfor joint MCS and resource block allocation in LTE Femtocelldownlink for multimedia applications with QoS guaranteesrdquoMultimedia Tools andApplications vol 76 no 3 pp 4035ndash40542017
[10] D Lopez-Perez X Chu A V Vasilakos and H Claussen ldquoOndistributed and coordinated resource allocation for interferencemitigation in self-organizing lte networksrdquo IEEEACMTransac-tions on Networking vol 21 no 4 pp 1145ndash1158 2013
[11] H Zhang H Liu J Cheng and V C M Leung ldquoDownlinkenergy efficiency of power allocation and wireless backhaulbandwidth allocation in heterogeneous small cell networksrdquoIEEE Transactions on Communications no 99 2017
[12] H Zhang W Zheng X Chu et al ldquoJoint subchannel andpower allocation in interference-limited OFDMA femtocellswith heterogeneous QoS guaranteerdquo in Proceedings of theIEEEGlobal Communications Conference (GLOBECOM rsquo12) pp4572ndash4577 December 2012
[13] H Elsawy E Hossain and M Haenggi ldquoStochastic geometryfor modeling analysis and design of multi-tier and cognitivecellular wireless networks a surveyrdquo IEEE CommunicationsSurveys amp Tutorials vol 15 no 3 pp 996ndash1019 2013
[14] H ElSawy A Sultan-Salem M-S Alouini and M Z WinldquoModeling and analysis of cellular networks using stochasticgeometry a tutorialrdquo IEEE Communications Surveys amp Tutori-als vol 19 no 1 pp 167ndash203 2017
[15] R Kwan and C Leung ldquoA survey of scheduling and inter-ference mitigation in lterdquo Journal of Electrical and ComputerEngineeringmdashSpecial Issue on LTELTE-AdvancedCellular Com-munication Networks pp 11ndash110 2010
[16] J G Andrews F Baccelli and R K Ganti ldquoA tractable approachto coverage and rate in cellular networksrdquo IEEE Transactions onCommunications vol 59 no 11 pp 3122ndash3134 2011
[17] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for OFDMA cellu-lar networksrdquo IEEE Transactions on Wireless Communicationsvol 10 no 12 pp 4294ndash4305 2011
[18] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for heterogeneouscellular networksrdquo IEEE Transactions on Communications vol60 no 7 pp 2029ndash2039 2012
[19] H Zhuang and T Ohtsuki ldquoA model based on poisson pointprocess for downlink K tiers fractional frequency reuse hetero-geneous networksrdquo Physical Communication vol 13 pp 3ndash122014 Special Issue on Heterogeneous and Small Cell Networks
[20] A A Gebremariam T Bao D Siracusa T Rasheed F Granelliand L Goratti ldquoDynamic strict fractional frequency reusefor software-defined 5G networksrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo16) pp 1ndash6May 2016
[21] N Lee D Morales-Jimenez A Lozano and R W HeathldquoSpectral efficiency of dynamic coordinated beamforming astochastic geometry approachrdquo IEEE Transactions on WirelessCommunications vol 14 no 1 pp 230ndash241 2015
[22] X Zhang and M Haenggi ldquoA Stochastic geometry analysisof inter-cell interference coordination and intra-cell diversityrdquoIEEE Transactions on Wireless Communications vol 13 no 12pp 6655ndash6669 2014
[23] F Baccelli andAGiovanidis ldquoA stochastic geometry frameworkfor analyzing pairwise-cooperative cellular networksrdquo IEEETransactions on Wireless Communications vol 14 no 2 pp794ndash808 2015
[24] J Yoon and G Hwang ldquoDistance-based inter-cell interferencecoordination in small cell networks stochastic geometry mod-eling and analysisrdquo IEEE Transactions on Wireless Communica-tions 2018
[25] B Błaszczyszyn M Jovanovicy and M K Karray ldquoHow userthroughput depends on the traffic demand in large cellularnetworksrdquo in Proceedings of the 12th International Symposiumon Modeling and Optimization in Mobile Ad Hoc and WirelessNetworks (WiOpt rsquo14) pp 611ndash619 May 2014
