2008 Patrick - Modeling and Computing Throughput Capacity of Wireless Multihop Networks

8/2/2019 2008 Patrick - Modeling and Computing Throughput Capacity of Wireless Multihop Networks

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Modeling and computing throughput capacity of wirelessmultihop networks q

Patrick Stuedi *, Gustavo Alonso

Department of Computer Science, ETH Zurich, 8092 Zurich, Switzerland

Available online 29 September 2007

Abstract

Capacity is an important property for QoS support in Mobile Ad Hoc Networks (MANETs) and has been exten-sively studied. However, most approaches rely on simplified models (e.g., protocol interference, unidirectional links,perfect scheduling or perfect routing) and either provide asymptotic bounds or are based on integer linear program-ming solvers. In this paper, we present a probabilistic approach to capacity calculation by linking the normalizedthroughput of a communicating pair in an ad hoc network to the connection probability of the two nodes in a so-called schedule graph GTN;E. The effective throughput of a random network is modeled as a random variableand its expected value is computed using Monte-Carlo methods. A schedule graph GTN;E for a given network isdirectly derived from the physical properties of the network like node distribution, radio propagation and channelassignment. The modularity of the approach leads to a capacity analysis under more realistic network models. In

the paper, throughput capacity is computed for various forms of network configurations and the results are comparedto simulation results obtained with ns-2. 2007 Elsevier B.V. All rights reserved.

Keywords: Ad hoc networks; Throughput capacity

1. Introduction

Capacity is typically studied by choosing a net-work model that facilitates analytical treatment.

In doing so, the problem has to be simplifiedby either making assumptions about the network

(e.g., symmetric links), radio propagation (e.g,isotropic signal propagation, protocol interfer-ence) or the size of the network (e.g., very largenumber of nodes). In this paper, we eliminate

many of these restrictions by looking at through-put capacity from a probabilistic perspective.Since the capacity of random networks must berandom as well, we model the achievablethroughput per communication pair in a multihopwireless network as a random variable. Theapproach is centered around a so-called schedulegraph GTN;E which is directly derived fromthe physical properties of the network. The effec-tive throughput capacity of a pair of nodes in an

1389-1286/$ - see front matter 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.comnet.2007.09.014

q The work presented in this paper was supported (in part) bythe National Competence Center in Research on Mobile Infor-mation and Communication Systems (NCCR-MICS), a centersupported by the Swiss National Science Foundation underGrant Number 5005-67322.* Corresponding author.

E-mail addresses: [email protected] (P. Stuedi), [email protected] (G. Alonso).

Available online at www.sciencedirect.com

Computer Networks 52 (2008) 116129

www.elsevier.com/locate/comnet
mailto:[email protected]:alonso@%20inf.ethz.chmailto:alonso@%20inf.ethz.chmailto:alonso@%20inf.ethz.chmailto:alonso@%20inf.ethz.chmailto:[email protected]


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ad hoc network is then shown to be related tothe connection probability of these two nodes inGTN;E. Due to its modularity, our approachis decoupled from specific network propertiessuch as the channel multiplexing scheme, the sig-

nal propagation and interference model, the rout-ing, or the node distribution. Thus, our approachcan be seen as a powerful tool to analyze anyform of interaction between the physical and log-ical properties of the network with regard tothroughput capacity.

1.1. Related work

Capacity and scheduling issues have been afocus of research for many years [9,7,12]. In con-trast to the consensus that accurate physical layer

models are important [3,21,25,1], many recentstudies are still based on a simplified interferencemodel such as the protocol model. In [6], theauthors use the protocol model to investigatethe interaction between channel assignment anddistributed scheduling in multi-channel, multiradiowireless mesh networks. Broadcast capacity ofmultihop wireless networks under protocol inter-ference is studied in [10]. The k-hop interferencemodel is an extension of the traditional protocolmodel in that it considers all nodes within a

hop distance of k from the receiver as interferingnodes. Such a model is studied in [19] to derivebounds for the scheduling complexity. The rela-tion between the k-hop neighborhood and theset of interfering nodes, however, is not clear.An interference model similar to the disk modeldescribed in this work is used in [14]. The authorsdescribe an improved packet scheduling algorithmbased on virtual coordinates. In [24], the authorsalso use a disk-based interference model but theirwork allows for different transmission and inter-ference range settings per node. In their seminalwork [9], Gupta and Kumar have studied capac-ity asymptotically for an increasing node density.They have shown that the throughput capacityk(n) for a network of n nodes within an area of[0, 1]2 is in the order of H1=

ffiffiffiffiffiffiffiffiffiffiffiffiffin log n

p. However,

asymptotic analysis typically omits the constantfactor that determines whether a realistic andfinite network will have a useful per nodethroughput. Recently, there has been some effortto compute concrete throughput values [2,4,23]using integer linear programming (ILP). However,

ILP makes it very difficult to model physical net-

work properties such as realistic signal propaga-tion, link asymmetry or interference. There issome work on capacity trying to use more realis-tic network assumptions. In [8], the result in [9]was extended for models including variable trans-

mission power. Bound attenuation functions andmultiple channels are studied in [7,12]. Joint con-gestion control and resource allocation, alsounder physical interference, has been investigatedin [20]. One of the first approaches to apply com-bined topology control and channel assignmentalgorithms to SINR-based interference models inmulti-hop wireless networks can be found in[15]. A fast scheduling algorithm for the physicalinterference model is proposed in [5]. Similar tointerference, an accurate modeling of signal prop-agation is fundamental when computing capacity

in wireless networks. Effects of shadowed radiopropagation on the packet success probability ofa fixed distance link have been analyzed in [26].In such networks, any variation in the signal pat-tern impacts the perceived interference at a givennode. Non-deterministic variation of signal powermay further lead to link asymmetry. This behav-ior was measured experimentally in [1]. IEEE802.11, the MAC protocol often mentioned incombination with ad hoc networks, allows fordata transmission only if there exists a bi-direc-

tional connection between the two communicatingnodes since data packets need to be acknowl-edged by the receiving node. Effects of asymmet-ric links on higher network layers wereinvestigated in [25].

