KEY WORDS: power control; scheduling; TD/CDMA; multihop wireless ad hoc network
1. Introduction
Wireless ad hoc networks have been the topic of exten-sive research recently. The interests in such networksare due to their ability to provide wireless networking
*Correspondence to: Lijun Qian, CeBCom Research Center, Electrical Engineering Department, Prairie View A&M University,Prairie View, TX 77446, U.S.A.†E-mail: [email protected]
Contract/grant sponsor: U.S. Army Research Office; contract/grant number: W911NF-04-2-0054.
capability in scenarios where no fixed wired infrastruc-ture is available (e.g., disaster relief efforts, battlefields,etc.). The lack of fixed infrastructure introduces greatdesign challenges. One way to reduce the difficulty isby organizing nodes into clusters and assigning certain
nodes management functions [1], such as transmissioncoordination. These nodes are called cluster heads.It has been shown that proper clustering in wirelessad hoc networks reduces the complexity of link-layerand routing protocol design significantly and improvesthe scalability of the protocols [2]. In addition, clus-tering increases the network capability of supportingQuality-of-Service (QoS) [3]. Clustering is alsodesirable because of practical reasons. For instance,in a battlefield deployment, a cluster may be naturallyformed by a set of soldiers equipped with wireless com-munication devices and a tank serving as cluster head.
In order to resolve the issue of low end-to-endthroughput in a multihop ad hoc network, innovativeMedium Access Control (MAC) protocols are indis-pensable. In wireless ad hoc networks, MAC protocolsplay a critical role in optimizing bandwidth efficiencyand ensuring QoS due to the broadcast nature ofwireless channels. There are mainly three types ofMAC protocols, namely random access with collisionresolution based schemes (such as CSMA/CA),reservation based schemes (such as D-PRMA [4]),and signal orthogonalization based schemes (such asTDMA or CDMA). For a thorough review of the MACprotocols for wireless ad hoc networks, see for exampleReference [5] and the references therein. Due to theirpoor scalability in a multihop ad hoc network, randomaccess protocols are not an efficient solution [6]. InReference [7], it is demonstrated that CDMA-basedMAC protocols achieve a significant increase innetwork throughput at no additional cost in energyconsumption compared to 802.11x MAC protocols.
In this research work, we restrict our interestsin clustered TD/CDMA wireless ad hoc networks.It is assumed that the wireless ad hoc network isorganized into clusters and each cluster has a clusterhead with higher than average network resources suchas power. All users/nodes within the cluster share thesame frequency band and each user/node is assigneda randomly generated orthogonal code. On top ofthat, time is split into equal-sized slots where onlyscheduled users/nodes are allowed to transmit in eachslot. The cluster head functions as a manager and isresponsible for scheduling the transmissions withinthe cluster. It is assumed that the communication linksamong cluster heads (inter-cluster communications)have sufficient bandwidth such that the bottleneck ofend-to-end traffic between nodes in different clustersresides within clusters. Hence, scheduling intra-clustertransmissions is the main concern in this paper.
An example of a clustered TD/CDMA wirelessad hoc network is shown in Figure 1. There are two clus-ters with cluster heads CH1 and CH2, respectively. It isassumed that the intra-cluster route is given for a trafficsession: rI = A → E → G → F → CH1. Data traf-fic is forwarded in a multihop fashion. Figure 1 alsoshows a schedule for intra-cluster traffic transmissions.
Power control is employed in a wireless ad hocnetwork to control transmission range and keep thenetwork fully connected [8]. It is a physical layer func-tion. However, transmission power has a direct impacton multiple access of nodes by affecting receivedsignal-to-interference ratio (SIR) at receivers. Hence,power control is strongly coupled with scheduling
Fig. 1. A clustered TD/CDMA wireless ad hoc network. CH, cluster head.
POWER CONTROL AND SCHEDULING IN MULTIHOP TD/CDMA WIRELESS NETWORKS 793
and has additional functions of reducing unnecessaryinterference among concurrent transmissions inTD/CDMA-based systems [9]. Power control andscheduling is of paramount importance for ensuringthe success of multiple simultaneous transmissionsand maximizing network utility in TD/CDMA wirelessad hoc networks, and is the focus of this paper. Inthis work, we are interested in traffic sessions withminimum rate constraints. The goal is to study powercontrol and scheduling schemes that maximize certainnetwork utility functions while guarantee minimumrate of traffic sessions, given the routes and minimumrate constraints of those sessions. Although theproposed power control and scheduling schemes focuson intra-cluster traffic transmissions where a centralcontroller (cluster head) is available, fully distributedversions of schemes are also developed for scenarioswhere no central controller is available.
The rest of the paper is organized as follows: Sec-tion 2 presents an overview of the works that are closelyrelated to our problem. Section 3 states the wirelessnetwork model and formulates the joint power controland scheduling problem with QoS constraints. Both op-timal solution and low complexity approximations areproposed, together with several algorithms that serve aslower bounds. The proposed algorithms are evaluatedby extensive discrete-event simulations in Section 4.Distributed schemes and other various implementationissues are discussed in Section 5. Section 6 containsthe concluding remarks.
2. Related Works
A power control and scheduling problem has beensolved in Reference [10] for TDMA ad hoc networkson a per frame basis and each link is assigned to a num-ber of slots in a given frame. The authors assume thateach slot has fixed data rate. Using the concept of vir-tual links, assigning one slot to each virtual link satis-fies the end-to-end session rate requirements. The jointfeasibility problem is proven to be NP-complete andcentralized approximation algorithms are provided. Inour study, we assume variable data rate from slot to slotdue to channel fluctuations.
A centralized joint routing, scheduling, and powercontrol problem is formulated for TD/CDMA ad hocnetworks and an approximation algorithm is derivedin Reference [11]. However, a simplified interferencemodel is adopted, where no interference is assumedamong different links. In Reference [12], a centralizedjoint routing, scheduling and power control problem
is solved for multihop base stations where data rate isassumed to be a linear function of SIR (in low SIRregime). In our work, a general interference model isadopted, where each transmitting node in the networkis assumed to cause interference at any receiving nodes,even if they are far apart. The data rate is calculated asa concave function of the SIR, which covers the entirerange of SIR.
