modeling multi-cell ieee 802.11 wlans with …arxiv:0903.0096v2 [cs.ni] 14 mar 2009 modeling...

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Modeling Multi-Cell IEEE 802.11 WLANswith Application to Channel Assignment

Manoj K. Panda and Anurag KumarDepartment of Electrical Communication Engineering

Indian Institute of Science, Bangalore – 560012.Email: {manoj,anurag}@ece.iisc.ernet.in

Abstract—We provide a simple and accurate analytical modelfor multi-cell infrastructure IEEE 802.11 WLANs. Our modelapplies if the cell radius, R, is much smaller than the carriersensing range, Rcs. We argue that, the condition Rcs >> R

is likely to hold in a dense deployment of Access Points (APs)where, for every client or station (STA), there is an AP veryclose to the STA such that the STA can associate with the APat a high physical rate. We develop a scalablecell level modelfor such WLANs with saturated AP and STA queues as wellas for TCP-controlled long file downloads. The accuracy of ourmodel is demonstrated by comparison withns-2 simulations. Wealso demonstrate how our analytical model could be applied inconjunction with a Learning Automata (LA) algorithm for opt i-mal channel assignment. Based on the insights provided by ouranalytical model, we propose a simple decentralized algorithmwhich providesstatic channel assignments that areNash equilibriain pure strategies for the objective of maximizing normalizednetwork throughput. Our channel assignment algorithm requiresneither any explicit knowledge of the topology nor any messagepassing, and provides assignments in only as many steps as thereare channels. In contrast to prior work, our approach to channelassignment is based on thethroughput metric.

Index Terms—throughput modeling, fixed point analysis, chan-nel assignment algorithm, Nash equilibria, learning automata

I. I NTRODUCTION

This paper is concerned withinfrastructure modeWirelessLocal Area Networks (WLANs) that use the Distributed Co-ordination Function (DCF) Medium Access Control (MAC)protocol as defined in the IEEE 802.11 standard [1]. SuchWLANs contain a number of Access Points (APs). Everyclient station (STA) in the WLAN associates with exactly oneAP. Each AP, along with its associated STAs, defines acell.Each cell operates on a specific channel. Cells that operate onthe same channel are calledco-channel. Thus, in our setting,DCF is used only forsingle-hopcommunication within thecells, and STAs can access the Internet only through theirrespective APs, which are connected to the Internet by a high-speed wire line local area network. Figure 1 depicts such amulti-cell infrastructure WLAN.

To support the ever-increasing user population at high accessspeeds, WLANs are resorting to dense deployments of APswhere, for every STA, there exists an AP close to the STAwith which the STA can associate at a high Physical (PHY)rate [2]. However, as the density of APs increases, cell sizesbecome smaller and, since the number of non-overlapping

Local Server

Cell 1

Cell 2

Cell 3

Cell 5

Cell 4

Internet

Backbone Link AP STA

Telco/LAN Router

Fig. 1. A multi-cell infrastructure WLAN: DCF is used only forcommunications within the cells. Connectivity to nodes outside thecell is provided over a separate backbone which connects theAPs tothe Internet through a local server and a LAN router.

channels is limited1, co-channel cells become closer. Nodesin two closely located co-channel cells can suppress eachother’s transmissions via carrier sensing and interfere witheach other’s receptions causing packet losses. Thus, capacitymight sometimes degrade with increased AP density [3], [4].Clearly, effective planning and management are essential forachieving the benefits of dense deployments of APs. It hasbeen demonstrated that a dense deployment of APs, alongwith careful channel assignment and user association control,can enhance the capacity by as much as 800% [2]; but thetechnique has been tested on a small scale2. Large-scaleWLANs are difficult to plan and manage since good networkengineering models are lacking.

Much of the earlier work on modeling WLANs deals withsingle-AP networks or the so-calledsingle cells[5], [6], [7].

1For example, the number of non-overlapping channels in 802.11b/g is 3and that in 802.11a is 12.

2In [2], to provide connectivity at high PHY rates, 24 APs weredeployedin an area which could have been covered by a single AP at a low PHY rate.

http://arxiv.org/abs/0903.0096v2

The existing performance analyses of multi-cell WLANs(i.e., WLANs consisting of multiple APs) are mostly basedeither on simulations or small-/medium-scale experiments[2], [3], [4]. Studying large-scale WLANs by simulationsand experiments is both expensive and time-consuming. It is,therefore, important to develop an analytical understandingof such WLANs in order to derive insights into the systemdynamics. The insights thus obtained can be applied: (i)to facilitate planning and management, and (ii) to developefficient adaptive schemes for making the WLANsself-organizingand self-managing. In this paper, we first developan analytical model for 802.11-based multi-cell WLANs andthen apply our model to the task of channel assignment.

Our Contributions: We make the following contributions:• We identify a condition, which we call thePairwise

Binary Dependence(PBD) condition, under which multi-cell WLANs can be modeled at thecell level(see A.3 inSection IV).

• We develop a scalable cell level model for multi-cellWLANs with arbitrary cell topologies(Section V).

• We extend the single cell TCP analysis of [8] to multipleinterfering cells (Section V-B).

• We demonstrate how our model could be applied inconjunction with a Learning Automata (LA) algorithmfor optimal channel assignment (Section VIII).

• Based on the insights provided by our analytical model,we propose a simple decentralized algorithm which canprovide faststatic channel assignments (Section IX).

Unlike the node levelmodels [9], [10] or thelink levelmodels [11], [12] reported earlier in the context ofmulti-hop ad hocnetworks, our cell level model does not requiremodeling the activities of every single node (resp. link). Thus,the complexity of our cell level model increases with thenumber of cells rather than the number of nodes (resp. links).However,our model can account for the number of nodes ineach cell which may differ across the cells.

We argue that the PBD condition is likely to hold in adense deployments of APs (see Sections III and IV-A). Hence,our cell level model can be applied to obtain afirst-cutunderstanding of large-scale WLANs.

Our cell level model is based on the channel contentionmodel of Boorstyn et al. [9] and the transmission attemptmodel of [7]. Thus, our approach is similar to that of [10].However, our model of transmission attempt process is simpler(see the discussion following Equation 2) and the closed-formexpressions for collision probabilities and cell throughputs thatwe derive are new. We also provide new insights.

Our channel assignment algorithm provides assignmentsthat areNash equilibria in pure strategiesfor the normalizednetwork throughput maximizationobjective (see Section IX).Furthermore it provides an assignment in onlyM steps whereM denotes the number of available channels. In contrast toprior work, our approach to channel assignment is based on thethroughput metric(see the last paragraph in Section II). Our

channel assignment algorithm is motivated by our analyticalmodel which requires the knowledge of the topology of APs.However, our channel assignment algorithm requires neitherany topology information nor any message passing. To bespecific, our channel assignment algorithm takes the help ofthe network to discover an optimal assignment (in the sensementioned above) in a completely decentralized manner.

The remainder of this paper is organized as follows. InSection II, we discuss the related literature. In Section III,we provide the motivation for our simple cell level model. InSection IV, we provide our network model and summarize ourkey modeling assumptions. The analytical model is developedin Section V. In Section VI, we validate our model bycomparing withns-2 simulations. In Section VII, we applyour model to compare three different channel assignments fora 12-cell network and emphasize the impact of carrier sensingin 802.11 networks. In Section VIII, we demonstrate howour analytical model could be applied along with a LearningAutomata (LA) algorithm for optimal channel assignment. Wepropose a simple and fast decentralized channel assignmentalgorithm in Section IX, and summarize the conclusions inSection X. Two lengthy derivations have been deferred to theappendices at the end of the paper.

II. RELATED L ITERATURE

Modeling of multi-hop ad hoc networksis closely relatedto that of multi-cell WLANs. In both cases, thehidden nodeand theexposed nodeproblems [13] are known to be the twomain capacity-degrading factors. Multi-hop ad hoc networksand multi-cell WLANs are much harder to model than singlecells precisely because of the presence of hidden and exposednodes. Since the evolution of activities of each node is, ingeneral, statistically different from that of the others’,oneneeds to model the system at thenode level[9], [10], or at thelink level [11], [12], i.e., the activities of every single node orlink and the interactions among them need to be modeled.

In the context of CSMA based multi-hop packet radionetworks, Boorstyn et al. [9] proposed a Markovian modelwith Poisson packet arrivals and arbitrary packet length distri-butions. Wang and Kar [11] adopted the node level Boorstynmodel to develop a link level model for 802.11 networks andstudied the fairness issues. As a simplification, they assumeda fixed contention window. Garetto et al. [10] extended theBoorstyn model to multi-hop 802.11 networks. They computedthe packet loss probabilities and the throughputs per nodeaccounting for the details of the 802.11 protocol. In particular,they incorporated the evolution of contention windows, whichwas missing in [11], by applying the analysis of [7]. However,a direct application of the Boorstyn model required twodifferent activation rates, namely,λ andg, which complicatedtheir model (see the discussion following Equation 2). Wedevelop a simpler model by slightly modifying the Boorstyncontention model and making it particularly suitable for 802.11networks.