1.2. Contribution

The contributions of this paper are as follows:

The paper presents an abstract model to com-pute throughput capacity in multihop wirelessnetworks. By combining the model withMonte-Carlo methods, the paper proposes anew way of throughput capacity computationfor more realistic network configurations withcomplex random properties. Our approach offirst transforming the physical properties of thenetwork into a graph representation has twoadvantages: it makes the actual throughput com-putation independent of low level networkdetails and at the same time facilitates the anal-ysis of various physical and logical effects with

regard to throughput capacity.

P. Stuedi, G. Alonso / Computer Networks 52 (2008) 116129 117


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By linking throughput capacity of multihop wire-less networks with the connection probability in aschedule graph GTN;E, the paper proposes away of analyzing capacity in sparse and partiallydisconnected random networks. This might be

particularly helpful with regard to throughputcalculations in mobile scenarios where the move-ment of the nodes often leads to temporarily bro-ken paths.

The paper further presents and discusses an algo-rithm for a conflict-free channel assignmentunder arbitrary interference models, includingSINR-based interference.

2. Network model

As a first step, we turn the physical properties ofwireless multihop networks into a so-called schedulegraph GTN;E. Examples of physical propertiesare node locations or perceived signal strengths. Ina schedule graph, N is the set of nodes in the net-work and E denotes a set of directed edges betweenthe nodes such that the existence of a sequence ofnodes n0, n1, . . . , nk with ni 2 N 8i 6 k andni; ni1 2 E 8i < k states that there is also a sche-dule of channel assignments w(n0, n1), w(n1, n2), . . .

w(nk1, nk) such that node n0 is able to consecutivelytransmit data to node nkat a rate kn0;nk > 0. The idea

behind building a schedule graph is to create anabstraction that allows us to later on reasonabout the achievable capacity of the underlyingwireless network. In this section, we first definesome common properties in order to then graduallydevelop the graph representation by assigning twosets Dn Un of nodes to each node n, withDn N. Nodes within the particular sets corre-spond to the different forms of interaction nodescan have, such as unidirectional and bi-directional

communication. A list of all the notations used inthis paper, including the aforementioned sets ofnodes, can be found in Table 1.

We parameterize the network using the followingfive properties: The set of NnodesN, a node distri-bution d, a signal propagation #, a channel assign-ment w and an interference model j. We assumexn 2 R

2 to be the coordinate1 of node n, identifyingthe nodes position with respect to an area A, andwe consider the set N of nodes as being distributed

in A according to some probability function d :A ! 0; 1. Throughout this paper, we use P torefer to the collection of all possible subsets of a set.

Let us start by defining how signals are propa-gated. A node n in the network is supposed to trans-

mit with a signal power P

t

n 2 0;1. We use thetuple notation (n , n) to refer to the transmissionfrom a node n to a node n. For a certain signalpropagation function #, Pnn #P

tn ; jxn xnj 2

0;Ptn denotes the power of the received signal atnode n due to the transmission (n , n). In thesimplest case, # is a direct function of the distance.The path loss radio propagation model, forexample, defines #pl(p, l) = p (l/l0)

q for some pathloss exponent q, and l0 as a reference distancefor the antenna far-field. A more sophisticatedmodel is the log-normal shadowing radio

propagation [18]:

#shp; l p l=l0q 10X=10; 1

where X is a gaussian random variable with zeromean and standard deviation r and q is the afore-mentioned path loss exponent. In case of r equal0, there is no random effect and #sh #pl. In thiswork, we assume the physical signal propagationto be symmetric. Thus, the gaussian random vari-able X involved in the computation of Pnn is the

same as the one involved in the computation ofPn n2. From practical measurements, however,

it is known that the signal strengths Pnn andPnn (corresponding to transmissions of two identi-cal radio transmitters) may not always be equal.This is due to tiny differences of the radio hardwareand is taken into account in our model by the powerdistribution Ptn.

Whether a signal from a node n can be decodedcorrectly at node n in the absence, or the presence,of concurrent transmissions, is determined by theinterference model. In the literature, two main inter-

ference models have been proposed [9]: the protocoland the physical interference model. In the protocolmodel, a transmission from a node n is said to bereceived successfully by another node n if no noden closer to the destination node is transmittingsimultaneously. However, in practice, nodes outsidethe interference range of a receiver might still causeenough cumulative interference to prevent the recei-ver from decoding a message from a given sender.This behavior is captured by the physical model,

1 The model could also be applied to R3. 2 Therefore Ptn P tn ) Pnn Pnn.

118 P. Stuedi, G. Alonso / Computer Networks 52 (2008) 116129


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where a communication between nodes n and n issuccessful if the SINR (Signal to Interference andNoise Ratio) at v (the receiver) is above a certainthreshold.