The authors in Reference [13] proposed a joint powercontrol and scheduling scheme based on a utility func-tion of instantaneous power or instantaneous data rate.A degree-based greedy scheduling and an iterativepower control algorithm using a penalty function ap-proach are suggested to maximize the utility functionwhile guarantee minimum and maximum link datarates. The algorithm in Reference [13] focused on asnapshot of a set of wireless links. Another work oninstantaneous power control in wireless ad hoc net-works is Reference [9]. In this study, we focus on long-term average data rate and minimum average data raterequirements for traffic sessions in a routed wirelessad hoc network.
A randomized policy is given to solve the multi-commodity flow problem given the long-term linkcapacity as weight in wireless networks [14]. Thena dynamic policy (throughput-optimal policy) isproposed for unknown arrival and channel statisticsand is proven to perform better than the randomizedpolicy. The dynamic policy is a non-linear andnon-convex optimization problem that is very difficultto solve. A steepest ascent search is suggested as asub-optimal centralized solution. As pointed out laterin this paper, throughput-optimal policies maximizethe effective rate of data flows. However, no fairnessamong users/flows is addressed in such policies. Inaddition, no minimum rate constraint is considered inReference [14]. In this paper, both throughput-optimaland proportional-fair (PF) scheduling algorithms aretaken into consideration. Furthermore, a token countermechanism is introduced to maintain minimum rate oftraffic flows whenever feasible. A distributed approx-imation is also proposed in Reference [14] assumingthat the link gains between a node and its neighborsare known. Our proposed distributed algorithm usescontrol channel to exchange link gain information. Inthe simulation of Reference [14], the link gains arecalculated only based on distances between nodes. Nofading is considered and the locations of nodes are as-sumed to be known. In our simulation study, channel ismodeled to have both shadowing and rayleigh fading.
Power allocation and scheduling has been exten-sively studied for WLAN. In Reference [15], a fully
distributed algorithm for scheduling packet transmis-sions is proposed such that different flows are allocatedbandwidth in proportion of their weights. The paperproposes a distributed fair scheduling (DFS) approachobtained by modifying the distributed coordinationfunction (DCF) in IEEE 802.11 standard. A fairscheduling mechanism, distributed elastic round robin(DERR) is proposed in Reference [16]. DERR issuitable for IEEE 802.11 wireless LANs operated inthe ad hoc mode and capable of avoiding collisionsthrough a random mapping between allowance andIFS. DERR outperforms 802.11e in terms of delayand throughput. In Reference [17], an enhanced timer-based scheduling control algorithm is proposed toeffectively manage the delay budget in IEEE 802.11e.Simulation results show that the proposed algorithmoutperforms the simple scheduler control algorithm indelay and jitter in infrastructure mode. Although thereare a lot of work on power allocation and schedulingfor WLAN, most of them studied infrastructure modeand focused on random access part (DCF) in 802.11.Furthermore, to our best knowledge, very few of thepapers considered ad hoc mode and none of themconsidered multihop scenarios.
3. Joint Power Control and SchedulingWith Minimum Rate Constraints
In this paper, we assume that the routes for the multipleend-to-end traffic sessions are given. All the links con-tained in the routes form the set of ‘active links.’ Eachactive link is uniquely identified by its transmitter andreceiver. In other words, transmitter i and receiver i arethe transmitter and receiver of active link i. The re-ceived SIR at the ith receiver from the ith transmitter(received SIR of the ith active link) is defined by
γi = hiipi
1L
∑j �=i hijpj + σ2
(1)
where hii is the link gain from transmitter i to its des-ignated receiver i. hij is the link gain from transmitterj to receiver i (active link i’s designated receiver). pi
and pj are the transmission power of transmitters i andj, respectively. σ2 is the background (receiver) noise.L is the spreading gain for spread spectrum systems.
In this paper, we assume that each link has variablerate. This rate is bounded by the feasible rate region.The link gains (channel quality) may fluctuate dramat-ically from one slot to another slot. In other words, thedata rates of the active links are different from slot toslot during the traffic sessions. A scheduling scheme
should take advantage of channel fluctuations, that is,it should be ‘channel-aware.’
The instantaneous data rate of each active link canbe evaluated by Shannon capacity formula (for AWGNchannel)
Ri = Wi log2(1 + γi) (2)
where Wi is the bandwidth occupied by the transmis-sion from the ith transmitter to its designated receiver.Note that this formula gives the achievable rate (upperbound) of the AWGN channel. However, it is justifiedby the fact that with the current modulation and cod-ing technology it can be closely approximated in mostpractical scenarios [18].
The interference model adopted here assumes thateach transmitting node in the network causes interfer-ence at any receiving nodes, even if they are far apart.This model is considered more realistic than the onewhich assumes that transmitting nodes only cause in-terference to their neighbors. This is because the aggre-gate interference from a large number of nodes may notbe negligible even if interference from each of them issmall. The instantaneous data rate will be determinedsolely by the received SIR.
3.1. Problem Formulation
In this work, we will focus on end-to-end traffic ses-sions with QoS constraints. Specifically, we are inter-ested in traffic sessions with minimum rate constraints.The goal of this research work is to study joint powercontrol and scheduling schemes that maximize certainutility functions while guarantee minimum rate of traf-fic sessions, given the routes and minimum rate con-straints of those sessions. A guarantee on minimum rateis arguably the simplest possible QoS guarantee. There-fore, we believe it is natural that mobile users wouldexpect such an assurance. Other reasons of ensuringminimum rate are
(1) Some applications need a minimum rate in orderto perform well. For example, streaming audio andvideo can become unusable if the data rate is toolow.
(2) Even for static TCP-based applications such as webbrowsing, if the data rate is too low then we typ-ically get a large queue buildup which can leadto TCP timeouts and poor performance. Such ef-fects were discussed by Chakravorty et al. in Ref-erence [19].
(3) Providing a minimum rate guarantee can help tosmooth out the effects of a variable wireless chan-nel.