In the context of multi-cell infrastructure WLANs, Nguyenet al. [14] proposed a model for dense 802.11 networks. To

keep their interference analysis tractable, they assumed allthe APs in the WLAN to be operating on the same channel.Recently, Bonald et al. [15] applied the concept of “exclusionregion” to model a multi-AP WLAN as a network of multi-class processor-sharing queues with state-dependent servicerates. The concept of “exclusion domains” has also beenapplied in [12] to study the long-term fairness properties oflarge networks. The concept of exclusion says that, among aset of links that are in conflict, at most one can be active atany point of time. As argued in [12] and [15], exclusion canbe enforced in 802.11 networks by the RTS/CTS mechanism.However, suppression of interferers by RTS/CTS is not perfectbecause: (1) RTS frames can also collide and they must beretransmitted to enforce exclusion, (2) the overheads due toRTS/CTS that are transmitted at low rates and the capacitywastage due to RTS collisions might be significant especiallywhen data payloads are transmitted at high rates, and mostimportantly, (3) RTS/CTS cannot completely eliminate hiddennode collisions since nodes that cannot decode the CTSframes may collide with DATA frames that are longer thanthe Extended Inter Frame Space (EIFS). Thus,“exclusion”is strongly dependent on the virtual carrier sensing enforcedby the RTS/CTS mechanism and is, essentially, a modelingapproximation which ignores the possibility of hidden nodecollisions. The ineffectiveness of RTS/CTS is well known [16]and Basic Access can outperform RTS/CTS at high data rateseven when hidden nodes are present [17].

Channel assignment in WLANs has been extensively stud-ied (see, e.g., [18], [19], [20], [21], [22], [23] and thereferences therein). Much of the existing work on channelassignment proposes to minimize the global interference poweror maximize the global Signal to Noise and Interference Ratio(SINR) without taking into account the combined effect of thePHY and the MAC layers. Due to carrier sensing, nodes in802.11 networks get opportunity to transmit for only a fractionof time, and this must be accounted for when computing theglobal interference power or SINR. Such an approach is foundonly in [18] where the authors propose to maximize a quantitycalled “effective channel utilization”. In reality, however, endusers are more interested in the “throughputs” and, ideally, theobjective should be to maximize thesum utilityof throughputs.A heuristic method for computing throughputs can be found in[23] where the authors assume long term fairness among thenodes that are mutually interfering. The heuristic adoptedin[23] says that, ignoring collisions and channel errors, a nodeobtains approximately1

nof the maximum capacity wheren

denotes the cardinality of the maximumclique3 to which thenode belongs. In reality, the size of the maximum clique towhich a node belongs can be very different from the size ofthe maximum clique to which one of its interfering neighborbelongs, and this leads to inconsistent predictions for nodethroughputs. Nevertheless, the authors in [23] claim that,theirheuristic can predict theglobal throughput quite well eventhough individual node throughputs may not be. We remark

3A clique is a set of mutually interfering nodes.

that, the multi-cell model in [15] does not actually model theDCF contention and adopts a model of capacity sharing similarto the heuristic of [23].

In summary, due to the inherent complexity of the problem,simplifying assumptions have often been made to developtractable analytical model. The common simplification is toignore hidden node collisions through suitable assumptions.We also ignore hidden node collisions to develop our model.However, unlike [15], our model can be applied either withthe Basic Access mechanism or with the RTS/CTS mechanismprovided that the PBD condition holds (see A.3 in Section IV).To our knowledge, an accurate throughput based approachwhich accounts for PHY-MAC interactions does not exist.Thus, we first develop a simple and accurate throughput modelfor multi-cell WLANs and then apply our model to the taskof channel assignment.

III. M OTIVATION FOR A CELL LEVEL MODEL

In a dense deployment of APs with denser user population,it seems practically impossible to apply a node or a link levelmodel for planning and managing the network. However, wecan exploit a specific characteristics of dense deployments.Let R denote thecell radius, i.e.,R is the maximum distancebetween an AP and the STAs associated to it. LetRcs denotethecarrier sensing range, i.e.,Rcs is the distance up to whichcarrier sensing is effective4. We observe that,Rcs >> R islikely to hold in a dense deployment of APs where, for everySTA, there is an AP very close to the STA. WithRcs >> R,the network model can be simplified in the following ways:

1) Since any transmitter ‘T’ is within a small distanceR from its receiver ‘R’, a node ‘H’ that is beyond adistanceRcs from ‘T’ (i.e., a hidden node) is unlikelyto interfere with ‘R’5, i.e., carrier sensing would avoidmuch of the co-channel interference and we may ignorecollisions due to hidden nodes.

2) If Node-1 in Cell-1 can sense the transmissions by Node-2 in Cell-2, then it is likely that all the nodes in Cell-1can sense the transmissions from all the nodes in Cell-2and vice versa, i.e., we may assume thatnodes belongingto the same cell have an identical view of the rest of thenetwork and interact with the rest of the network in anidentical manner. Figure 2 summarizes this idea.

The assumption that the AP and all its associated STAs havean identical view of the network has been applied in a denseAP setting [2] where the authors approximate STA statisticsby statistics collected at the APs for efficiently managing theirnetwork. We adopt this idea of [2] to develop an analyticalmodel. We identify the locations of the STAs with the locationsof their respective APs, and treat a cell as a single entity,thus yielding a scalablecell levelmodel. Unlike a node levelmodel, the complexity of a cell level model increases withthe number of cells rather than number of nodes. A simple

4Precise definition of carrier sensing range can be found in [13] and [24].5Ignoring noise, we have, SINR≥

“

RcsR

”ν, whereν is called thepath

loss exponentand takes a value between 2 to 4.

Fig. 2. A dense AP network: Small circular areas represent cells withradiusR. APs are shown as tiny circles at the centers of the cells. Thebig circular disk represents the area covered by the carriersensingrangeRcs of one of the APs. Observe that, withRcs >> R, othercells are either almost completely covered or almost not covered at allthrough carrier sensing. This indicates that, withRcs >> R, nodesbelonging to the same cell “perceive” identical medium activities, atleast approximately.

cell level model is particularly suitable for the task of channelassignment since channels are assigned to cells rather thantonodes. The foregoing simplification is extremely useful at thenetwork planning stage when the locations of the STAs are notknown but the locations of the APs and the expected numberof users per cell might be known. Furthermore, since much ofthe traffic in today’s WLANs is downlink, i.e., from the APsto the STAs, a large fraction of channel time is occupied bytransmissions from the APs. It is then reasonable to developamodel based only on the topology of the APs and the expectednumber of users per cell assuming that the users are locatedclose to their respective APs.

The appropriate locations for placing the APs can bedecided using standard RF tools and thephysical topologyof the APs can be determined. Given the physical topologyof the APs and a channel assignment, we can obtain thelogical topology, i.e., the topology of the co-channel APscorresponding to every channel. Given a logical topology, ouranalytical model can predict the cell throughputs accountingfor the expected number of users per cell. The cell throughputsthus obtained can provide thegoodnessof the assignment toa channel assignment algorithm based on which the algorithmcan determine abetter channel assignment. Given the logicaltopology with the new channel assignment, our model canprovide the modified feedback and this procedure can berepeated several times to arrive at an optimal assignment. Weexplain this approach in Section VIII where we apply ourmodel in conjunction with a Learning Automata (LA) basedalgorithm for optimal channel assignment.

IV. N ETWORK MODEL AND ASSUMPTIONS

We consider scenarios withRcs >> R. Based on ourearlier discussion, we assume that simultaneous transmissions

by nodes that are farther thanRcs from each other resultsin successful receptions at their respective receivers. Wealsoassume that simultaneous transmissions by nodes that arewithin Rcs always lead to packet losses at their respectivereceivers, i.e., we ignore the possibility ofpacket capture6.For simplicity, we assumesymmetryamong the nodes andof the propagation environment. In particular, this means that(i) nodes use identical protocol parameters (e.g., contentionwindows, retry limit etc.), and (ii) if Node-1 can sense Node-2, then Node-2 can also sense Node-1.

We say that two nodes aredependentif they are withinRcs; otherwise, the two nodes are said to beindependent.Two cells are said to be independent if every node in a cellis independent w.r.t. every node in the other cell; otherwise,the two cells are said to be dependent. Two dependent cellsare said to becompletely dependentif every node in a cellis dependent w.r.t. every node in the other cell. In this broadsetting, our key assumptions are the following:A.1 Only non-overlapping channels are used.A.2 Associations of STAs with APs are static. This implies

that the number of STAs in a cell is fixed.A.3 Pairwise Binary Dependence (PBD):Any pair of cells

is either independent or completely dependent.A.4 The STAs are so close to their respective APs that packet

losses due to channel errors are negligible.A.5 The EIFS deferral has been disabled in the sense that

medium access always starts after a Distributed Inter-Frame Space (DIFS) deferral.

A. Discussion of the PBD Condition and Assumption A.5

The PBD condition is crucial to the analytical model beingdeveloped in this paper. It includes the possibility that twogiven co-channel cells can be independent, i.e., two co-channelcells can be so far apart that activities in one cell do notaffect the activities in the other cell. However, if two cellsare dependent, the PBD condition rules out the possibilitythat only a subset of nodes in one cell can sense a subset ofnodes in the other cell. We note that, in a dense deploymentof APs, due to small cell radiusR andRcs >> R, completedependence would hold up toRcs and independence wouldhold beyondRcs, i.e., the PBD condition would hold, at leastapproximately. The PBD condition is a geometric propertythat enables modeling at the cell level since,if the PBDcondition holds, the relative locations of the nodes withinacell do not matter. Also, any interferer ‘I’, which must bewithin Rcs of a receiver ‘R’, will also be withinRcs of thetransmitter ‘T’ if the PBD condition holds. Hence, either (i)‘I’ can start transmitting at the same time as ‘T’ causingsynchronouscollisions at ‘R’, or (ii) ‘I’ gets suppressed byT’s RTS (or DATA) transmission followed by R’s CTS (orACK) transmission since CTS and ACK frames are givenhigher priority through SIFS (< DIFS < EIFS). Thus, ifthe PBD condition holds, nodes do not require deferral by

6The difficulty in modeling packet capture lies in analytically computingthe capture probabilities. Given the capture probabilities, packet capture canalso be incorporated along the lines of [25].