In this work, we assume interference models to bedefined by a binary interference functionj : NNPN ! f0; 1g with

jn

;n;I

1 The signal ofn can be

decoded at node n

under a set I of interferers

0 otherwise:

8>>>>>>>>>: 2

The interference function for the protocol model [9]is

jprotocoln; n;I 1 () dn; n

> dn; n 8n 2 I: 3

For the physical interference model [9], the interfer-

ence function is

jsinrn; n;I 1 ()

Pnn

Pn P

n2IPnn> bsinr;

4

for some threshold bsinr and Pn as the thermal noise

perceived at node n.We now assign two sets of nodes to each node

n 2 N, namely, Dn and Un,

Dn fn 2 Njjn; n; ; 1g 5

is the set of nodes that can be correctly decoded atnode n in the absence of any other concurrenttransmission,

Un fn 2 Dnjjn

; n;In 1g 6

contains all nodes n that can be correctly decodedat node n in the presence of a set of nodes In 0 trans-mitting concurrently as node n . For later use wedefine D fn; njn 2 Dng to be the set of trans-missions in the network when interference is ig-nored, and U fn; njn 2 Dng to be the set oftransmissions in the network if interference is

considered.

Table 1Mathematical notations

Symbol Semantic

Parameters for GTN;E)N Set of nodes in the networkd Node distribution function

# Signal Propagation functionw Channel assignment functionj Interference model

GTN;E internalX Set of coordinates xn for each node nC Set of channelsPtn Transmission power of node nPnn Signal power perceived at node n due to the transmission of node n

Pn Thermal noise perceived at node nDn Set of nodes that can be decoded at node n without noiseUn Set of nodes that can be decoded at node n under noiseE Set of directed edges in a schedule graph

Parameters for kGTN;E Schedule graphx(n, n) Weight function indicating the number of channels used on a link Set of source destination pairsg Routing function

k internalP Set of paths participating in communication

Bn ;n Lowest number of channels between any two neighbors along a pathT Used channels, T= jCj in GTN;Efn ;n Achievable throughput capacity along a pathk Expected throughput capacity



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3. Scheduling

Which transmissions in the network occur simul-taneously is determined by the scheduling algo-rithm. In our model, we assume the medium to be

divided into a set of channelsT

. Each channelwi2 T can be seen as a set of directed transmissions(n , n), with n 2 Dn, between two nodes n and n.For the sake of simplicity, we use w(n , n) to referto the set of channels used by the transmission(n , n). We further use wn

Sn:n2Dn

wn; n torefer to all the channels where node n acts as atransmitter.

Scheduling transmissions in multi-hop wirelessnetworks so that no two transmissions scheduledwithin the same channel interfere, is trivial for theprotocol model, but turns out to be more difficult

under the physical interference model. In general,the problem of scheduling is related to the tradi-tional graph coloring problem, except that the verti-ces in the graph to be colored refer to thetransmissions in the network and the edges in thegraph refer to the interference conflicts. Two verti-ces conflict if their corresponding transmissions can-not be scheduled simultaneously. We call such agraph a conflict graph.

Under the physical interference model (Eq. (4)),conflicts between two transmissions cannot be

determined without considering all other transmis-sions. As an example, two nodes n and n mayinterfere with a transmission from node n to noden, even if node n cannot successfully decode the sig-nals of either n or n in the absence of interference.For a node n to belong to Un, jphysicaln; n;N nCnn must compute to 1, given Cnn containsall nodes n acting as a sender in a transmission thatconflicts with the transmission (n , n). In practice, ofcourse, one wants to find the minimum set Cnn ofconflicts for a transmission (n , n) because this min-imizes the number of channels to be used at a laterpoint in time. How to compute the minimum set ofconflicts for a given set of transmissions D in thephysical interference model is shown in Algorithm1. For a given transmission (n , n), the algorithmoperates by gradually testing the SINR ratio withan increasing set of interferers, starting with thenode contributing the lowest signal power. At thepoint where the cumulated interference of a noden leads to a SINR ratio smaller than bSINR, alltransmitting nodes n with Pnn000P Pn n00 areconsidered as interferers and their associated trans-

missions are defined as conflicts with (n , n).

Algorithm 1. Conflict graph under physicalinterference

Input: Set of all transmission DOutput: Set of conflicts C fe; eje; e 2 Dg

1: C : ;;2: for all e : n; n 2 D do3: L : sort(N fn; ng) such that n 0 n $Pnn < Pnn4: M : ;;5: for all n 2 L6: M : M [ fng7: ifjsinrn

; n;M 0 then8: Q : fn; n j n 2 Dn g9: for all e 2 Q do

10: C : C [ fe

; e; e; e

g;11: end for12: end if13: end for14: end for

We have shown how a conflict graph can bebuilt for the physical interference model. Basedon the conflict graph, efficient coloring algorithmsmight be used to assign channels to the transmis-sions (represented as nodes in the conflict graph).Finding the minimum number of channels, how-ever, is an NP hard problem and thus is not feasi-ble for large networks [11,17]. We decided to applya Greedy channel assignment algorithm. Algorithm2 assigns channels to transmissions in a greedy

e1

e1

e2

e3

Network Interference Conflicts Channels

e2

e3n

1

8 6

2

2

3

6

e4

e4

01

0 1

n2

Fig. 1. Channel assignment under physical interference. Thestraight line arrows represent the transmissions. The dottedarrows denote signals which contribute to the interference noiseof transmission e1. The weight assigned to an edge corresponds tothe signal strength. We assume the thermal noise P* used in Eq.(4) to be 1. According to Algorithm 1, nodes in the grey area areconsidered as the smallest set of nodes such that the remainingcumulative interference does not prohibit transmission e1 frombeing established. There is no conflict between transmissions e1and e3 because node n1 is not included in the grey area. Note thatboth conflict graph and channel assignment are considered as

snapshots from the perspective of e1.