POWER CONTROL AND SCHEDULING IN MULTIHOP TD/CDMA WIRELESS NETWORKS 795
(4) Providing a minimum rate can allow us to ensurethat a slot-based TD/CDMA service is no worsethan circuit-based data systems such as wirelinedialup or 3G1X wireless service.
(5) By setting minimum data rate differently for dif-ferent users we can ensure service differentiation.
Given the routes of multiple end-to-end traffic ses-sions with minimum rate constraints, our approach fol-lows the Gradient algorithm with Minimum/MaximumRate constraints (GMR) developed in Reference [20].Let’s define the long-term average rate vector R =(R1, . . . , RN ) assuming that there are N active linksresulted from routing, and each of the active link hasminimum rate constraint (Ri
min). The joint power con-trol and scheduling problem is formulated as the fol-lowing optimization problem(P.1)
maxR∈R,p∈P
U(R) (3)
subject to
Ri ≥ Rimin (4)
where the instantaneous rate is determined by Equa-tions (1) and (2). R is the rate region, which is the setof long-term service rate vectors which the system iscapable of providing. P is the set of allowable powervector defined by
pi ≤ pmaxi ∀ i (5)
where pmaxi is the maximum allowable transmission
power of transmitter i. The utility function is of theform
U(R) =∑
i
Ui(Ri) (6)
where each Ui(x) is an increasing concave continuouslydifferentiable function defined for x ≥ 0.
A node cannot transmit and receive simultaneously.This primary conflict [11] is resolved by setting the linkgain matrix appropriately. For example, if node i is se-lected to transmit in the current slot, the correspondinglink gains where node i is the receiver will be set tozero.
The multihop nature of the problem (P.1) will beincluded in the choices of the utility functions as well.For example, queue backlog weighted average rate at
each node will be included in the utility function forthroughput optimal criterion, which is directly relatedto the paths resulted from routing [14,21]. In otherwords, the order of the transmissions is implicitlyincluded in the problem formulation. It also reflectsin the fact that the links on the same route require thesame minimum rate whereas links on different routestypically have different minimum rate requirements.
3.2. Centralized Solution
Before introducing the Multi-link Gradient algorithmwith Minimum Rate constraints (MGMR) to solve theoptimization problem (P.1), we observe some usefulproperties of the optimal solution.
3.2.1. Optimal power control
Theorem 1. The optimal scheme has the propertythat each transmitting node transmits at full power, thatis, pi = pmax
i for some subsetS of the nodes and pi = 0for the complementary set S.
Proof. Assume that there are N active links(transmitter-receiver pairs) in the network. Let prcv
ii andprcv
ij be the instantaneous received power at receiver i
from transmitter i and j, respectively. For simplicity, weexpress prcv
ii and prcvij in units of the background noise
σ2. In order to meet the minimum rate constraints ofall active links, we must have for each active link i
prcvii
1L
∑j∈{1,2,...,N},j �=i p
rcvij + 1
≥ γ tari (7)
where γ tari is the required SIR of link i. If a desired data
flow rate is specified by a certain application, say, Rtari ,
then γ tari can be expressed as
γ tari ≥ 2
Rtari
Wi − 1, i = 1, 2, . . . , N (8)
The feasible SIR vectors specified in Equation (7) isadapted from that in the cellular wireless networks tomultihop wireless networks. Given the peak receivedpower p
A given SIR vector is feasible if Equation (9) can besatisfied with equality with 0 ≤ θij ≤ 1 for all i andj. We hence examine the solution to the set of linearequations
θiiprcv,maxii
γ tari
= 1
L
∑j∈{1,2,...,N},j �=i
θijprcv,maxij + 1 (10)
which can be further rewritten as
θiiprcv,maxii
(1 + L
γ tari
)=
∑j∈{1,2,···,N}
θijprcv,maxij + L
(11)
It can be seen by inspection that the solution is of theform θiip
rcv,maxii (1 + L
γ tari
) = C where C is a global pa-
rameter. The value of C can be obtained by substitut-ing the postulated solution in Equation (11) to obtain
C = C∑
j
γ tarj
γ tarj +L
+ L which gives the final solution
θii = γ tari
(L + γ tari )prcv,max
ii
L[1 − ∑
j∈{1,2,···,N}γ tarj
γ tarj +L
]
(12)
Define αi = γ tari
γ tari +L
we see that
θii = Lαi/prcv,maxii
1 − ∑j αj
(13)
Clearly, 0 ≤ αi < 1. Since we require 0 ≤ θii ≤ 1,Equation (13) results in the following feasibility con-ditions to meet the required SIRs.
∑j
αj + Lαi
prcv,maxii
≤ 1 ∀i (14)
Note the simple linear form of the feasible SIRs in termsof the αi.
Note that the utility function Ui(Ri) is a concavefunction and Ri is a linear combination of the instanta-neous data rate Ri. The instantaneous data rate is againa concave function of the SIR. Since Equation (14) islinear in αi, it is more convenient to consider the R,α relationship which is now convex. The optimizationproblem (P.1) becomes
maxR∈R,p∈P
∑i
Ui(Ri) (15)
subject to
∑j
αj + Lαi
prcv,maxii
≤ 1, αi > 0 ∀i (16)
Equation (16) specifies 2N constraints on the feasibleαi. From standard theorems on convex maximizationwith linear constraints, it is easy to see that the opti-mum occurs at corner point of Equation (16) due tothe joint-convexity of Equation (15) in the αi. Cornerpoints of Equation (16) have exactly N of the 2N con-straints binding, that is, some subset of the αi are null,while the complementary set saturate their respectiveconstraints in the first equation of (16). Combining thisobservation with Equation (13) results in θii = 1 forthe complementary set, thus proving the theorem. �
Note that similar observations are obtainedunder various different contexts and assump-tions [12,14,22,23]. Specifically, the results reportedin Reference [12] may be viewed as a special case ofthe above theorem where the data rate is assumed alinear function of SIR instead of the more general formthat adopted in this paper. Theorem 1 reveals the bang-bang characteristics of the nodes’ transmission powerin order to maximize the network’s utility. In each timeslot, selected transmitting nodes will use the maximumtransmission power, while other nodes remain silent.