EIFS; deferral by DIFS would suffice. Thus, we assume thatcontention for medium access always begins after deferral byDIFS and we do not model the impact of EIFS. We remarkthat certain commercial products, e.g., Atheros cards, haveuser-configurable EIFS and EIFS can be easily disabled insuch devices.

V. A NALYSIS OF MULTI -CELL WLAN S WITH ARBITRARY

CELL TOPOLOGY

In this section, we develop a cell level model for WLANsthat satisfy the PBD condition. We provide a generic modeland demonstrate the accuracy of our model by comparingwith simulations of specific cell topologies pertaining to:(i)networks with linear or hexagonal layout of cells (see Figures3(a)-3(c)), and (ii) networks with arbitrary layout of cells(see Figure 3(d)). We index the cells by positive integers1, 2, . . . , N , in some arbitrary fashion whereN denotes thenumber of cells. Two completely dependent co-channel cellsare said to beneighbours. Note that,two completely dependentcells operating on different non-overlapping channels arenotneighbors. Let N = {1, 2, . . . , N} denote the set of cellsandNi (⊂ N ) denote the set of neighboring cells of Cell-i(i ∈ N ). Note thati /∈ Ni. Given the location of the APs and achannel assignment, we can obtain the set of neighboring cellsfor each cell. The key to modeling the cell level contentionis the cell levelcontention graphG which is obtained byrepresenting every cell by a vertex and joining every pairof neighbors by an edge. Figures 3(a)-3(d) also depict thecontention graphs corresponding to each cell topology.

In Section V-A, we model the case where nodes are in-finitely backlogged and are transferring packets to one or morenodes in the same cell using UDP connection(s). Notice thatsingle-hop direct communications among the STAs in the samecell without involving the AP are also allowed. In Section V-B,we extend to the case when STAs download long files throughtheir respective APs usingpersistentTCP connections.

A. Modeling with Saturated MAC Queues

Due to the PBD condition, nodes belonging to the same cellhave an identical view of the rest of the network. When onenode senses the medium idle (resp. busy) so do the other nodesin the same cell and we say that a cell is sensing the mediumidle (resp. busy). Since the nodes are saturated, whenever acellsenses the medium idle, all the nodes in the cell decrementtheir back-off counters per idle back-off slot that elapsesintheir local medium7 and we say that the cell is in back-off.If the nodes were not saturated, a node with an empty MACqueue would not count down during the “medium idle” periodsand the number ofcontendingnodes would be time-varying.With saturated AP and STA queues, the number of contendingnodes in each cell remains constant.

We say that a cell transmits when one or more nodes inthe cell transmit(s). When two or more nodes in the samecell transmit, anintra-cell collision occurs. Consider Figure

7Nodes belonging to different co-channel cells can have different views ofthe network activity.

1 2 3 4

(a)

1 2 3 4 5

(b)

1

7

5 2

34

6

(c)

43

5

6 72

1

(d)

Fig. 3. Examples of multi-cell systems: (a) four linearly placed co-channelcells, (b) five linearly placed co-channel cells, (c) seven hexagonally placedco-channel cells, and (d) seven co-channel cells with an arbitrary cell topology.The cell level contention graphs have also been shown where the dots representthe cells. Neighbors have been joined by edges. For example (a), the pairs{1, 2}, {2, 3} and {3, 4} are dependent and the pairs{1, 3}, {1, 4} and{2, 4} are independent.

3(a). There are periods during which all the four cells are inback-off. We model these periods, when none of the cells istransmitting, by the stateΦ whereΦ denotes theempty set. Thesystem remains in StateΦ until one or more cell(s) transmit(s).When a cell transmits, its neighbors sense the transmissionafter a propagation delay and they defer medium access. Wethen say that the neighbors areblockeddue to carrier sensing.However, two neighboring cells can start transmitting togetherbefore they could sense each other’s transmissions resulting insynchronous inter-cellcollisions.

We observe that a cell can be in one of the three states:(i) transmitting, (ii) blocked, or (iii) in back-off. Modelingthe synchronous inter-cell collisions requires a discretetimeslotted model. However, this would require a large state spacesince the cells change their states in an asynchronous manner.For example, consider Figure 3(a) and suppose that Cell-1 starts transmitting and blocks Cell-2 after a propagationdelay. However, Cell-3 is independent of Cell-1 and can starttransmitting at any instant during Cell-1’s transmission.Thus,the evolution of the system is partly asynchronous and partlysynchronous. To capture both, we follow atwo-stageapproachalong the lines of [10]. In the first stage, we ignore inter-cellcollisions and assume that blocking due to carrier sensing isimmediate. We develop a continuous time model as in [9] toobtain the fraction oftime each cell is transmitting/blocked/inback-off. In the second stage, we obtain the fraction ofslotsin which various subsets of neighboring cells can start trans-mitting together. This would allow us to compute the collisionprobabilities accounting for synchronous inter-cell collisions.We combine the above through a fixed-point equation andcompute the throughputs using the solution of the fixed-pointequation. We define the following [7]:

ni := number of nodes in Cell-i

βi := (transmission) attempt probability (over the back-offslots) of the nodes in Cell-i

γi := collision probability as seen by the nodes in Cell-i(conditioned on an attempt being made)

Let Ai(T ) denote the cumulative number of attempts madeby a tagged node (any node due to symmetry) in Cell-i up totimeT . LetCi(T ) denote the cumulative number of collisionsas seen by the tagged node up to timeT . LetBi(T ) denote thecumulative number of back-off slots elapsed in Cell-i up totime T . Then, the attempt probabilityβi and the (conditional)collision probabilityγi of the tagged node are more preciselydefined by

βi := limT→∞

Ai(T )

Bi(T ); γi := lim

T→∞

Ci(T )

Ai(T ).

Note thatβi is the attempt probability of the nodes in Cell-i, irrespective of whether the nodes in the other cells canalso attempt. This is a simplification and can be viewed asan extension to thedecoupling approximationintroduced in[5]. Using the decoupling approximation and the analysis in[7], the attempt probabilityβi of the nodes in Cell-i, ∀i ∈ N ,can be related toγi as

βi = G(γi) :=1 + γi . . .+ γK

i

b0 + γib1 . . .+ γki bk + . . .+ γK

i bK(1)

whereK denotes theretry limit andbk, 0 ≤ k ≤ K, denotesthe mean back-off sampled afterk collisions.

The First Stage:When Cell-i and some (or all) of its neigh-boring cells are in back-off their (cell level) attempt processescompete until one of the cells, say, Cell-j, j ∈ Ni ∪ {i}, trans-mits. Since, we ignore inter-cell collisions in the first stage,the possibility of two or more neighboring cells attemptingtogether is ruled out. When Cell-i wins the contention, we saythat it has becomeactive. When Cell-i becomes active, it gainsthe control over its local medium by immediately blocking itsneighboring cells that are not yet blocked. We assume that thetime until Cell-i goes from the back-off state to the active stateis exponentially distributed with mean1

λi. The activation rate

λi is given by

λi =1− (1− βi)

ni

σ(2)

whereσ denotes the duration of a back-off slot (in seconds)and1 − (1 − βi)

ni is the probability that there is an attemptin Cell-i per back-off slot. Notice that we have convertedthe aggregate attempt probability in a cell per back-offslotto an attempt rate over back-offtime. Also notice that, ourassumption of exponential “time until transition from the back-off state to the active state” is the continuous time analogue ofthe assumption of geometric “number of slots until attempt”in the discrete time models of [6] and [7].

Discussion 5.1:In [10], the authors use an unconditionalactivation rateλ over all times as well as a conditional

activation rateg over the back-off times and relate the tworates through a throughput equation which makes their modelunnecessarily complicated. The activation rateλ in our modelis conditional on being in the back-off state. Thus, we use asingle activation rate and our model is much simpler than thatof [10]. Our modified approach can also be applied to simplifythe node level model of [10].

When Cell-i becomes active, it remains active and its neigh-bors remain blocked due to Cell-i until Cell-i’s transmissionfinishes and an idle DIFS period elapses. The active periods ofCell-i are of mean duration1

µi. When Cell-i becomes active

through a successful transmission (resp. an intra-cell collision)its neighbors remain blocked due to Cell-i for a success timeTs (resp. acollision timeTc)8. Hence, 1

µiis given by

1

µi

=

(

niβi(1− βi)ni−1

1− (1− βi)ni

)

· (Ts)

+

(

1−niβi(1 − βi)

ni−1

1− (1 − βi)ni

)

· (Tc) (3)

whereniβi(1− βi)

ni−1

1− (1− βi)niis the probability that Cell-i becomes

active through a success given that it becomes active.Due to carrier sensing, at any point of time, only a set

A (⊂ N ) of mutually independent cells can be active together,i.e., A must be an independent set(of vertices) of thecontention graphG9. From the contention graphG, we candetermine the set of cellsBA that get blocked due toA, andthe set of cellsUA that remain in back-off, i.e., the set ofcells in which nodes can continue to decrement their back-offcounters. Note thatA, BA andUA form a partition ofN , i.e.,A, BA andUA are pairwise disjoint andN = A∪ BA ∪ UA.

We takeA(t), i.e., the setA of active cells at timet, as thestate of the multi-cell system at timet. It is worthwhile nowto mention theinsensitivityresult of Boorstyn et al. [9] whichsays that the product-form solution provided by their modelis insensitive to the packet length distribution and dependsonly on the mean packet lengths. Applying their insensitivityargument, we take the active periods of Cell-i to be i.i.d. ex-ponential random variables with mean1

µi. Due to exponential

activation rates and active periods, at any timet, the rate oftransition to the next state is completely determined by thecurrent stateA(t). For example, Cell-j, j ∈ UA, joins the setA (and its neighboring cells that are also inUA join the setBA) at a rateλj . Similarly, Cell-i, i ∈ A, leaves the setA (andits neighboring cells that are blocked only due to Cell-i leavethe setBA) to join the setUA at a rateµi. In summary, theprocess{A(t), t ≥ 0} has the structure of a Continuous TimeMarkov Chain (CTMC). This CTMC contains a finite numberof states and is irreducible. Hence, it is stationary and ergodic.