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way, so that no two transmissions e1, e2 will bescheduled using the same channel if there exists aconflict between the two transmissions (e1; e2 2C). Algorithm 2 further assigns channels in a trafficproportional way, meaning that each node pair

(n

,n

), with n

2D

n, is assigned exactly as manychannels as there are flows occupying the wirelesslink. The function l() in Algorithm 2 refers tothe number of flows of a link. Conflict graph andchannel assignment for the physical interferencemodel are illustrated in Fig. 1 in a small examplenetwork.

Algorithm 2. Greedy channel assignmentInput: Set of all transmission D, set of conflicts COutput: Set of channels {w0, w1, . . . , wT1} withwi D

1: for all e 2 D do2: for i: 0; i< l(e) do3: Q : feje; e 2 Cg4: X : ;;5: for all e 2 Q do6: X : X [ {wjje

2 wj}7: end for8: k: freechannel(X);9: wk : wk[ {e};10: end for11: end for

freechannel(Q)1: X : sortQ such that wi 0 wj $ idwi i+ 1 then5: break;6: end if7: i: id(w);8: end for

9: return i+ 1;

4. Schedule graph

Coming back to the definition ofUn, we can saythat a node n belongs to Un if In0 in Eq. (6) isdefined as the set of nodes transmitting in the samechannel as node n . Given a schedule and the set Unfor each node, we define a so-called schedule graphas a directed and weighted graph GTN;E, where

E denotes the set of directed edges with

E fn; n 2 NNjn 2 Un ^ n 2 Dn g: 7

The set E includes all transmissions (n , n) whosesignals can be decoded correctly at node n underinterference, while the reverse signal might only becorrectly decoded if there is no interference. Note

that Eq. (7) models the acknowledgment as an infi-nitely small packet not occupying the medium. Thesubscript T indicates the number of channels used(Algorithm 2). The weight of an edge n; n 2 E isgiven by x(n , n),

xn; n X

c2wn ;n

jn; n;Ic: 8

Here, Ic denotes the set of nodes transmitting duringchannel c, or Ic fn

2 Njc 2 wng.It follows directly from the definition of a sche-

dule graph GTN;E that for any path n0, n1, . . . , nk with ni 2 N 8i 6 k and ni; ni1 2 E 8i < k there is also a corresponding schedule of channelassignments w(n0, n1), w(n1, n2), . . . , w(nk1, nk) ina way that node n0 is able to consecutively transmitdata to node nk at a rate strictly greater than zero.We will make use of this property later on to reasonabout the achievable capacity of the underlyingphysical network.

5. Throughput capacity

Throughout this section, an ad hoc network isrepresented by its schedule graph GTN;E andthe corresponding weight function x. Capacity isthen defined over a set of communication pairs,

fn; n 2 NNjn 6 ng: 9

More precisely, we say that a schedule graphGTN;E with a communication pattern has athroughput capacity ofkn;n if a communication pair(n , n) 2 can expect an end-to-end throughput of

kn

;n bits per second. Important to the computationof throughput capacity is the routing functiong : NN ! PE. Hence, for a given sourcedes-tination pair (n , n) the resulting route simply con-sists of the set3 of edges included in the sequencee0, e1, . . . , ek1, with ei ni; ni1 2 E; n0 n

andnk= n.

3 In practice, a route would be modeled as a sequence ratherthan as a set, however, since we assume no loops and the order ofthe edges in a route is not important for the computation of k we

prefer the set notion which simplifies further treatment.



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We now want to analyze the expected throughputk of a communication pair (n , n) 2 . Since both thenetwork and its graph representation GTN;E arerandom, obviously the resulting throughput pernode pair can also be considered to be random.

Based on this, the approach we follow is of a proba-bilistic nature. For any node pair (n, n ) 2 , wemodel throughput capacity as a random variablefn;n : P ! 0;1 to then compute the expectedvalue Efn ;n of fn;n, with Efn;n kn;n. Considerthe fact that in a schedule graph, a path betweentwo nodes also reflects a schedule of channels.Throughput capacity is a concave metric, meaningthat the available throughput for a certain sourcedestination pair is always determined by the nodewith the lowest bandwidth, the so-called bottleneck.Hence, let Bn;n mine2gn;nxe be a random vari-

able indicating lowest number of channels availablebetween two nodes along the path from n to n. Onecan easily verify that the resulting throughput capac-ity along the path cannot be bigger than WBn;n=T,where Wis the maximum transmission rate equal toall nodes and T= jCj is the number of channels usedin total. The throughput capacity may however befurther diminished when considering all the traffic taking place in the network. For this purpose letus define a so-called load function l : E ! 0;N,indicating to what extent a certain edge e 2 E is

shared with other ongoing traffic, or, more formally:

le X

ign;nn;n2

1ie; 10

where 1i : E ! f0; 1g is the set membership func-tion. If we want to take all ongoing traffic into ac-count we therefore have to consider l whilecomputing Bn;n, or

Bn

;n

0 gn; n ;

mine2gn;nxele otherwise:

8


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In other words, we approximately compute the ex-pected value of f for a given set of parameters bysampling over k realizations of the underlying ran-dom network, with Xi as a concrete set of nodeplacements in the area A.

6. Capacity of static networks

To validate the model, we compute the through-put capacity of two simple, static scenarios, static inthe sense that the network topology as well as thecommunication pattern is fixed. The throughput ofsuch fixed network configurations can be seen asthe conditional expected value E[fjX] of the randomvariable f under a concrete node placement X. For a

fixed channel assignment w, E[fjX] simply computesas EfjX 1jj

Pn;n2fn;njXX

i, where Xi is the

given set of coordinates of the nodes. For bothexamples we will consecutively derive E[fjX] bygoing through the basic steps of Sections 2 and 5.