3.2.2. Scheduling algorithms
As highlighted by Theorem 1, the joint power con-trol and scheduling problem is reduced to a schedul-ing problem given the bang-bang characteristics of theoptimal transmission power. We address the schedul-ing problem by first reviewing two important types ofscheduling algorithms. Then the Multi-link Gradientscheduling algorithm with Minimum Rate constraints(MGMR) will be proposed to solve the optimizationproblem (P.1).
One type of scheduling algorithm considered inthis paper is the throughput-optimal scheduling, suchas the scheduling algorithms proposed in Refer-ences [14,21,24,25], where a weighted sum of userrates is maximized for each scheduling interval. Thischoice has provable stability properties shown inmuch previous work in various contexts involvingdata scheduling and resource allocation. The weightsmay be chosen to optimize one of many possibleperformance measures, including average queuelength, delay, or corresponding percentiles, and othersimilar criteria. A version of this type of algorithms
POWER CONTROL AND SCHEDULING IN MULTIHOP TD/CDMA WIRELESS NETWORKS 797
that guarantees queue stability, that is, boundednessof queue lengths when feasible, is specified as the ratechoice that satisfies
R∗ = arg maxR∈R
Q · R
where R, Q are rate and queue vectors of the userset respectively, and R is the rate region, or theset of feasible rate vectors. Minimum/maximuminstantaneous rate guarantees may be satisfied byrestricting the rate regionR appropriately.
Another type of scheduling algorithms consideredin this paper is the fair scheduling, such as the propor-tional fair scheduling proposed in References [26,27]and further analyzed in References [28,29]. PF schedul-ing algorithm was proposed and implemented by Qual-Comm for 3G1X EVDO (HDR) downlink. PF algo-rithm provides fairness among users such that in thelong run each user receives the same number of timeslots of services. At the same time, PF also takes advan-tage of channel variations (user diversity). However,since PF schedules users one-at-a-time, it needs to bemodified for a multihop scenario.
A review of throughput-optimal and proportionalfair scheduling is given in Reference [30]. The authorsalso proposed a combination of throughput-optimalscheduling and congestion control for cellular systems.
In this paper, we are interested in proposing andstudying the multi-link version of the throughput-optimal and the PF algorithms for multihop wirelessad hoc networks, called multi-link throughput optimal(MQR) and multi-link proportional fair (MPF), respec-tively. We are particularly interested in their modifiedversions that accommodate QoS constraints requiredby multiple traffic sessions. MQR and MPF are modi-fied to satisfy minimum rate constraints using a tokencounter mechanism inspired by the scheme developedfor cellular systems [20], thus they are named multi-link throughput optimal with minimum rate (MQRMR)and multi-link proportional fair with minimum rate(MPFMR), respectively.
We now formulate the MGMR, which seeks to solvethe optimization problem (P.1).
MGMR. In a time slot k, select the active links
arg maxR∈R
∑i
eaiTi(k)U ′i(Ri(k))Ri(k) (17)
where Ri(k) is the current average service rate receivedby link i, Ti(k) is a ‘token counter’ for link i, and ai > 0is a parameter. The values of average rate Ri are updated
as in the PF algorithm [26,27]:
Ri(k + 1) = (1 − β)Ri(k) + βRi(k)
where β > 0 is a small fixed parameter, and Ri(k) isthe instantaneous data rate if link i was actually servedin slot k and Ri(k) = 0 otherwise. The token counterTi is updated as follows:
Ti(k + 1) = max{
0, Ti(k) + Rimin − Ri(k)
}(18)
We prove the optimality of the MGMR algorithm bystudying the dynamics of user throughputs and tokencounters under the MGMR algorithm when parametersβ and ai are small. Namely, we consider the asymptoticregime such that β converges to 0, and each ai = βαi
with some fixed αi > 0. We study the dynamics offluid sample paths (FSP), which are possible trajec-tories (r(t), τ(t)) of a random process which is a limitof the process (R(t/β), βT (t/β)) as β → 0. (Thus, r(t)approximates the behavior of the vector of throughputsR(t) when β is small and we ‘speed-up’ time by the fac-tor 1/β; τ(t) approximates the vector T (t) scaled downby factor β, and with 1/β time speed-up.) The main re-sult is a ‘necessary throughput convergence’ conditionstated in the following theorem:
Theorem 2. Suppose FSP (r, τ) is such that
r(t) → R∗ as t → ∞
and τ(t) remains uniformly bounded for all t ≥ 0. Then,R∗ is a solution to the problem (P.1) and, moreover,R∗ ∈ Rcond ∩R∗ �= ∅.
Rate region R is a convex closed bounded polyhe-dron in the positive orthant. ByR∗ we denote the subsetof maximal elements of R: namely, v ∈ R∗ if condi-tions v ≤ u (component wise) and u ∈ R imply u = v.Clearly, R∗ is a part of the outer boundary of R. ThesubsetRcond ⊆ R of elements v ∈ R satisfying condi-tions Rmin
i ≤ vi ≤ Rmaxi for all i, is also a convex closed
bounded set.The proof of Theorem 2 follows that in Reference
[20], and it is given in Reference [31]. Theorem 2 saysthat if FSP is such that the vector of throughputs r(t)converges to some vector R
∗ as t → ∞, then R∗ is nec-
essarily a solution to the problem (P.1). This impliesthat if the user throughputs converge, then the corre-sponding stationary throughputs do in fact maximizethe desired utility function, subject to the minimumrate constraints.
The token counter Ti provides the key mechanismtrying to ensure that the active link i received (longterm) service rate stays above Ri
min. The dynamics ofthe token counter process Ti(k) (see Equation (18)) isbriefly described and interpreted as follows. There isa virtual ‘token queue’ corresponding to each flow i.The tokens ‘arrive in the (token) queue’ (i.e., Ti is in-cremented) at the rate Ri
min per slot. If active link i isserved in slot k, then Ri(k) tokens are ‘removed fromthe queue’ (i.e., Ti is decremented). Thus, if in a certaintime interval, the average service rate of flow i is lessthan Ri
min, the token queue size Ti has ‘positive drift,’and therefore, the chances of flow i being served in eachtime slot gradually increase. If the average service rateof flow i stays close to Ri
min, Ti will stay around zeroand will not affect scheduling decisions.