8For theBasic Access(resp.RTS/CTS) mechanism,Ts corresponds to thetime DATA-SIFS-ACK-DIFS (resp. RTS-SIFS-CTS-SIFS-DATA-SIFS-ACK-DIFS) andTc corresponds to the time DATA-DIFS (resp. RTS-DIFS). WhenDATA payload sizes are not fixed,Ts andTc are to be computed using theexpected payload size.

9An independent set of vertices of a graphG is a set of vertices ofG suchthat no two vertices in the set are connected by an edge.

Φ2

3

4

1,3

2,4 1,4

λ

λλ

λλ

λ

λ

λ

µ

µ

µ

µµ

µ

µ

µ1

1

1

2

2

3

4λ 4 λ 4

µ3 3 3µ1

1

4

1

4

2

2

4

1

Fig. 4. The CTMC describing the cell level contention for thefourlinearly placed cells given in Figure 3(a).

The set of all possible independent sets which constitutes thestate space of the CTMC{A(t), t ≥ 0} is denoted byA.For a given contention graph,A can be determined. For thetopology given in Figure 3(a), we haveN = {1, 2, 3, 4} andA = {Φ, {1}, {2}, {3}, {4}, {1, 3}, {1, 4}, {2, 4}} where werecall thatΦ denotes the empty set. The CTMC{A(t), t ≥ 0}corresponding to this example is given in Figure 4.

It can be checked that the transition structure of the CTMC{A(t), t ≥ 0} satisfies the Kolmogorov Criterion for re-versibility (see [26]). Hence, the stationary probabilitydistri-bution π(A),A ∈ A, satisfies the detailed balance equations,

π(A)λi = π(A ∪ {i})µi , (∀i ∈ UA)

and the stationary probability distribution has the form

π(A) =

(

∏

i∈A

ρi

)

π(Φ), (∀A ∈ A) (4)

where ρi := λi

µiand π(Φ) (which denotes the stationary

probability that none of the cells is active) is determined fromthe normalization equation

∑

A∈A

π(A) = 1. (5)

Combining Equations 4 and 5, we obtain

π(A) =

(

∏

i∈A

ρi

)

∑

A∈A

∏

j∈A

ρj

, (∀A ∈ A) (6)

Convention:A product∏

over an empty index set is taken tobe equal to 1. Notice that, with this convention, Equation 6holds forπ(Φ) as well.

The Second Stage:We now compute the collision proba-bilities γi’s accounting for inter-cell collisions. Note thatγi

is conditional on an attempt being made by a node in Cell-i. Hence, to computeγi, we focus only on those states inwhich Cell-i can attempt. Clearly, Cell-i can attempt in State-A iff it is in back-off in State-A, i.e., iff i ∈ UA. In allsuch states a node in Cell-i can incur intra-cell collisionsdue the other nodes in Cell-i. Furthermore, some (or all) ofCell-i’s neighbors might also be in back-off in State-A. Ifa neighboring cell, say, Cell-j, j ∈ Ni, is also in back-offin State-A, i.e., if j ∈ UA, then a node in Cell-i can incurinter-cell collisions due to the nodes in Cell-j. The collisionprobabilityγi is then given by Equation 7 (appears at the top ofthe next page). A formal derivation of Equation 7 is providedin Appendix A.

Fixed Point Formulation: Equations 2, 3, 6, 7 andρi := λi

µi

can express theγi’s as functions of only theβi’s. Togetherwith Equation 1, they yield anN -dimensional fixed pointequation where we recall thatN is the total number of cells.The N -dimensional fixed point equation can be numericallysolved to obtain the collision probabilitiesγi’s, the attemptprobabilitiesβi’s and the stationary probabilitiesπ(A)’s. In allof the cases that we have considered, the fixed point iterationswere observed to converge to the same solutions irrespectiveof starting points. However, we have not yet been able toanalytically prove the uniqueness of solutions.

Calculating the Throughputs: The stationary probabilitiesof the CTMC{A(t), t ≥ 0} can provide the fraction of timexi for which Cell-i is unblocked. A cell is said to be unblockedwhen it belongs to eitherA or UA. Thus,∀i ∈ N ,

xi =∑

A∈A : i∈A∪UA

π(A). (8)

Definition 5.1: Let Gi, i ∈ N , denote the subgraph obtainedby removing Cell-i and its neighboring cells inNi from thecontention graphG. For a given contention graphG, let ∆ bedefined as follows:

∆ :=∑

A∈A

∏

j∈A

ρj

(9)

Let ∆i denote the∆ corresponding to the subgraphGi.An important observation which facilitates the computation

of the xi’s is given by the following theorem.Theorem 5.1:The fraction of timexi for which Cell-i is

unblocked is given by

xi =(1 + ρi)∆i

∆, ∀i ∈ N . (10)

Proof: See Appendix B.Let Θi denote the aggregate throughput of Cell-i in a given

multi-cell network and letΘni,singlecell denote the aggregatethroughput of Cell-i if it was an isolated cell containingni nodes. BothΘi andΘni,singlecell are in packets/sec. WeapproximateΘi by

γi =

∑

A∈A : i∈UAπ(A)

[

1− (1− βi)ni−1

∏

j∈Ni : j∈UA(1− βj)

nj

]

∑

A∈A : i∈UAπ(A)

(∀i ∈ N ) (7)

Θi = xi ·Θni,singlecell

=(1 + ρi)∆i

∆·Θni,singlecell , (11)

and Θi divided by ni gives the per node throughputθi inCell-i, i.e., θi = Θi

ni(packets/sec).

Discussion 5.2:Equation 11 is justified as follows. If Cell-i is indeed independent of every other cell in the network,it is never blocked and does not incur inter-cell collisions.Then, we havexi = 1 andΘi = Θni,singlecell . However, ingeneral, Cell-i gets blocked due to its neighbors for a fractionof time 1 − xi and remains unblocked for a fraction of timexi. If we ignore the time wasted in inter-cell collisions, thetimes during which Cell-i is unblocked would consist onlyof the back-off slots and the activities of Cell-i by itself.Thus, we approximate the aggregate throughput of Cell-i, overthe times during which it is unblocked, byΘni,singlecell andΘni,singlecell multiplied with xi gives the aggregate through-put Θi of Cell-i in the multi-cell network.

Clearly, Equation 11 is an approximation since the timewasted in inter-cell collisions have been ignored. However,we prefer to keep the approximation because: (1) it is quiteaccurate when compared with the simulations (see SectionVI), and (2) it can be efficiently computed sinceΘni,singlecell

follows from a single cell analysis and the∆ as well as the∆i’s can be computed using efficient algorithms [27].

Complexity of the Model: In general, obtaining the statespaceA by searching for all possible independent setsAcould be computationally expensive and the complexity growsexponentially with the number of vertices in the contentiongraph [27]. For realistic topologies, where connectivity in thecontention graph is related to distance in the physical network,efficient computation ofA is possible up to several hundredvertices in the contention graph [27]. Thus, a cell level modelis extremely helpful in analyzing large-scale WLANs withhundreds of cells since, unlike a node level model, each vertexin the contention graph now represents a cell10.

Large ρ Regime: Let η (resp.ηi) denote the number ofMaximum Independent Sets11 (MISs) of G (resp. Gi) (seeDefinition 5.1). Notice that,ηi is also equal to the numberof MISs of G to which Cell-i belongs. From Equation 6 it iseasy to see that, asρi → ∞, ∀i ∈ N , we have,

π(A) →

1

ηif A is an MIS,

0 otherwise.

10Note that, the state spaceA depends only on the cell topology and doesnot depend on the number of nodes in each cell. It also does notdepend onwhether the nodes use same/different protocol parameters.

11A maximum independent set of a graph is an independent set of the graphhaving maximum cardinality.

Also, from Equations 8, 9 and 10, we observe that, asρi → ∞, ∀i ∈ N , we have,

xi →ηiη

,

where we recall thatxi is the fraction of time for which Cell-iis unblocked. The quantity

xi =Θi

Θni,singlecell

can be interpreted as the throughput of Cell-i, normalizedwith respect to Cell-i’s single cell throughput. We define thenormalized network throughputΘ by

Θ :=

N∑

i=1

xi. (12)

Clearly, asρi → ∞, ∀i ∈ N , the cells that belong to everyMIS of the contention graph obtain normalized throughput 1and the cells that do not belong to any MIS obtain normalizedthroughput 0. Similar observations have also been made in[11]. We further observe that, asρi → ∞ for all i ∈ N , onlyan MIS of cells can be active at any point of time. Since anMIS is always active, asρi → ∞, ∀i ∈ N , we have,

Θ → α(G) ,

whereα(G) denotes the cardinality of an MIS ofG and is alsocalled theindependence numberof G. Notice thatα(G) is ameasure ofspatial reusein the network.

B. Extension to TCP Traffic

We now extend the analysis of Section V-A to the morerealistic case when users access a local proxy server viapersistentTCP connections. Our extension to TCP-controlledlong file downloads is based on the single cell TCP-WLANinteraction model of [8]. The model proposed in [8] has beenshown to be quite accurate when: (1) the local proxy serveris connected with the AP by a relatively fast wired LAN suchthat the AP in the WLAN is the bottleneck, (2) every STA hasa single persistent TCP connection, (3) there are no packetlosses due to buffer overflow, (4) the TCP timeouts are setlarge enough to avoid timeout expirations due to Round TripTime (RTT) fluctuations, and (5) the delayed ACK mechanismis disabled. We keep the above assumptions in this paper.