The first network topology we consider consistsof three nodes being distance d apart from eachother, as shown in Fig. 2a. To simplify the analysis,we use the more primitive protocol interferencemodel as described in Eq. (3). Let us further assumen 2 Dn for all n

5 n. According to jprotocol, the set

of senders Un is modeled in a way that a node n

belongs to Un if, and only if, no other concurrenttransmission with a signal stronger than Pnn isreceived by node n. Hence, the graph GT onlydepends on how the different channels are assignedto the nodes. We now want to illustrate the outcomeofE[fjX] for three possible channel assignments. Wekeep track of all states and sets of the networkmodel for each of the three channel assignments inTable 2.

In all the configurations we assume shortest pathrouting and only assign channels to edges that arealso used when considering the traffic pattern .In the case of one common channel w(n , n) =w(n, n ) for all nodes n, n , no transmission canFig. 2. Static topologies.

Table 2

States for the triangle scenario

T w(n, n) Un Bn;n E[fjX]

1 w(B, A) = ; ; (A, B) 0 0w(C, B) = ; (B, C)w(A, C) = ; (C, A)w(A, B){0}w(B, C) = {0}w(C, A) = {0}

2 w(B, A) = ; UA ; (A, B) BA,B= 0 1/6Ww(C, B) = ; UB ; (B, C) BB,C= 1w(A, C) = ; UC fBg (C, A) BC,A = 0w(A, B) = w(C, A) = {0}

w(B, C) = {1}

3 w(B, A) = ; UA fCg (A, B) BA,B= 1 1/3Ww(C, B) = ; UB fAg (B, C) BB,C = 1w(A, C) = ; UC fBg (C, A) BC,A = 1w(A, B) = {0}w(B, C) = {1}w(C, A) = {2}

4 w(B, A) = ; UA fCg (A, B) BA,B= 2 1/3Ww(C, B) = ; UB fAg (B, C) BB,C= 1

5 w(A, C) = ; UC fBg (C, A) BC,A = 1w(A, B) = {0, 1}w(B, C) = {2}

w(C, A) = {3}



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correctly be decoded at any receiver (Eq. (3)) whichleads to Un ;;E ;, Sn;n 0, fn;n 0 and finallyto E[fjX] = 0. In the presence of two separate chan-nels (T= 2), two directed links can be established(among the potential 6). Along with a communica-tion pattern = {(A, B), (B, C), (C, A)} (see Table2), E[fjX] is W 1/3 (0 + 1/2 + 0) = 1/6 W. Iftransmissions are spread over three channels,

GTN;E becomes fully connected and E[fjX]equals W 1/3 (1/3 + 1/3 + 1/3) = 1/3 W. Addinga fourth edge, e.g., to the transmission betweennode A and B, does not increase the capacity anyfurther. This is because the increase in the bottle-neck (BA, B= 2) is compensated by the increase ofthe total amount of used channels (T= 4).

The situation is slightly different for the scenarioin Fig. 2b since node B acts as a router and some ofits bandwidth is consumed by traffic sent from A toC. The case T= 1 is trivial and comparable with thecorresponding case in the triangle scenario. Assign-ing two channels to the four edges results in twoestablished links (UA fBg and UC fBg). Con-sidering the traffic pattern , it is sufficient for onepath to be established (e.g., SB,C= 1) and the result-ing capacity E[fjX] computes to 1/6 W. Two of thethree routes can be established if three channels areused (T= 3), which results in E[fjX] = 2/9 W. Thecapacity of the given traffic pattern can be furtherimproved by assigning one channel per transmissionpair. All the routes can be established with a bottle-neck of Bn;n 1;8n; n

2 and E[fjX] equals

1/4

W. Obviously, the same channel assignment

results in a different capacity if another traffic pat-tern is used, like, e.g. = {(A, C), (B, C), (C, A)}.Since the link between node B and C is used twice,the values for BA,C BB,C reduce to 1/2. For such atraffic pattern, a 5-channel assignment performs bet-ter, as shown in Table 3.

7. Capacity of larger networks

In this section, we analyze throughput capacityof various types of communication patterns and net-work topologies. To simplify the notation we willrefer to E[f] as k for the rest of the paper. For eachanalyzed configuration we also provide results takenfrom simulations with ns-2 [22] under the very sametopology and communication setup. Throughoutthis section, we use a path loss radio propagationas defined by #pl and a SINR-based interferencemodel, j

sinr, as described in Eq. (4). Since we use

#pl, the threshold for a node n to be part ofDn only

depends on the distance between the two nodes. Wehave set Pn of Eq. (4) such that n

2 Dn ()jxn xnj 6 250. To avoid mixing up capacity mea-surements with routing issues, packets within ns-2simulations are forwarded using pre-computedshortest path routes. We further have set theMAC data rate in ns-2 to 1 Mbit/s. This is necessarysince operating 802.11 at higher rates results in dras-tically reduced efficiency and makes the measure-ments difficult to compare as the per-packet

overhead dominates the overall cost. This is due

Table 3States for the chain scenario

T w(n, n) Un Bn;n E[fjX]

2 w(A, B) = w(C, B) = {0} UA fBg (A, B) BA,B= 0 1/6Ww(B, C) = {1} UB ; (B, C) BB,C = 1w(B, A) = ; UC fBg (C, A) BC,A = 0

3 w(A, B) = {0} UA fBg (A, B) BA,B= 1 2/9Ww(B, C) = {1} UB fAg (B, C) BB,C= 1w(C, B) = w(B, A) = {2} UC fBg (C, A) BC,A = 0

4 w(A, B) = {0} UA fBg (A, B) BA,B= 1 1/4Ww(B, A) = {1} UB fA;Cg (B, C) BB,C= 1w(B, C) = {2} UC fBg (C, A) BC,A = 1w(C, B) = {3}

(A, C) BA,B= 1/2 1/6W(B, C) BB,C= 1/2(C, A) BC,A = 1

5 w(A, B) = {0} UA fBg (A, C) BA,B= 1 1/5Ww(B, A) = {1} UB fA;Cg (B, C) BB,C= 1

w(B, C) = {2,3} UC fBg (C, A) BC,A = 1w(C, B) = {4}



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to the fixed length 802.11 preamble used by thehardware for bit synchronization.