The special cases of the MGMR algorithm (based ondifferent choices of the utility function, U(R) ) includethe following
(1) MPFMR algorithm: The multi-link proportionalfair with minimum rate (MPFMR) algorithm corre-sponding to utility functions U(R) = ∑
i log(Ri),and the scheduling rule is
arg maxR∈R
∑i
eaiTi(k) Ri(k)
Ri(k)(19)
(2) MMTMR algorithm: The multi-link maximumthroughput with minimum rate (MMTMR) algo-rithm corresponding to utility functions U(R) =∑
i Ri, and the scheduling rule is
arg maxR∈R
∑i
eaiTi(k)Ri(k) (20)
(3) MQRMR algorithm: The multi-link throughputoptimal with minimum rate (MQRMR) algo-rithm corresponding to utility functions U(R) =∑
i QiRi, where Qi is the queue backlog at thetransmitter of link i, and the scheduling rule is
arg maxR∈R
∑i
eaiTi(k)QiRi(k) (21)
In this study, we also considered a set of schedulingalgorithms that solve a similar optimization problemas (P.1), however, without the minimum rate constraint(Equation (4)). The resulted special cases are
(1) MPF algorithm: The multi-link proportional fair(MPF) algorithm corresponding to utility functionsU(R) = ∑
(3) MQR algorithm: The multi-link throughput op-timal (MQR) algorithm corresponding to utilityfunctions U(R) = ∑
i QiRi, where Qi is the queuebacklog at the transmitter of link i, and the schedul-ing rule is
arg maxR∈R
∑i
QiRi(k) (24)
3.3. Low Complexity Approximations
In this part, we attempt to provide a greedy, low-complexity, approximate solution to the optimizationproblem (P.1) discussed before. The optimal solutionneeds to sort all the possible combinations of activelinks. In order to run the scheduler in real-time, lowcomplexity approximations are needed. We hence pro-pose the following simple scheduling scheme (greedyalgorithms that rank active links by their respectivemeasure) that may be more suitable for practical im-plementation.
3.3.1. Greedy algorithms
In each time slot
(1) Create a list by sorting active links in decreasingorder of the measure vi assuming no interferencefrom other active links while computing R0
i .(2) Add active link j, in order starting from the top of
the list, while maintaining and updating the valueof Φ = ∑
i≤j vi, where Ri now takes into accountinterference from all added active links.
(3) Stop if adding the next active link reduces Φ, andallow transmission of all added active links at theirpeak powers and rates as computed.
POWER CONTROL AND SCHEDULING IN MULTIHOP TD/CDMA WIRELESS NETWORKS 799
Table I. Scheduling algorithms for TD/CDMA wireless ad hoc networks.
Throughput-optimal Proportional fair
Multi-link algorithms Without min rate MQR MPFWith min rate MQRMR MPFMR
One-at-a-time algorithms Without min rate QR PFWith min rate QRMR PFMR
Implementation Queue backlog needed Average rate neededComments Session rates maximized. Take advantage of diversity and
No fairness considered guarantee long-term fairness
The measure vi for different algorithms are vi =eaiTi
R0i
Ri, for MPFMR; vi = eaiTiR0
i , for MMTMR; vi =eaiTiQiR
0i , for MQRMR; vi = R0
i
Ri, for MPF; vi = R0
i ,
for MMT; vi = QiR0i , for MQR.
We also considered several algorithms that will serveone active link in each time slot. These algorithms serveas the lower bound for performance comparison.
3.3.2. One-at-a-time algorithms
Create a list by sorting active links in decreasing or-der of the measure vi assuming no interference fromother active links while computing R0
i . Serve the top on
the list. vi = R0i
Ri, for PF; vi = R0
i , for MT; vi = QiR0i ,
for QR; vi = eaiTiR0
i
Ri, for PFMR; vi = eaiTiR0
i , for
MTMR; vi = eaiTiQiR0i , for QRMR.
The various scheduling algorithms considered in thispaper are summarized in Table I.
4. Performance Evaluation
One benchmark algorithm is the optimal (centralized)MGMR algorithm given in the previous section. It givesthe best possible performance. Other benchmark al-gorithms are the one-at-a-time algorithms, which willserve as lower bounds. We will compare with these al-gorithms to evaluate the gains of different optimal/sub-optimal multi-link algorithms. Round Robin and fully
simultaneous transmission are considered too far fromoptimal and perform very poorly in most of the cases,and are thus ignored here.
4.1. Simulation Setup
In order to quantify the performance gain by applyingoptimal/sub-optimal scheduling algorithms, discrete-event simulations using OPNET have been performedto evaluate them in multihop TD/CDMA wirelessad hoc networks. Networks of two types of topologiesand corresponding routing configurations are tested,see Figures 2 (linear topology) and 3 (network withcrossover traffic). It is assumed that routes are given forfixed destinations and marked with arrows in the Fig-ures. There is one route (rI ) for destination node F inthe linear network. There are three routes (rII , rIII , andrIV ) for destination nodes L, J, K, respectively. Thelinks on the routes are indexed with numerical numbers.
The routing setups represent important scenarios inmultihop wireless ad hoc networks. The linear modelis considered as the simplest case of relaying traffic se-quentially and represents intra-cluster traffic to a fixeddestination (cluster head). Figure 3 shows a generalmodel where there are multiple data collection nodessuch as cluster heads or data gathering gateways inwireless sensor networks.
In order to quantify the performance of different al-gorithms, all the nodes generate traffic such that thenetwork is fully loaded, that is, each node will haveenough data to transmit at any time slot. It is also as-sumed that the traffic sources are Poisson with different
Fig. 3. A TD/CDMA wireless ad hoc network with crossovertraffic.
inter-arrival time for different traffic sessions. Packetlength is exponentially distributed with mean 1024 bits.