In [8], the authors propose to model a single cell havingan AP and an arbitrary number of STAs with long-livedTCP connections by an “equivalent saturated network” whichconsists of a saturated AP and a single saturated STA. “Thisequivalent saturated model greatly simplifies the modelingproblem since the TCP flow control mechanisms are nowimplicitly hidden and the total throughput can be computed

using the saturation analysis [8].” Using the equivalent satu-rated model of [8], the analysis of Section V-A can be appliedto TCP-controlled long file downloads, takingni = 2, ∀i ∈ N .We explain the intuition behind this straightforward extensionin the following.

The equivalent saturated model of [8] is quite accurate inthe single cell scenario because of the following reasons. ForTCP-controlled long file transfers in WLANs, the AP hasto serven ≥ 1 STA(s) by sending TCP DATA (resp. TCPACK) packets tond downloading (resp.nu uploading) STAs(nl + nu = n). However, since DCF ispacket fair, the APdoes not get any prioritized access to the medium. Hence,in steady state, the AP’s MAC queue never becomes empty,i.e., the AP is saturated12. Furthermore, since it is assumedthat there are no buffer losses and no TCP timeouts, initially,the TCP windows keep growing and, in steady state (i) theTCP windows stay at their maximum, say,Wmax, for everyconnection, and (ii) the sum of the number of packets in theAP and in the STAs is equal tonWmax. Most of these packetsreside in the AP queue due to the DCF contention mentionedabove and only a few STA queues remain non-empty at anypoint of time. It turns out that, the mean number of STAswith non-empty MAC queues≈ 1.5 for sufficiently largen[8] (n ≥ 5 suffices)13. Thus, then STAs can be approximatedby a single saturated STA and the equivalent saturated modelcan still predict the throughputs quite accurately. In fact, theequivalent saturated model can also accurately predict thecollision probability of the AP which is indeed saturated14.

Our extension to multiple cells is based on our key obser-vations that,

1. As in single cells, the AP continues to be saturated inmulti-cell scenarios as well.

2. The number of non-empty STAs changes only at the endof successful transmissions [28]; when the AP (resp. aSTA) succeeds, this number increases (resp. decreases)by one.

3. Whenever a cell is blocked, its activities arefrozen, andhence, the number of non-empty STAs cannot changeduring the blocked periods. Thus, we can strip off theblocked periods and put the unblocked periods togetherover which time the cell would behave like a single cell.

Applying the third observation above, the TCP downloadthroughput of a cell, say, Cell-i, in the multi-cell scenariocan be obtained by multiplying the single cell TCP downloadthroughput of Cell-i with the fraction of timexi for whichCell-i is unblocked. Clearly, due to blocking, the likelihoodof TCP timeouts increases. However, in our simulations, wehave observed that,if the minimum value of Retransmission

12This observation holds in steady state even for then = 1 STA case ifthe maximum TCP window is large enough. In fact, for then = 1 STA case,in steady state, the STA is also saturated [7]. Thus, for then = 1 STA case,the equivalent saturated model is exact.

13In fact, applying the Markov model of [28], it can be formallyprovedthat the expected number of non-empty STAs is equal to 1.5.

14Our simulations indicate that, the equivalent saturated model cannotaccurately predict the collision probabilities of the STAs.

TABLE ISINGLE CELL RESULTS: COLUMNS 2-3 CORRESPOND TOn SATURATED

NODES. COLUMNS 4-7 CORRESPOND TOTCP DOWNLOADS WITH n STAS

n γsatana θsatana γAP

sim γAPana θAP

sim θAPana

(pkts/sec) (pkts/sec) (pkts/sec)

1 0 801.78 0.0578 0.0586 454.01 456.532 0.0586 349.94 0.0538 0.0586 456.06 456.533 0.1077 236.09 0.0533 0.0586 456.09 456.534 0.1473 176.63 0.0528 0.0586 456.17 456.535 0.1812 140.29 0.0531 0.0586 456.05 456.536 0.2100 115.89 0.0530 0.0586 456.10 456.537 0.2348 98.43 0.0536 0.0586 456.88 456.538 0.2565 85.35 0.0531 0.0586 456.97 456.5310 0.2927 67.11 0.0531 0.0586 456.02 456.53

TimeOut (RTO) is set to 200ms (which is the default valuein ns-2.31), the analytical model is capable of predicting APstatistics quite well both in the single cell and the multi-cellscenarios. In fact, cells switch between the blocked and theunblocked states at a fast time scale of few packet transmissiontimes which takes only a few milliseconds. Hence, timeoutexpirations are rare given that there are no buffer losses15.

VI. RESULTS AND DISCUSSION

We carried out simulations usingns-2.31[29]. We createdthe example topologies given in Figure 3(a)-3(d). We chosethe cell radii and the distances among the cells such that thePBD condition holds. Nodes were randomly placed within thecells. The saturated case was simulated with high rate CBRover UDP connections. For the TCP case, we created one TCPdownload connection per STA. Each TCP connection was fedby an FTP source with the TCP source agent attached directlyto the AP to emulate a local proxy server. The AP bufferwas set large enough to avoid buffer losses. The EIFS deferraland the delayed ACK mechanism were disabled. Each casewas simulated 20 times, each run for 200 sec of “simulatedtime”. We report the results for “Basic Access”. Similar resultswere obtained with “RTS/CTS”. We took 11 Mbps data rateand packet payloads of 1000 bytes. The function “fsolve()” ofMATLAB was used for solving theN -dimensional fixed pointequation.

Table I summarizes the results for a single cell. Columns2-3 (resp. 4-7) correspond to the saturated case (resp. TCPdownload case) withn saturated nodes (resp. 1 AP andnSTAs). The analytical results for the saturated case wereobtained using [7] and that for the TCP case were obtainedusing [8]. These single cell results obtained from knownanalytical models serve as the basis of our multi-cell results.The analytical throughputs per-node (resp. of AP) in Column3 (resp. Column 7) multiplied with thexi’s obtained from ourmulti-cell analysis provide the analytical throughputs per-node(resp. of AP) in the multi-cell cases (see Equation 11).

15In certain cases, where some cells remain severely blocked,their blockedperiods can be very long and TCP timeouts can occur for their respectiveconnections. As we show in Section VI, our analytical model correctlyidentifies such severely blocked cells by predicting close to zero throughputsfor them.

TABLE IIRESULTS FOR THEFOUR L INEARLY PLACED CELLS GIVEN IN FIGURE

3(A) WHEN EACH CELL CONTAINSn = 5 NODES

Cell γsim γana θsim θana θ∞index (pkts/sec) (pkts/sec) (pkts/sec)

1 0.2351 0.2399 94.48 97.41 93.532 0.3005 0.3146 41.21 46.66 46.763 0.2999 0.3146 41.66 46.66 46.764 0.2359 0.2399 93.99 97.41 93.53

TABLE IIIRESULTS FOR THEFIVE L INEARLY PLACED CELLS GIVEN IN FIGURE

3(B) WHEN EACH CELL CONTAINS n = 5 NODES


1 0.1882 0.1897 129.35 131.35 140.292 0.3321 0.3975 8.69 8.64 03 0.1892 0.1925 123.35 126.41 140.294 0.3321 0.3975 8.72 8.64 05 0.1884 0.1897 129.31 131.35 140.29

TABLE IVRESULTS FOR THESEVEN HEXAGONALLY PLACED CELLS GIVEN IN

FIGURE 3(C) WHEN EACH CELL CONTAINS n = 10 NODES


1 0.2335 0.8896 0.003 0.02 02 0.3045 0.3158 31.97 32.35 33.563 0.3061 0.3158 31.93 32.35 33.564 0.3055 0.3158 32.05 32.35 33.565 0.3054 0.3158 31.86 32.35 33.566 0.3070 0.3158 32.00 32.35 33.567 0.3058 0.3158 31.95 32.35 33.56

TABLE VRESULTS FOR THESEVEN ARBITRARILY PLACED CELLS GIVEN IN

FIGURE 3(D) WHEN CELL-i, 1 ≤ i ≤ 7 CONTAINS ni = i+ 1 NODES

Cell ni γsim γana θsim θana θ∞index i (pkts/sec) (pkts/sec) (pkts/sec)

1 2 0.0669 0.0666 320.66 325.26 349.942 3 0.1163 0.1163 216.19 219.65 236.093 4 0.2764 0.3280 12.48 12.97 04 5 0.3105 0.3318 34.29 40.20 46.765 6 0.2505 0.2585 83.77 84.92 77.266 7 0.3574 0.3787 28.67 32.40 32.817 8 0.3062 0.3139 56.87 59.21 56.90

A. Results for the Saturated Case

Tables II-IV summarize the results for the example multi-cell cases depicted in Figures 3(a)-3(c), respectively, wheneach cell containsn saturated nodes. Table V summarizes theresults for the example case given in Figure 3(d) when Cell-i,1 ≤ i ≤ 7, containsni = i + 1 saturated nodes. Quantitiesdenoted with a subscript“sim” (resp. “ana” ) correspondto results obtained fromns-2 simulations (resp. fixed pointanalysis). In each case,θ∞ represents the throughput per-nodeobtained by takingρi → ∞, ∀i ∈ N . We report only the meanvalues for our simulation results. The 99% confidence intervalswere observed to be within 5% of the mean values.