7.1. Chain

In a first comparison we look at a configurationof a chain of n nodes. Each node is 200 m awayfrom its neighbor. The first node acts as a sourceof data traffic, the last node is the traffic sink. Datais sent as fast as the MAC allows. We use Greedy as

the channel assignment algorithm. Since there areno random components involved, k is a direct func-tion of the channels needed, and computes to 1/4 asthe chain grows. From Fig. 3a, we see that the valueofk lies above the throughput measured with ns-2,especially when the chain becomes large. This isdue to the overhead of headers, RTS, CTS andACK packets but also because in reality nodes failto achieve an optimal schedule. The results obtainedwith our model match those presented in [13], wherethe authors discuss throughput capacity measure-ments taken from ns-2 simulations with respect totheoretical upper bounds.

As a more realistic scenario, we now investigaterandom communication patterns in chain topolo-gies. For this purpose, we assign a random destina-tion dn 2 N fng to every node n 2 N. Fig. 3bshows the effect of such a traffic pattern on through-put. The plot shows quite a close match between kand the measurements obtained with ns-2. This isnot too surprising since we know from Fig. 3a thatthe throughput of an 802.11 chain tallies with thetheoretical limit if the chain length is short. Under

a random communication pattern the average path

length in a chain is far below the maximum valueofn 1, for a chain of length n. Furthermore, over-lapping communication paths reduce capacity (Bn;nin our model) due to the forwarding load inflictedupon the nodes, especially if the chain becomeslong.

7.2. Grid

We look at grid topologies where each node is

200 m away from its closest neighbor and the nodescommunicate using a random communication pat-tern. Fig. 4a shows the capacity in the grid topologyfor a cross communication pattern: source nodes inthe first column have a destination assigned in thecorresponding row in the last column, and sourcenodes in the first row have a destination assignedin the corresponding column in the last row. FromFig. 4a we see that the model based computationpredicts a higher throughput capacity than the onemeasured with ns-2. This is because the cross com-

munication pattern is actually composed of end-to-end chain communications exactly like the sce-nario used to compute Fig. 3a. We know that forlarge chain communications, 802.11 throughputcapacity is far below the information-theoreticcapacity,4 for reasons explained in Section 7.1. Inthe grid topology with cross communication thisbehavior is amplified. Fig. 4b shows the capacityin the grid for a random communication pattern

2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Nodes

NormalizedCap

acity[0,1

]

NS2

using Greedy

0 10 20 30 40 500.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Nodes

NormalizeCapa

city[0,1

]

using Greedy

NS2

Fig. 3. Chain topology.

4 Information-theoretic capacity refers to the capacity that canbe achieved with optimal routing and scheduling decisions, which

possibly require global knowledge.



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where each node gets assigned a random destina-tion. Similarly to the chain example, the gapbetween ns-2 measurements and results obtainedthrough our model disappears slightly when com-munication becomes random. Random communica-tion reduces the average path length and thereforediminishes the impact of the suboptimal channelassignment and the header overhead inherent in802.11.

7.3. Random topology

We consider random topologies of n nodes dis-tributed uniformly within an area of 1000

1000 m2. Here, we have configured Pn such thatthe transmission range of the nodes equals 200 m.Each node n acts as a traffic generator and has arandom destination assigned, chosen uniformlyout ofNfng. Fig. 5a shows the throughput capacityk in contrast with ns-2 simulation measurements.The result supports the trend already observed inthe previous configurations of the chain and thegrid: randomness improves 802.11 throughput

capacity with respect to k. This might particularlybe the case in dense networks where the demandfor channels is high due to the high node degree,leaving less room for an optimal channelassignment.

20 40 60 80 100 120 140 160 180 2000.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Nodes

NormalizedCapacity

[0,1

]

using Greedy

NS2

0 20 40 60 80 1000.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

3

Channels

NormalizedCapacity[0,1

]

Random

Greedy

Fig. 5. Random topology.

0 50 100 150 2000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Nodes

NormalizedCapa

city[0,1

]

Greedy

NS2

0 50 100 150 2000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Nodes

NormalizedCapa

city[0,1

]

NS2

using Greedy

Fig. 4. Grid topology.



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Algorithm 3. RandomEdge Channel Assignment

Input: The maximum number of channels T*

Output: Channel assignment w and number ofused channels

1: O : E;2: i: 0;3: while O 6 ; do4: e : ANYfe 2 Eg;5: O : O feg;6: w(e) : i;7: i: (i+ 1)MOD T*;8: end while9: if jNj < T then10: return jNj;11: else

12: return T*;13: end if

The model for throughput computation pro-posed in this work allows to easily exchange anyof its components, such as, e.g., the scheduling algo-rithm or the radio propagation. In the followingpart, we will study the difference between a greedy

channel assignment and arandom

channel assign-ment, with respect to throughput capacity. Moreparticularly, we use the RandomEdge channelassignment (Algorithm 3), which assigns a set ofmaximum T channels in a round robin mannermodulo T to all transmission pairs (n, n ) withn 2 Dn. At each round, one transmission pair ispicked on a random basis. Fig. 5b shows thethroughput capacity when using RandomEdge in arandom topology of 200 nodes distributed withinan area of 2000 2000 m2. We consider a randomcommunication pattern. The result ofFig. 5b clearly

shows that there is an optimum in terms of the

Fig. 6. Random networks and their schedule graphs under different channel assignment strategies.