In this simulation study, we will use the time-averaged service rate as the criterion to compare dif-ferent algorithms for fully loaded networks. Individualas well as total average rates are considered for com-parison. It will quantify the traffic carrying capabilityof the entire network.
In order to measure the QoS-support capability forspecific traffic sessions, we also define the effective ratealong a route/path (Reff
r ) as the minimum average rateamong all the links in the path r, that is,
Reffr = min
i∈rRi. (25)
Higher effective rate of a path implies higher QoS-support capability.
Four routes/paths are of interests here. There isroute one (rI ) from node A to node F in the lin-ear network, whereas there are three routes travers-ing through the network in Figure 3 with crossovertraffic, namely, rII : A → D → E → H → I → L,rIII : B → E → G → J , and rIV : C → F → H →K. Suppose there are each traffic session along eachroute, and their respective minimum rate require-ments are Rmin
I = 160 kbps, RminII = 90 kbps, Rmin
III =190 kbps, and Rmin
IV = 100 kbps. The goal is to exam-ine various algorithms and decide whether they couldsupport the required minimum rate.
In the simulation, we further make the following as-sumptions:
(1) The scheduling decision is made by a central con-troller in every time slot. We use 1.6667 ms timeslot as defined in 3G1xEV-DO (HDR) [32].
(2) It is assumed that the link gains have the followingform
hij(k) = d−4ij (k)Aij(k)Bij(k) (26)
where dij(k) is the distance from the jth transmitterto the ith receiver at time instant k, Aij is a log-normal distributed stochastic process (shadowing).Bij is a fast fading factor (Rayleigh distributed).
(3) It is assumed that dij(k) is a uniformly distributedrandom variable between 150 and 250 m.
(4) It is assumed that the standard deviation of Aij is8 dB [[33].
(5) It is assumed that the Doppler frequency is 8 Hz,corresponding to pedestrian mobile users [33].
(6) It is assumed that all users share 1.25 MHz band-width.
(7) It is assumed that the maximum allowable trans-mission power pmax = 200 mW for all nodes.
(8) Simulation time = 40 000 slots.
In order to study the detailed behavior of each al-gorithm, the slot occupancy rate of each link i (ηi) isalso an important quantity. It is defined as the percent-age of slots assigned to link i. Note that in Multi-linkalgorithms, one slot may be assigned to multiple linkssimultaneously.
4.2. Linear Network
The results of the linear network are summarized in Ta-ble II. We observe that the throughput-optimal familyof algorithms (QR, MQR, MQRMR) have achievedbetter effective rates (Reff
rI) than that of the propor-
tional fair family of algorithms (PF, MPF, PFMR,MPFMR) for a single traffic session. In general, the
Table II. Effective rate and total average rate (both in kbps) in thelinear network.
POWER CONTROL AND SCHEDULING IN MULTIHOP TD/CDMA WIRELESS NETWORKS 801
Fig. 4. Comparison of PF-family of algorithms in a linear TD/CDMA wireless ad hoc network.
throughput-optimal family of algorithms tend to bal-ance the average rate along each traffic session/flow aslong as the system is feasible because the optimizationcriterion (network utility function) address the queuebacklog together with the average data rate. On theother hand, the proportional fair family of algorithmstry to assign each link similar amount of slots (in thelong-term) and thus, will not balance the average ratealong the routes. However, they tend to achieve highertotal average data rate (R) because they take advan-tage of the wireless channel fluctuations and give moreslots to links with better channel quality than that ofthe throughput-optimal family of algorithms.
We also observe that the multi-link algorithms out-perform the one-at-a-time counterparts as expected.For example, the MPF outperform PF 30% in effectiverate and 39% in total average rate, respectively. Theresults also show that the greedy algorithm (for exam-ple, MQR (G)) performs very closely to the optimalalgorithm (MQR (O)).
The proposed token counter mechanism helps to liftthe minimum rate, and hence the effective rate. PFMRhas lifted the minimum rate from PF’s 95.5 kbps to155.8 kbps, while MPFMR has lifted the minimum ratefrom MPF’s 123.7 kbps to 170.1 kbps. Of course, thisis achieved by assigning more slots to links that violatethe minimum rate constraints. As a result, the links thatmay get higher service rates will be assigned less slots,
which result in lower total average data rate. This effectcan be better observed in Figure 4.
In Figure 4, the average rate (in kbps) and percent-age of slot occupancy of all five links in the linearnetwork are plotted when PF-family of algorithmsare employed. It is clear that multi-link algorithms(MPF and MPFMR) outperform their one-at-a-timecounterpart (PF and PFMR) by allowing multiple linkstransmit at the same slot. The plot also show that link 1needs help to achieve the minimum rate. PFMR andMPFMR use the token counter mechanism to assignmore slots to link 1 than PF and MPF, from 29% to51% and from 45% to 62%, respectively. As a result,other links will receive less slots assignments and thusless average rates.
Figure 5 shows the average rate (in kbps) and per-centage of slot occupancy of all five links in the lin-ear network when throughput-optimal family of algo-rithms are employed. They tend to balance the averagerate along the route as discussed before.
4.3. Network With Balanced Crossover Traffic
The simulations of network with crossover trafficreveals similar observations as those obtained inthe linear network. Figures 6 and 7 show the averagerate (in kbps) of all the links along each of thethree routes of the PF-family of algorithms and the
Fig. 5. Comparison of throughput-optimal family of algorithms in a linear TD/CDMA wireless ad hoc network.
throughput-optimal family of algorithms, respectively.As long as the network load is feasible, the throughput-optimal family of algorithms provide higher effectiverate than the PF-family of algorithms. On the other
hand, the PF-family of algorithms provide higher totalaverage rate than the throughput-optimal family ofalgorithms. Note that if the total average rate is theonly concern, then the MMT and MMTMR algorithms
Fig. 6. Comparison of PF-family of algorithms with balanced crossover traffic sessions.
POWER CONTROL AND SCHEDULING IN MULTIHOP TD/CDMA WIRELESS NETWORKS 803
Fig. 7. Comparison of throughput-optimal family of algorithms with balanced crossover traffic sessions.
should be used. However, these algorithms considerneither queue occupancy nor fairness among nodes.