We show the plots corresponding to Table V in Figures5 and 6 which compare the collision probabilityγ and thethroughput per nodeθ, respectively. In Figures 5 and 6, wealso show the relevant single cell results obtained from Table

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

1 2 3 4 5 6 7

Colli

sion P

robabili

ty

Cell Index

ns-2 simulationanalyticalsingle cell

Fig. 5. Comparing collision probabilityγ for the example scenarioin Figure 3(d) when Cell-i, 1 ≤ i ≤ 7, containsni = i+1 saturatednodes.

0

50

100

150

200

250

300

350

1 2 3 4 5 6 7

Thro

ughput per

node (

pack

ets

/sec)

Cell Index

ns-2 simulationanalytical

infinite ρ approximationsingle cell

Fig. 6. Comparing throughput per nodeθ for the example scenarioin Figure 3(d) when Cell-i, 1 ≤ i ≤ 7, containsni = i+1 saturatednodes.

I, i.e., the results one would expect had the seven cells beenmutually independent. Referring Tables II-V, and Figures 5and 6, we make the following observations:O-1.) Collision probabilities (resp. throughputs) in the multi-cell scenarios are always higher (resp. lower) than the corre-sponding single cell values (see Figures 5 and 6) because (a)inter-cell collisions can be significant, and (b) due to inter-cellblocking, cells get opportunity to transmit only a fractionoftime.O-2.) Our analytical model is quite accurate (less than 10%error in most cases) in predicting the collision probabilitiesand throughputs. However, our model always over-estimatesthe throughputs since the time wasted in inter-cell collisionshave been ignored. Ignoring inter-cell collisions in the firststage of the model also over-estimates the fraction of timespent in back-off. Thus, the collision probabilities are alsoover-estimated.O-3.) The relative mismatch between the analytical model andthe simulation is observed to be the worst for cells that remain

blocked most of the time. For example, consider the first rowof Table IV which corresponds to Cell-1 in Figure 3(c). SinceCell-1 is dependent with respect to every other cell, it obtainsvery few attempt opportunities in the simulations and itscollision probability had to be averaged over very few samples.The same argument applies to the collision probability forCell-3 in Figure 5.O-4.) Our analytical model correctly identifies the severelyblocked cells, e.g., Cell-2 and Cell-4 in Table III, Cell-1in Table IV, and Cell-3 in Table V obtain significantly lessthroughputs than the other cells in the respective networks.Furthermore, our model works well with either equal orunequal number of nodes per cell.O-5.) Throughput distribution among the cells can be very un-fair even over long periods of time. Furthermore, introductionof a new co-channel cell can drastically alter the throughputdistributions. For example, compare Tables II and III. Cell-2and Cell-4 severely get blocked if Cell-5 is introduced to thefour cell network given in Figure 3(a).O-6.)The throughput of a cell cannot be accurately determinedbased only on thenumberof interfering cells. Consider, for ex-ample, Figure 3(d). Cell-3 and Cell-4 each have two neighborsbut their per node throughputsθ are quite different (see Figure6). In particular,θ4 > θ3 even thoughn3 = 4 < n4 = 5. Thisis due to Cell-7 which blocks Cell-6 for certain fraction of timeduring which Cell-4 gets opportunity to transmit whereas Cell-1 and Cell-2 are almost never blocked and Cell-3 is almostalways blocked due to Cell-1 and Cell-2. Thus,topology playsthe key role and heuristic methods based only on the numberof neighbors as in [23] would fail to capture the individualcell throughputs.

B. Results for the TCP Download Case

Referring Columns 4-7 of Table I it can be seen that The APstatistics is largely insensitive to the number of STAs. Also,O-7.) The collision probability of the AP in a single cell withany number of STAs is approximately equal to the collisionprobability in a single cell with two saturated nodes. Thiscan be verified by comparing Columns 4 and 5 with Row 2Column 2 in Table I.O-8.) The AP throughput does not change with the num-ber of STAs. In fact, the AP throughput with any numberof downloading STAs is equal to the per node throughputin a single cell containing two saturated nodes with pay-load size LTCP−DATA+LTCP−ACK

2 whereLTCP−DATA andLTCP−ACK denote the size of TCP DATA and TCP ACKpackets.

ObservationsO-7 and O-8 are well-known and they formthe basis for the equivalent saturated model of [8]. ObservationO-7 has led to the conclusion in [3] that the collision probabil-ity of the nodes in a WLAN containingm mutually interfering(i.e., dependent) APs is equal to the collision probabilityina single cell containing2m saturated nodes since each cellcan be assumed to consist of two saturated nodes (a saturatedAP and a saturated STA). Tables VI-VIII summarize the APstatistics for the topologies in Figures 3(a), 3(b) and 3(d),

TABLE VIRESULTS FOR THEAP CORRESPONDING TOFIGURE 3(A) WHEN EACH

CELL CONTAINS 1 AP AND n = 5 STAS

Cell γsim,AP γana,AP θsim,AP θana,AP θ∞,AP

index (pkts/sec) (pkts/sec) (pkts/sec)

1 0.1038 0.1033 306.33 318.73 304.352 0.1560 0.1574 153.16 169.18 152.183 0.1555 0.1574 153.06 169.18 152.184 0.1038 0.1033 306.41 318.73 304.35

TABLE VIIRESULTS FOR THEAP CORRESPONDING TOFIGURE 3(B) WHEN EACH




1 0.0728 0.0775 381.21 387.16 456.532 0.1793 0.1950 75.16 85.62 03 0.0744 0.0832 340.24 346.47 456.534 0.1786 0.1950 75.23 85.62 05 0.0728 0.0775 381.15 387.16 456.53

TABLE VIIIRESULTS FOR THEAP CORRESPONDING TOFIGURE 3(D) WHEN EACH




1 0.0610 0.0670 421.70 425.83 456.532 0.0604 0.0670 421.92 425.83 456.533 0.2010 0.2528 33.79 38.50 04 0.1561 0.1685 141.80 156.41 152.185 0.0987 0.1028 317.39 329.06 304.356 0.1551 0.1644 158.55 172.64 152.187 0.1061 0.1099 301.15 314.10 304.35

respectively. Since the AP statistics does not change with thenumber of STAs, we report the results withn = 5 STAs ineach case. We show the plots corresponding to Table VIII inFigures 7 and 8 which compare the collision probabilityγand the throughput per nodeθ, respectively. Referring TablesVI-VIII, and Figures 7 and 8, we conclude that the foregoingobservations (O.1-O.6) for the saturated case carry over toTCP-controlled long file transfers as well. Furthermore, wegeneralize the conclusion of [3] as follows:O-9.) The collision probability in a WLAN depends, not onlyon the number of interfering APs but also on the fractionof time for which the neighboring APs can cause collisions.In general, collision probability does not grow as twice thenumber of interfering APs and depends on the cell topology.

Referring Tables VI-VIII we extend the validity of theequivalent saturated model of [8] as follows:O-10.) The equivalent saturated model of [8] proposed in thecontext of a single cell, preserves its desirable properties, i.e.,it predicts the AP statistics quite well when extended to amulti-cell WLAN that satisfies the PBD condition.

C. Variation withρ

Till now, we have discussed the results corresponding tothe payload size of 1000 bytes. We now examine how theresults vary with the payload size. Figures 9 and 10 depict thevariation with payload size of analytically computed collisionprobabilities and normalized cell throughputs, respectively, for

0

0.05

0.1

0.15

0.2

0.25

0.3

1 2 3 4 5 6 7

AP

Colli

sion P

robabili

ty

Cell Index

ns-2 simulationanalyticalsingle cell

Fig. 7. Comparing collision probabilityγ for the example scenarioin Figure 3(d) when each cell contains an AP andn = 5 STAs. STAsare downloading long files through their respective APs using TCPconnections.

0

100

200

300

400

500

600

1 2 3 4 5 6 7

AP

Thro

ughput (p

ack

ets

/sec)

Cell Index

ns-2 simulationanalytical

infinite ρ approximationsingle cell

Fig. 8. Comparing throughput per nodeθ for the example scenarioin Figure 3(d) when each cell contains an AP andn = 5 STAs. STAsare downloading long files through their respective APs using TCPconnections.

the seven cell network of Figure 3(d) when each cell containsn = 10 saturated nodes. Similar results were obtained for theTCP case as well. From Figures 9 and 10, we observe that:O-11.) The results are largely insensitive to the variation inpayload size. Moreover, as the payload size increases, thenormalized cell throughputs become closer to the normalizedthroughputs under the infiniteρ approximation, i.e., theybecome closer tox1 = x2 = 1, x3 = 0, x4 = x6 =13 , x5 = x7 = 2

3 . Hence, except for very small payloadsizes, the infiniteρ approximation can be expected to providefairly accurate predictions. Furthermore, for sufficiently largepayload sizes, we have,Θ ≈ α(G).

VII. A S IMPLE DESIGN EXAMPLE

We consider a 12-cell network as shown in Figure 11 andapply our throughput model to compare three different channelassignments as shown in Figures 12(a)-12(c). Notice that the“logical” contention graphs for the three assignments in Fig-

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0 1000 2000 3000 4000 5000 6000

Colli

sion P

robabili

ty γ

Payload Size in bytes

γ1γ2γ3γ4γ5γ6γ7

Fig. 9. Variation of collision probabilities with payload size for theseven cell network in Figure 3(d) when each cell containsn = 10

saturated nodes.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Norm

aliz

ed C

ell

Thro

ughput

Payload Size in bytes

x1x2x3x4x5x6x7

Fig. 10. Variation of normalized cell throughputs with payload sizefor the seven cell network in Figure 3(d) when each cell containsn = 10 saturated nodes.