[26] H H Yang and T Q S Quek ldquoSIR coverage analysis incellular networks with temporal traffic a stochastic geometryapproachrdquo Corrossion 2018
[27] S M Yu and S-L Kim ldquoDownlink capacity and base stationdensity in cellular networksrdquo in Proceedings of the 11th Interna-tional SymposiumandWorkshops onModeling andOptimizationin Mobile Ad Hoc and Wireless Networks (WiOpt rsquo13) pp 119ndash124 May 2013
[28] YWang M Haenggi and Z Tan ldquoThemeta distribution of thesir for cellular networks with power controlrdquo Corrossion 2017
[29] M S Omar S A Hassan H Pervaiz et al ldquoMulti-objectiveoptimization in 5g hybrid networksrdquo IEEE Internet of ThingsJournal 2018
[30] Y Leo A Busson C Sarraute and E Fleury ldquoCall detail recordsto characterize usages and mobility events of phone usersrdquoComputer Communications vol 95 pp 43ndash53 2016
[31] I L Cherif Spectral and Energy Efficiency in 5G WirelessNetworks [PhD thesis] Univeristy Paris Saclay 2016
[32] S Mukherjee Analytical Modeling of Heterogeneous CellularNetworks Analytical Modeling of Heterogeneous Cellular Net-works Geometry Coverage and Capacity Cambridge Univer-sity Press 2014
[33] ldquoTechnical specification group radio access network base sta-tion (bs) radio transmission and reception 2012 Release 1123gpp ts 36104 v112rdquo 2012
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Wireless Communications and Mobile Computing 15
Transactions on Wireless Communications vol 12 no 11 pp5800ndash5812 2013
[5] C-H Lee C-Y Shih and Y-S Chen ldquoStochastic geometrybased models for modeling cellular networks in urban areasrdquoWireless Networks vol 19 no 6 pp 1063ndash1072 2013
[6] W Lu and M D Renzo ldquoStochastic geometry modeling ofcellular networks analysis simulation and experimental valida-tionrdquo in Proceedings of the 18th ACM International Conferenceon Modelling Analysis and Simulation of Wireless and MobileSystems (MSWiM rsquo15) pp 179ndash188 New York NY USANovember 2015
[7] Y L Lee T C Chuah J Loo and A Vinel ldquoRecent advancesin radio resource management for heterogeneous LTELTE-Anetworksrdquo IEEE Communications Surveys amp Tutorials vol 16no 4 pp 2142ndash2180 2014
[8] F Mhiri K Sethom and R Bouallegue ldquoA survey on inter-ference management techniques in femtocell self-organizingnetworksrdquo Journal of Network and Computer Applications vol36 no 1 pp 58ndash65 2013
[9] X Chen L Li and X Xiang ldquoAnt colony learning methodfor joint MCS and resource block allocation in LTE Femtocelldownlink for multimedia applications with QoS guaranteesrdquoMultimedia Tools andApplications vol 76 no 3 pp 4035ndash40542017
[10] D Lopez-Perez X Chu A V Vasilakos and H Claussen ldquoOndistributed and coordinated resource allocation for interferencemitigation in self-organizing lte networksrdquo IEEEACMTransac-tions on Networking vol 21 no 4 pp 1145ndash1158 2013
[11] H Zhang H Liu J Cheng and V C M Leung ldquoDownlinkenergy efficiency of power allocation and wireless backhaulbandwidth allocation in heterogeneous small cell networksrdquoIEEE Transactions on Communications no 99 2017
[12] H Zhang W Zheng X Chu et al ldquoJoint subchannel andpower allocation in interference-limited OFDMA femtocellswith heterogeneous QoS guaranteerdquo in Proceedings of theIEEEGlobal Communications Conference (GLOBECOM rsquo12) pp4572ndash4577 December 2012
[13] H Elsawy E Hossain and M Haenggi ldquoStochastic geometryfor modeling analysis and design of multi-tier and cognitivecellular wireless networks a surveyrdquo IEEE CommunicationsSurveys amp Tutorials vol 15 no 3 pp 996ndash1019 2013
[14] H ElSawy A Sultan-Salem M-S Alouini and M Z WinldquoModeling and analysis of cellular networks using stochasticgeometry a tutorialrdquo IEEE Communications Surveys amp Tutori-als vol 19 no 1 pp 167ndash203 2017