In order to verify the feasibility of the network load,all the queues at all the nodes have to be bounded.
A sample of the queue occupancy of all five nodesalong rII using algorithms MQR in the networkwith crossover traffic is given in Figure 8. All queuelengths are bounded below 105 bits through the entire
Fig. 8. Queue occupancy of all links of rII using algorithms MQR in a TD/CDMA wireless ad hoc network with crossover traffic.
simulation, which demonstrate the feasibility of thenetwork load and the throughput-optimal nature of theMQR algorithm.
4.4. Network With Unbalanced CrossoverTraffic
The above experiments show that the throughput-optimal family of algorithms outperforms the PF-family of algorithms in terms of effective rate of trafficsessions. However, it is noticeable that the throughput-optimal family of algorithms provide no fairnessamong the nodes, and thus may have serious unhealthybehavior when some malicious nodes take advantageof that and send large amount of data into the network.
A simple example is created to demonstrate thisdamaging effect. Instead of balanced traffic loads alongthe three routes (rII , rIII , and rIV ), node A injected alot of traffic into the network, to be exact, an order ofmagnitude higher than the other traffic sessions. The re-sults are listed in Table III. It is obvious that because nofairness has been considered by the throughput-optimalfamily of algorithms, they perform poorly with the ef-fective rate of rIII and rIV far below the required min-imum rate. On the other hand, the PF-family of algo-rithms still provides required minimum rate for all thetraffic sessions and suppresses the disturbance causedby the malicious node. All the multi-link PF-family ofalgorithms are able to support all the minimum rate re-quirements. However, in the throughput-optimal fam-ily of algorithms, only MQRMR is able to support allthe minimum rate requirements because of the tokencounter mechanism. This result also indicates that thetoken counter mechanism indeed can help maintain thefair share of the traffic sessions specified by their min-imum rate requirements.
Table III. Effective rates of route II, III and IV and total average rate(all in kbps) in the network with unbalanced traffic.
The centralized solution needs a central controller andglobal information of all the link gains. It may be im-plemented, for example, in a clustered wireless ad hocnetwork with ‘strong’ cluster heads where centralizedcontrol is not far-fetched. However, it is very difficultto obtain the knowledge of all the link gains in manyother cases and thus, it is impractical to implement acentralized solution.
5.1. Distributed Implementation
A distributed implementation is proposed in this sec-tion where only local information is used to performthe power control and scheduling decisions at eachtransmitting node individually. At the start of each timeslot, neighboring nodes will exchange information us-ing control/signaling channel. The procedures are asfollows:
(1) At the beginning of each time slot, each node i inthe potential transmitter set S select to transmit ornot by flipping a coin. (This is motivated by thework of References [14] and [34].)
(2) Each node that decide to transmit will send a probepacket using power equal to the maximum trans-mission power pmax.
(3) Each receiver detects the probe packets from alltransmitting nodes nearby, and estimates the cor-responding channel gain. The receiver then sendsa packet including information of all the estimatedlink gains using power equal to the maximum trans-mission power pmax.
(4) Each node i in the potential transmitter set S detectsthe packets from the receivers within its transmis-sion range. From each of these receivers, node i
obtains the list of all possible interfering transmit-ters and their link gains toward the receiver.
(5) Each node i in the potential transmitter set S willtransmit to one of the neighboring receivers wherevi (e.g., vi = Ri/Ri for MPF) is maximized.
(6) Update the token counter according to Equa-tion (18) for the algorithms using the token countermechanism.
Note that each node need to keep a table of all the tokenqueue length (for MGMR algorithms) and average ratefor all outgoing active links.
5.2. Simulation Results
In this simulation study, only local information is avail-able to each node by exchanging control messages with
POWER CONTROL AND SCHEDULING IN MULTIHOP TD/CDMA WIRELESS NETWORKS 805
Fig. 9. Gain/loss of distributed algorithms over their centralized counterparts: (a) Total average rate, (b) Effective rate. 1, MPFMR(rII ); 2, MQRMR (rII ); 3, MPFMR (rIII ); 4, MQRMR (rIII ); 5, MPFMR (rIV ); 6, MQRMR (rIV ).
its neighbors as described above. The overhead of theinformation exchange includes a 1-byte (8 bits) probepacket and the reply from the receiver (which may con-tain multiple bytes). The exact size of the reply dependson the number of probes that the receiver get. Each linkgain in the reply is counted as 1 byte assuming that thelink gain is quantized using a 256-level quantizer. Theother parameters of the simulation are the same as inSection 4. MPFMR and MQRMR algorithms are se-lected for comparison in the network with balancedcrossover traffic.
The percentage of rate gain/loss of distributed al-gorithms over their centralized counterparts is shownin Figure 9. The experiment reveals somewhat sur-prising results. The total average rate achieved by thedistributed algorithms is about 30% higher than theircentralized counterparts in spite of lack of centralizedcontrol and global information. This surprising result ismainly due to the greedy nature of local decisions madeby each transmitting node. Because there is no globalinformation about queue backlogs or average rate, nei-ther throughput-optimal nor fairness can be guaranteedin the distributed algorithm. The same greedy nature oflocal decisions also results in the reductions in most ofthe effective rates. The centralized algorithm still out-performs the distributed schemes in terms of the mini-mum rate requirements (which is not directly shown in
Figure 9). The goal of the scheduling is not just max-imize the utilities, but also satisfy all minimum rateconstraints. In this specific scenario, route II containslinks with good channel conditions whereas route IIIand route IV need help to satisfy the minimum rate con-straints along the routes. The centralized MPFMR andMQRMR are able to satisfy all minimum rate require-ments. The distributed algorithms perform poorly inthis respect, the effective rate along route III (MPFMR)and the effective rate along route IV (MQRMR) are farbelow requirements. Of course, the total average ratesgain due to the sacrifice in effective rates.
The overhead in all the cases is roughly the same16%. This simple experiment demonstrates that theproposed distributed implementation achieves accept-able performance (in terms of total average rate andeffective rate comparing to the corresponding central-ized algorithms) while keeps the overhead low.