13/14

number of channels T to be used when assigningthem randomly to the transmissions. If only a fewchannels are used, all nodes transmit simultaneouslyand no transmission can be correctly decoded. If toomany channels are used, most of the transmissions

can actually be decoded, but since many transmis-sions are scheduled in separate channels, bandwidthis wasted. In fact, RandomEdge achieves a maxi-mum throughput of about 0.0012 with an inputset of around 50 channels, which is less than thethroughput capacity of %0.0016 achieved by theGreedy channel assignment. Note that while Greedyassigns the channels in a conflict-free way, Random-Edge does not. This is also shown in Fig. 6 based ona snapshot of 200 nodes, distributed randomlywithin an area of 2000 2000 m2. Fig. 6a illustratesthe random topology and Fig. 6bd refers to the

corresponding schedule graphs under the specificchannel assignment algorithms. In the networkgraph, the dots represent the nodes and the edgesrepresent the possible transmissions in the absenceof interference (U). The schedule graph in Fig. 6cis the result of a RandomEdge channel assignmentusing a fixed set of 50 channels (which in Fig. 5was shown to be the optimum). As can be observedfrom Fig. 6b and c, Greedy (Algorithm 2) dropsedges unused by the routing, but maintains full con-nectivity while RandomEdge, with 50 channels, pro-

duces a partially disconnected schedule graph. If weuse more channels (Fig. 6d), the connectivity of theschedule graph improves, but more bandwidth iswasted due to the increase in the channels used.

7.4. Summary

Section 7 has shown that throughput capacity,computed based on the model proposed in thispaper, can be used as a reasonable approximationfor the potential throughput capacity of arbitrarynetwork configurations. In general, the simulationresults and the model based computations show asimilar behavior. In most of the cases, the through-put capacity computed by the model is slightlyhigher than the one measured with ns-2. This, how-ever, is natural since the ns-2 simulations are basedon 802.11 which entails a suboptimal channel assign-ment and packet overhead. It is anyway importantto note that the model should not be seen as athroughput capacity predictor for 802.11 based mul-tihop wireless networks, but rather as an approxima-tion of the potential throughput capacity of such a

network in an information-theoretical sense.

8. Conclusions

In this work, we presented a model for studyingthroughput capacity of wireless multi-hop networksunder realistic settings. Contrary to existing work,

looking at capacity from an asymptotic perspectivebased on simplified network models (e.g., protocolinterference, unidirectional links, perfect schedulingor straight line routing), our approach analyzescapacity for finite networks under more realistic net-work configurations. In our model, the effectivethroughput of a random network is considered asa random variable depending on the node distribu-tion, the communication pattern, the radio propa-gation, channel assignment, etc. Expected valuesof that random variable are then computed usingMonte-Carlo methods. The various components of

the model can easily be exchanged to study anyform of physical and logical interaction (e.g., sha-dow fading radio propagation, physical interfer-ence, random scheduling, etc.) with regard tothroughput capacity. While the idea of treatingthroughput capacity as the expected value of a wellmodeled random variable serves as the basis for thiswork, the general concept can also be applied toother network properties. In that sense, the paperalso suggests a new approach to ad hoc networkanalysis in cases where pure analytical approaches

fall short and protocol specific network simulationsare not generic enough. This is of particular impor-tance considering the increasing computing powerof todays hardware. For instance, although thecomputational costs of our model is O(n3), we wereable to compute all the results within a few minutesusing a cluster of 32 machines and JOpera [16] as agrid engine.

References

[1] D. Aguayo, J. Bicket, S. Biswas, G. Judd, R. Morris, Link-level measurements from an 802.11b mesh network, SIG-COMM (2004).

[2] P. Barbrand, D. Yuan, Maximal throughput of spatial tdmain ad hoc networks, in: 3rd Scandinavian Workshop onWireless Ad-hoc Networks, 2003.

[3] C. Bettstetter, C. Hartmann, Connectivity of wireless mul-tihop networks in a shadow fading environment, in:MSWIM03: Proceedings of the 6th ACM internationalworkshop on Modeling analysis and simulation of wirelessand mobile systems, ACM Press, New York, NY, USA,2003, pp. 2832.

[4] P. Bjorklund, P. Varbrand, D. Yuan, Resource optimizationof spatial tdma in ad hoc radio networks: a column

generation approach, in: IEEE INFOCOM, March 2003.



14/14

[5] G. Brar, D.M. Blough, P. Santi, Computationally efficientscheduling with the physical interference model for through-put improvement in wireless mesh networks, in: Mobi-Com06: Proceedings of the 12th Annual InternationalConference on Mobile Computing and Networking, ACMPress, New York, NY, USA, 2006, pp. 213.

[6] A. Brzezinski, G. Zussman, E. Modiano, Enabling distrib-uted throughput maximization in wireless mesh networks: apartitioning approach, in: MobiCom06: Proceedings of the12th Annual International Conference on Mobile Comput-ing and Networking, ACM Press, New York, NY, USA,2006, pp. 2637.

[7] O. Dousse, P. Thiran, Connectivity vs capacity in dense adhoc networks, in: Proceedings of the of IEEE INFOCOM,Hong Kong, 2004.