Fig. 11. A 12-cell network: Cell indices have been marked. Alsoshown is the cell levelphysical contention graph assumed for thenetwork.

ures 12(a)-12(c) are different from the “physical” contentiongraph of the network shown in Figure 11 which is basedon the physical separation among the cells. Two physicallydependent cells become logically independent if they areassigned different channels and the corresponding edge willbe missing in the logical contention graph. Our objective isto

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Fig. 12. Comparing three channel assignments for a 12-cell network: The cell levellogical contention graph are also shown for each assignment.

determine the best, among the three assignments, in terms ofthe normalized network throughputΘ (see Equation 12) andthe fairness index reported in [30]. Considering fairness at thecell level, the fairness indexJ is given by [30]

J :=(∑N

i=1 xi)2

N(∑N

i=1 x2i )

,

where we recall thatxi is the normalized throughput of Cell-i. Note that,0 ≤ J ≤ 1, and higher values ofJ imply morefairness.

Observe that co-channel cells in the first assignment havetopology as in Figure 3(a). There are three subsystem ofco-channel cells each consisting of four cells. The infiniteρ approximation predicts that, the two cells at the middle(resp. at the end) for each subsystem of co-channel cellswould obtain normalized throughput≈ 1

3 (resp.≈ 23 ). The

normalized network throughput for the first assignment is6and the fairness index is0.45. In the second assignment,each cell obtains normalized throughput≈ 1

3 . The normalizednetwork throughput for the second assignment is4 and thefairness index is1. Thus, the fairness has attained its maximumin the second assignment. However, this assignment is notdesirable sinceΘ for this assignment is only23 of Θ in thefirst assignment. In the third assignment, each cell obtainsnor-malized throughput≈ 1

2 . The normalized network throughputfor the third assignment is6 and the fairness index is equalto 1. Clearly, the third assignment is the best one among thethree.

Note that, an SINR based model is likely to prefer the firstassignment to the third. However, interference by nodes withinRcs can be taken care by carrier sensing, and hence, suchinterference should not be given undue weights. If required,cells that are very close to each other may be assigned thesame channel to form “big” cells. Unlike a cell which coversa large area by a single AP and provides connectivity at lowPHY rates to STAs that are located at the edge of the cell,a cluster of mutually interfering small co-channel cells eachcontaining an AP to ensure connectivity at high PHY ratescan provide high capacity and fairness. This emphasizes theimpact of carrier sensing in 802.11 networks.

VIII. A PPLYING THE ANALYTICAL MODEL ALONG WITH A

LEARNING ALGORITHM

In the 12-cell network of Figure 11, with 3 non-overlappingchannels, we need to examine312 possibilities to determinethe optimal assignment. It is also not clear if the normalizednetwork throughput for the 12-cell network can be actuallymore than 6 with some other assignment. Under the infiniteρ approximation, the maximum value ofΘ is equal to themaximum independence number over all possible logical con-tention graphs. Clearly, the problem of finding an assignmentthat maximizesΘ is NP-hard and, in general, it is not desirableto examine all the possibilities.

We now demonstrate how our analytical model could beapplied along with a Learning Automata (LA) algorithmcalled the Linear Reward-Inaction(LR−I) algorithm [31] toobtain optimal channel assignments. For every celli ∈ N ,we maintain anM -dimensional probability vectorpi whereM denotes the number of available channels. The(LR−I)algorithm proceeds in steps. Letk = 0, 1, 2, . . . be the stepindices. At each stepk, a channel is selected for Cell-i, i ∈ N ,according to the probability distribution

pi(k) :=(

pi,1(k), pi,2(k), . . . , pi,M (k))

,

where

pi,j(k) := the probability that Channel-j is selected forCell-i at stepk.

Let c(k) =(

c1(k), c2(k), . . . , cN(k))

be the channelassignment at stepk, where

ci(k) := the channel selected for Cell-i at stepk.

Given a channel assignmentc, we can obtain the logicalcontention graphG(c) and applying our throughput model,we can compute the normalized throughputxi(c) of everycell i with the assignmentc. We define thesum utility for anassignmentc by

U(c) :=

N∑

i=1

u(xi(c)), (13)

whereu(·) is some suitably definedincreasing concavefunc-tion. For a givenu(·), we compute the sum utility at stepkby

U(c(k)) =N∑

i=1

u(xi(c(k))) ,

and apply theLR−I algorithm which consists of the followingsteps [31]:

1. Begin withpi,j ∈ (0, 1), ∀i ∈ N , j = 1, 2, . . . ,M , suchthat pi · 1 = 1, where1 is theM -dimensional vectorwith all components equal to 1.

2. Update the probability vectors of Cell-i, ∀i ∈ N , as

pi(k + 1) = pi(k) + b U(c(k))(

δci(k) − pi(k))

,

whereδj denotes theM -dimensional probability vectorwith unit mass on Channel-j and0 < b < 1.

The parameterb is called thelearning parameteror thestep-size parameterwhich determines the convergence propertiesof the algorithm; with smallerb convergence is slower but thealgorithm may not converge to the desired optimum ifb is notsmall enough [31]. Notice that, to ensure non-negativity oftheprobability vectors after every update with anyb ∈ (0, 1), thesum utility U must satisfy0 ≤ U ≤ 1. To maximizeΘ =∑N

i=1 xi, we take theaveragenormalized network throughput

UΘ :=1

N

N∑

i=1

xi (14)

as the sum utilityU where we compute thexi’s applying theinfinite ρ approximation. Note that,0 ≤ UΘ ≤ 1. Other utilityfunctions can also be considered provided that0 ≤ U ≤ 1.Define

P := (p1,p2, . . . ,pN ).

Using the terminology of [31], the super vectorP willbe called astrategy for the channel assignment problem.A strategy P such that,∀i ∈ N , pi,j = 1 for some j,j = 1, 2, . . . ,M , will be called apure strategy. A strategyPthat is not a pure strategy is called amixed strategy. Under theLR−I algorithm,{P(k), k ≥ 0} is a Markov process with thepure strategies as the only absorbing states. Thus, in practice,theLR−I algorithm always converges to a pure strategy ratherthan to a mixed strategy [31]. Furthermore, by Theorem 4.1of [31], theLR−I algorithm always converges to one of theNash equilibria. Thus, in practice, channel assignment by theLR−I algorithm always provides an assignmentc∗ which isone of theNash equilibria in pure strategiesin the sense that

U(c∗) ≥ U(c),

for all c that differs fromc∗ by exactly one element, i.e.,changing the channel of one of the cells in the assignmentc∗

does not increase the sum utilityU .

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0 5000 10000 15000 20000 25000 30000 35000

Ave

rage N

etw

ork

Thro

ughput

Iteration

Evolution of Average Network Throughput

Fig. 13. Evolution of the average normalized network throughputUΘ

under theLR−I algorithm for the 12-cell network in Figure 11with M = 3 channels,b = 0.01, and pij =

1

3, ∀i ∈ N and j =

1, 2, . . . ,M .

A. Results for Channel Assignment by theLR−I Algorithm

Figure 13 shows the evolution of the average normal-ized network throughputUΘ under theLR−I algorithm withM = 3 channels andb = 0.01 for the 12-cell networkin Figure 11. We began with uniformunbiasedprobabilityvectorspi = (13 ,

13 ,

13 ), ∀i ∈ N . The algorithm converged to

the assignment16 c1 = c6 = c9 = 1, c4 = c7 = c12 = 3, c2 =c3 = c5 = c8 = c10 = c11 = 2. Notice thatUΘ convergesto 2

3 which corresponds toΘ = 8. It can be verified thatthis is the maximum possible value thatΘ can take for thegiven network with 3 channels17. We found that theLR−I

algorithm always converges to a globally optimum solution ifstarted with uniform unbiased probability vectors and ifb issufficiently small. However, starting with probability vectorsthat are highlybiased towards some assignment or ifb isnot small enough, it may not converge to a globally optimumsolution. We demonstrate this through Figures 14-17 for the7-cell example in Figure 3(d) withM = 2 channels.

Beginning with unbiased probability vectors andb = 0.01the algorithm may converge to a global optimum withUΘ = 1or Θ = 7 (Figure 14). There are two possible global optima,namely,c1 = c2 = c4 = c7 = 1, c3 = c5 = c6 = 2 andc1 = c2 = c4 = c7 = 2, c3 = c5 = c6 = 1. In Figure 15, weshow that, withb = 0.01, the algorithm may also convergeto a solution that is not a global optimum. In fact, for thespecific simulation run, the algorithm converged to the Nashequilibrium c1 = c2 = c5 = c6 = 1, c3 = c4 = c7 = 2with Θ = 6 which is not a global optimum. The assignmentc1 = c2 = c5 = c6 = 1, c3 = c4 = c7 = 2 is a Nashequilibrium in the sense that changing the channel of one of thecells does not increaseΘ, and hence, the sum utilityUΘ. With

16We do not provide the plots showing evolution of thepi,j ’s to save space.17Note that permutations of channels do not change the logicaltopology

and the associatedΘ’s would be equal. For example,c1 = c6 = c9 =

2, c4 = c7 = c12 = 1, c2 = c3 = c5 = c8 = c10 = c11 = 3 would alsogive Θ = 8.

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Ave

rage N

etw

ork

Thro

ughput

Iteration



for the 7-cell example in Figure 3(d) withM = 2 channels,b = 0.01,and pi1 = pi2 = 0.5, i = 1, 2, . . . , 7. In this example, theLR−I

algorithm converges to a global optimum.