6. Conclusions
In this paper, the joint power control and schedulingproblem for TD/CDMA wireless ad hoc networks isformulated using a utility function approach. Becausethe resulted optimal power control reveals bang-bangcharacteristics, that is, scheduled nodes transmit with
full power while other nodes remain silent, the jointpower control and scheduling problem is reduced to ascheduling problem. The MGMR is proposed to solvethe constrained optimization problem (P.1). Two spe-cial cases of MGMR are highlighted, namely, the multi-link version of the throughput-optimal and the pro-portional fair algorithms with a generic token countermechanism to satisfy the minimum rate requirements.By ensuring different minimum rate for different trafficsessions, service differentiation is also achieved.
If there is an admission control in the network tomonitor different traffic sessions and prevent maliciousnodes to occupy the network capacity, the throughput-optimal family of algorithms will be ideal for maximiz-ing session rates. However, such an admission controlmechanism may be very difficult to implement in real-ity, thus the PF-family of algorithms will be desirableto distribute the network resources fairly.
Note that the MGMR algorithm may be modified toaccommodate the maximum data rate constraints, bymodifying the way that the token counter updated. Thetoken counter Ti is updated as follows:
Ti(k + 1) = Ti(k) + Ritoken − Ri(k) (27)
where Ritoken = Ri
min if Ti(k) ≥ 0, and Ritoken =
Rimax if Ti(k) < 0. If Ri
max = ∞ for some i, the to-ken counter update rule becomes Equation (18). IfRi
min = 0 for some i, the rule is simplified for this i
to:
Ti(k + 1) = min{0, Ti(k) + Rimax − Ri(k)} (28)
Maximum data rate constraints may be necessary formobile device that has limited memory for buffering.
Acknowledgements
The authors would like to thank Mr. Ning Song for per-forming simulations in this study. We would also liketo thank OPNET Technologies, Inc. for providing theOPNET software. This research work is supported inpart by the U.S. Army Research Office under Cooper-ative Agreement No. W911NF-04-2-0054. The viewsand conclusions contained in this document are those ofthe authors and should not be interpreted as represent-ing the official policies, either expressed or implied, ofthe Army Research Laboratory or the U. S. Govern-ment.
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Authors’ Biographies
Lijun Qian is an assistant profes-sor in the Department of ElectricalEngineering at Prairie View A&MUniversity. He received his B.E. fromTsinghua University in Beijing, M.S.from Technion-Israel Institute ofTechnology, and Ph.D. from WINLAB,Rutgers—The State University of NewJersey. Before joining PVAMU, hewas a researcher at Networks and
Systems Research Department of Bell Labs in MurrayHill, NJ. His major research interests are in wirelesscommunications and mobile networks, network secu-rity and intrusion detection, control theory and gametheory.
Dhadesugoor R. Vaman has been work-ing on the all mobile management andcontrol based ad hoc wireless network ar-chitecture design, integrated power con-trol and scheduling based routing, appli-cation Quality of Service (QoS), Highaccuracy real time tracking of mobile el-ements (digital battlefield), and networkmanagement research issues for network
centric battlefield communications platform. Dr Vaman hasbeen addressing many issues of ad hoc wireless and wired net-work architectures since 1988 both as a Professor and as theCEO of a start-up company. He led the company towards im-plementation of commercial products for Business ISPs to of-fer differentiated services based on assuring QoS for applica-tions. Since September 2002, he has returned to the academicside of his career as the TI endowed chair professor at PrairieView A&M University. Dr Vaman has published over 130papers in journals and conferences (invited and non-invited);is the author of two books; and has lectured widely both na-tionally and internationally. He has been a key note speakerin many IEEE and other conferences, and industry forums.More recently, he was a key note speaker in IEEE conferenceon “Enabling Technologies for Smart Appliances,” January2005, Hyderabad, India. He is currently funded by the DODto establish a center of excellence for digital battlefield com-munications. He earned his Ph.D. degree from the City Uni-versity of New York, M.E.E. from the City College of NewYork, M.Tech. and B.E. from the Regional Engineering Col-lege, Warangal, India. Prior to his current position, he workedas the CEO of Megaxess, Professor of EECS & FoundingDirector of Advanced Telecommunications Institute, a USNavy Center of Excellence in Telecommunications, StevensInstitute of Technology; Member of Technical Staff in COM-SAT Labs and Network Analysis Corporation (CONTEL),and Systems Engineer at Space Application Center, ISRO.
Xiangfang Li is a Ph.D. student inthe Department of Electrical and Com-puter Engineering at Rutgers University.She received her B.S. and M.S. fromBeijing University of Aeronautics andAstronautics (BUAA), both in electri-cal engineering. Her major research in-terests are in wireless communicationsand mobile computing, especially in ra-
dio resource management, cross-layer design, and wirelessnetwork security.
Zoran Gajic received Dipl. Ing. (5-yearprogram) and Mgr. Sci. (2-year pro-gram) degrees in Electrical Engineeringfrom the University of Belgrade, andan M.S. degree in Applied Mathemat-ics and Ph. D. in Systems Science Engi-neering from Michigan State University.Dr. Gajic is a Professor of Electrical andComputer Engineering at Rutgers Uni-
versity. He has been teaching electrical circuits, linear sys-tems and signals, controls, and networking courses at thesame school since 1984. Dr Gajic’s research interests are incontrols systems, wireless communications, and networking.
He is the author or coauthor of more than 60 journal pa-pers, primarily published in IEEE Transactions on AutomaticControl and IFAC Automatica journals, and 7 books in thefields of linear systems and linear and bilinear control systemspublished by Academic Press, Prentice Hall, Marcel Dekker,and Springer Verlag. His textbook Linear Dynamic Systemsand Signals, Prentice Hall, 2003 has been translated into the
Chinese Simplified language. Professor Gajic has deliveredtwo plenary lectures at international conferences and pre-sented more than 100 conference papers. He serves on theeditorial board of the journal Dynamics of Continuous, Dis-crete, and Impulsive Systems, and was a guest editor of a spe-cial issues of that journal, on Singularly Perturbed DynamicSystems in Control Technology.