[8] J. Gomez, A. Campbell, A case for variable-range transmis-sion power control in wireless ad hoc networks, in: IEEEINFOCOM, March 2004.

[9] P. Gupta, P. Kumar, Capacity of wireless networks, Infor-mation Theory 46 (2) (2000) 388404.

[10] A. Keshavarz-Haddad, V. Ribeiro, R. Riedi, Broadcastcapacity in multihop wireless networks, in: MobiCom06:Proceedings of the 12th Annual International Conference onMobile Computing and Networking, ACM Press, NewYork, NY, USA, 2006, pp. 239250.

[11] S. Krumke, M. Marathe, Models and approximation algo-rithms for channel assignment in radio networks, WirelessNetworks 7 (6) (2001) 575584.

[12] P. Kyasanur,N.H. Vaidya, Capacity of multi-channel wirelessnetworks: impact of number of channels and interfaces, in:MobiCom05, ACM Press, New York, NY, USA, 2005.

[13] J. Li, C. Blake, D.S.D. Couto, H.I. Lee, R. Morris, Capacityof ad hoc wireless networks, in: MobiCom01, ACM Press,New York, NY, USA, 2001.

[14] H. Lim, C. Lim, J.C. Hou, A coordinate-based approach forexploiting temporal-spatial diversity in wireless mesh net-works, in: MobiCom06: Proceedings of the 12th AnnualInternational Conference on Mobile Computing and Net-working, ACM Press, New York, NY, USA, 2006, pp. 1425.

[15] T. Moscibroda, R. Wattenhofer, A. Zollinger, Topologycontrol meets sinr: the scheduling complexity of arbitrarytopologies, in: MobiHoc06: Proceedings of the SeventhACM International Symposium on Mobile Ad hocNetworking and Computing, ACM Press, New York, NY,USA, 2006, pp. 310321.

[16] Pautasso C. Pautasso, JOpera: an agile environment for webservice composition with visual unit testing and refactoring,in: VL/HCC, IEEE Computer Society, 2005, pp. 311313.www.jopera.ethz.ch.

[17] S. Ramanathan, E.L. Lloyd, Scheduling algorithms formultihop radio networks, IEEE/ACM Transactions onNetworks 1 (2) (1993) 166177.

[18] T.S.Rappaport, WirelessCommunications:Principles& Prac-tice, Prentice-Hall, Englewood Cliffs, New Jersey, 2002.

[19] G. Sharma, R.R. Mazumdar, N.B. Shroff, On the complexityof scheduling in wireless networks, in: MobiCom06: Pro-ceedings of the 12th Annual International Conference onMobile Computing and Networking, ACM Press, NewYork, NY, USA, 2006, pp. 227238.

[20] P. Soldati, B. Johansson, M. Johansson, Proportionally fairallocation of end-to-end bandwidth in stdma wirelessnetworks, in: MobiHoc06: Proceedings of the SeventhACM International Symposium on Mobile Ad hoc Net-working and Computing, ACM Press, New York, NY, USA,2006, pp. 286297.

[21] M. Takai, J. Martin, R. Bagrodia, Effects of wireless physicallayer modeling in mobile ad hoc networks, in: MobiHoc01:Proceedings of the 2nd ACM International Symposium onMobile Ad hoc Networking & Computing, ACM Press, NewYork, NY, USA, 2001, pp. 8794.

[22] The VINT Project. The NS network simulator, http://www.isi.edu/nsnam/ns, 2002.

[23] S. Toumpis, A. Goldsmith, Capacity regions for wirelessadhoc networks (2001).

[24] W. Wang, X.-Y. Li, O. Frieder, Y. Wang, W.-Z. Song,Efficient interference-aware tdma link scheduling for staticwireless networks, in: MobiCom06: Proceedings of the 12thAnnual International Conference on Mobile Computing andNetworking, ACM Press, New York, NY, USA, 2006, pp.

262273.[25] G. Zhou, T. He, S. Krishnamurthy, J.A. Stankovic, Impactof radio irregularity on wireless sensor networks, in: Mobi-Sys04: Proceedings of the 2nd International Conference onMobile Systems, Applications, and Services, ACM Press,New York, NY, USA, 2004, pp. 125138.

[26] M. Zorzi, S. Pupolin, Optimum transmission ranges inmultihop packet radio networks in the presence of fading,IEEE Transactions on Communication 43 (7) (1995).

Patrick Stuedi received a Diploma inComputer Science from the Swiss Fed-

eral Institute of Technology (ETH),Zurich in 2003. He is currently a Ph.D.student at the Institute of PervasiveComputing at ETH. His research inter-ests are in fundamental properties of adhoc networks, VoIP and QoS.

Gustavo Alonso is a Professor in theDepartmentof ComputerScience of ETHZurich, where he is the Head of the

Institute for Pervasive Computing andleads the Information and Communica-tions Research Group. His researchinterest include distributed systems,adaptive software, web services and ser-vice oriented architecture, wireless sensornetworks, and software systems in gen-eral. For more information on hisresearch and his group, check: www.iks.inf.ethz.ch.

http://www.jopera.ethz.ch/http://www.isi.edu/nsnam/nshttp://www.isi.edu/nsnam/nshttp://www.iks.inf.ethz.ch/http://www.iks.inf.ethz.ch/http://www.iks.inf.ethz.ch/http://www.iks.inf.ethz.ch/http://www.isi.edu/nsnam/nshttp://www.isi.edu/nsnam/nshttp://www.jopera.ethz.ch/

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2008 Patrick - Modeling and Computing Throughput Capacity of Wireless Multihop Networks

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