[15] R Kwan and C Leung ldquoA survey of scheduling and inter-ference mitigation in lterdquo Journal of Electrical and ComputerEngineeringmdashSpecial Issue on LTELTE-AdvancedCellular Com-munication Networks pp 11ndash110 2010
[16] J G Andrews F Baccelli and R K Ganti ldquoA tractable approachto coverage and rate in cellular networksrdquo IEEE Transactions onCommunications vol 59 no 11 pp 3122ndash3134 2011
[17] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for OFDMA cellu-lar networksrdquo IEEE Transactions on Wireless Communicationsvol 10 no 12 pp 4294ndash4305 2011
[18] T D Novlan R K Ganti A Ghosh and J G Andrews ldquoAnalyt-ical evaluation of fractional frequency reuse for heterogeneouscellular networksrdquo IEEE Transactions on Communications vol60 no 7 pp 2029ndash2039 2012
[19] H Zhuang and T Ohtsuki ldquoA model based on poisson pointprocess for downlink K tiers fractional frequency reuse hetero-geneous networksrdquo Physical Communication vol 13 pp 3ndash122014 Special Issue on Heterogeneous and Small Cell Networks
[20] A A Gebremariam T Bao D Siracusa T Rasheed F Granelliand L Goratti ldquoDynamic strict fractional frequency reusefor software-defined 5G networksrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo16) pp 1ndash6May 2016
[21] N Lee D Morales-Jimenez A Lozano and R W HeathldquoSpectral efficiency of dynamic coordinated beamforming astochastic geometry approachrdquo IEEE Transactions on WirelessCommunications vol 14 no 1 pp 230ndash241 2015
[22] X Zhang and M Haenggi ldquoA Stochastic geometry analysisof inter-cell interference coordination and intra-cell diversityrdquoIEEE Transactions on Wireless Communications vol 13 no 12pp 6655ndash6669 2014
[23] F Baccelli andAGiovanidis ldquoA stochastic geometry frameworkfor analyzing pairwise-cooperative cellular networksrdquo IEEETransactions on Wireless Communications vol 14 no 2 pp794ndash808 2015
[24] J Yoon and G Hwang ldquoDistance-based inter-cell interferencecoordination in small cell networks stochastic geometry mod-eling and analysisrdquo IEEE Transactions on Wireless Communica-tions 2018
[25] B Błaszczyszyn M Jovanovicy and M K Karray ldquoHow userthroughput depends on the traffic demand in large cellularnetworksrdquo in Proceedings of the 12th International Symposiumon Modeling and Optimization in Mobile Ad Hoc and WirelessNetworks (WiOpt rsquo14) pp 611ndash619 May 2014
[26] H H Yang and T Q S Quek ldquoSIR coverage analysis incellular networks with temporal traffic a stochastic geometryapproachrdquo Corrossion 2018
[27] S M Yu and S-L Kim ldquoDownlink capacity and base stationdensity in cellular networksrdquo in Proceedings of the 11th Interna-tional SymposiumandWorkshops onModeling andOptimizationin Mobile Ad Hoc and Wireless Networks (WiOpt rsquo13) pp 119ndash124 May 2013
[28] YWang M Haenggi and Z Tan ldquoThemeta distribution of thesir for cellular networks with power controlrdquo Corrossion 2017
[29] M S Omar S A Hassan H Pervaiz et al ldquoMulti-objectiveoptimization in 5g hybrid networksrdquo IEEE Internet of ThingsJournal 2018
[30] Y Leo A Busson C Sarraute and E Fleury ldquoCall detail recordsto characterize usages and mobility events of phone usersrdquoComputer Communications vol 95 pp 43ndash53 2016
[31] I L Cherif Spectral and Energy Efficiency in 5G WirelessNetworks [PhD thesis] Univeristy Paris Saclay 2016
[32] S Mukherjee Analytical Modeling of Heterogeneous CellularNetworks Analytical Modeling of Heterogeneous Cellular Net-works Geometry Coverage and Capacity Cambridge Univer-sity Press 2014
[33] ldquoTechnical specification group radio access network base sta-tion (bs) radio transmission and reception 2012 Release 1123gpp ts 36104 v112rdquo 2012