0

0.2

0.4

0.6

0.8

1

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

Ave

rage N

etw

ork

Thro

ughput

Iteration



for the 7-cell example in Figure 3(d) withM = 2 channels,b = 0.01,and pi1 = pi2 = 0.5, i = 1, 2, . . . , 7. In this example, theLR−I

algorithm converges to a local optimum.

b = 0.001, and beginning with unbiased uniform probabilityvectors, we observed that the algorithm always converged toaglobal optimum. Withb = 0.001, the algorithm converged to aglobal optimum even when we began with probability vectorsbiased towards the assignmentc1 = c2 = c5 = c6 = 1, c3 =c4 = c7 = 2 (Figure 16). However, when the initial probabilityvectors were highly biased towards the assignmentc1 = c2 =c5 = c6 = 1, c3 = c4 = c7 = 2, the algorithm converged tothe biased assignment which is not a global optimum (Figure17).

IX. A S IMPLE AND FAST ALGORITHM FOR MAXIMIZING

NORMALIZED NETWORK THROUGHPUT

We observe that theLR−I algorithm takes a large numberof iterations to converge and guarantees convergence only toNash equilibria. Clearly, for the 7-cell example withM = 2channels, anexhaustive searchwould have required exam-ination of only 27 = 128 possibilities whereas theLR−I

0

0.2

0.4

0.6

0.8

1

0 50000 100000 150000 200000 250000

Ave

rage N

etw

ork

Thro

ughput

Iteration



for the 7-cell example in Figure 3(d) withM = 2 channels,b =0.001, andp11 = p21 = p32 = p42 = p51 = p61 = p72 = 0.8. Inthis example, theLR−I algorithm converges to a global optimum.

0

0.2

0.4

0.6

0.8

1

0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

Ave

rage N

etw

ork

Thro

ughput

Iteration



for the 7-cell example in Figure 3(d) withM = 2 channels,b =

0.001, andp11 = p21 = p32 = p42 = p51 = p61 = p72 = 0.9. Inthis example, theLR−I algorithm converges to a local optimum.

algorithm takes many more steps to converge. The LinearReward-Penalty (LR−P ) algorithm of [21] and thesimulatedannealingalgorithm of [22] guarantee convergence to a glob-ally optimum solution as the number of iterations goes toinfinity. A greedyversion of simulated annealing algorithmin [22] is relatively faster but still takes a large numberof iterations to converge and guarantees convergence onlyto locally optimum solutions. To maximize the normalizednetwork throughputΘ we now propose a simple and fastdecentralized algorithm which can be easily implemented inreal networks.

We form a physical contention graphG in which everycompletely dependent pair of cells are neighbors (sincewe have not yet assigned channels) and our objective isto transformG to a logical contention graphG(c) by anassignmentc so that α(G(c)) is maximized. As noted inObservationO.11, for sufficiently large packet sizes, we haveΘ ≈ α(G). Hence, maximizingα(G(c)) would maximizeΘ.

Maximal Independent Set Algorithm (mISA) : We pro-pose the following channel assignment algorithm: (recall thatM is the number of available channels)

(1) Choose amaximal independent set (mIS)18 of cells,assign them Channel-1 and remove them from the graph.

(2) Increment the channel index and repeat the procedureon theresidualgraph until one channel is left.

(3) Assign Channel-M to all the cells in the residual graphafterM − 1 steps.

Notice that mISA takes onlyM steps. Also notice that, theresidual graph may becomenull (i.e., it might not have anyvertices left) afterM ′ < M steps. Clearly, mISA is basedon a classical graph coloring technique, but the novelty lies inour recognition of the notion of optimality that mISA provideswhich is given by the following

Theorem 9.1:The channel assignments by mISA are Nashequilibria in pure strategies for the objective of maximizingnormalized network throughputΘ as ρi → ∞, ∀i ∈ N .Furthermore, mISA provides a globally optimum solution ifM ≥ D(G) + 1 whereD(G) denotes the maximum vertexdegree in the physical contention graphG.

Proof: If the residual graph becomes null in less thanMsteps, then every cell would have a channel different from allits neighbors’ and, we have,Θ = N , i.e., mISA would providea globally optimum solution. Hence, to prove the first part ofthe theorem, we assume, without loss of generality, that theresidual graph afterM − 1 steps is not null.

Suppose thatNj cells are assigned Channel-j in Step-j, 1 ≤j ≤ M−1. Let Gj be the residual graph afterj steps,1 ≤ j ≤M−1. Let Θk denote the aggregate normalized throughput ofthe cells on Channel-k, 1 ≤ k ≤ M . Then, we have,Θj = Nj,1 ≤ j ≤ M − 1, due to independence, andΘM = α(GM−1).Hence,Θ =

∑Mk=1 Θk =

∑M−1j=1 Nj + α(GM−1). Suppose

that a cell on Channel-j, j 6= M , is moved to Channel-k,k 6= j. Then, Θj decreases by 1 butΘk can increase by atmost 1. Hence,Θ cannot increase. Suppose now that a cell onChannel-M is moved to Channel-j, 1 ≤ j ≤ M − 1. Clearly,any cell on Channel-M is dependent (in the original graphG)w.r.t. at least one of theNj cells on Channel-j since theNj

cells that are already on Channel-j, 1 ≤ j ≤ M − 1, form anmIS. Hence,Θj does not change butΘM can only decrease.Hence,Θ cannot increase by changing the channel ofonly oneof the cells and the first part of the theorem is proved.

For the second part of the theorem, notice that, every cellin the residual graphGj after j steps,1 ≤ j ≤ M − 1, mustbe dependent (in the original graphG) w.r.t. at least one cellon each channell, 1 ≤ l ≤ j − 1. This follows from themaximality of the independent sets of cells chosen in each ofthe firstM − 1 steps. In particular, every cell in the residualgraphGM−1 is dependent (in the original graphG) w.r.t. atleast one cell on each channelj, 1 ≤ j ≤ M − 1. Sincedifferent channels are assigned in each step, the vertex degree

18The lower case ‘m’ corresponds to “maximal” as opposed to theuppercase ‘M’ which corresponds to “maximum”.

of every cell inGM−1 restricted to the cells inGM−1 alonewould be≤ D(G) − (M − 1). If M ≥ D(G) + 1, then thecells in GM−1 would form an independent set and every cellwould have a channel different from its neighbors’. Thus, ifM ≥ D(G) + 1, then mISA would provide an assignment withΘ = N , i.e., a globally optimum solution.

Implementation of mISA: mISA can be implemented ina decentralized manner as follows. APs sample random back-offs using a contention windowW and contend for accessingthe medium using Channel-1. When an AP wins the contentionit keeps transmitting broadcast packets separated by ShortInterFrame Space (SIFS) for some durationT >> σW where werecall thatσ is the duration of a back-off slot. This emulatesthe infiniteρ situation since and AP after wining the contentiondoes not relinquishes the control over its local medium. Wehad observed that, asρi → ∞, ∀i ∈ N , only the cells thatbelong to an MIS obtain non-zero normalized throughputs.But this holds only in anensemble averagesense. If an AP,after wining the contention, does not relinquishes the controlover its local medium, in a particularsample path, an mISof APs (which may not be an MIS) wouldgrab the channelduringT 19. This is not surprising since with infiniteρi’s, theCTMC {A(t), t ≥ 0} becomes absorbing with the maximalindependent sets of cells as the only absorbing states and wecannot expect the time average to be equal to the ensembleaverage.

At time T , APs that could transmit consecutive broadcastpackets stop contending until time(M − 1) × T and APsthat remain blocked switch to Channel-2, sample fresh back-offs and keep contending until2T and so on. APs that remainblocked throughout the duration(M−1)×T stick to Channel-M . Thus, in every time durationT , a maximal independentset of APs would be assigned a channel. Normal networkoperation can begin after time(M − 1) × T . Notice that,mISA does not require any knowledge of AP topology and runsin a completely decentralized manner without any messagepassing. In addition, if there is a central controller to whichthe APs can communicate, mISA can be repeated several timesbefore normal network operation could begin. The centralcontroller, which obtains the global view of the channelassignments, can choose the best among the solutions providedby mISA. In absence of centralized control, mISA can beinvoked periodically. Thus, mISA can be easily implementedin real networks in a completely decentralized manner if thenumber of channels for every AP is the same and known.However, mISA requires loose synchronization among the APswhich necessitates some message passing before mISA couldbegin.

X. CONCLUSIONS ANDFUTURE WORK

In this paper, we identified a Pairwise Binary Dependence(PBD) condition that allows a scalable cell level modeling ofWLANs. The PBD condition is likely to hold at higher PHY

19If W is large enough, the possibility of two dependent APs transmittingtogether can be ruled out.

rates and denser AP deployments. We developed a cell levelmodel both under saturation condition and for TCP-controlledlong file downloads. Our analytical model was shown tobe quite accurate, insightful and capable of comparing fewdesign alternatives. Thus, we believe that our modeling frame-work is a significant step toward gaining “first-cut” analyticalunderstanding of WLANs with dense deployments of APs.We demonstrated how our analytical model could be appliedalong with the Linear Reward-Inaction learning algorithm foroptimal channel assignment. Based on the insights providedbyour model, we also proposed a simple decentralized algorithmcalled mISA which can provide channel assignments that areNash equilibria in pure strategies in only as many steps asthere are channels. Although mISA is based on a standardgraph coloring technique, its applicability to the consideredproblem is strengthen by and its practical implementabilityis guided by a rigorous analytical model developed in thispaper. Developing simple and practical algorithms for generalobjective functions, other than maximizing the normalizednetwork throughput, is a topic of our ongoing research.

REFERENCES

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[2] R. Murty, J. Padhye, R. Chandra, A. Wolman, and B. Zill, “DesigningHigh Performance Enterprise Wi-Fi Networks,” in5th USENIX Sympo-sium on Networked Systems Design and Implementation NSDI’08, 2008.

[3] M. A. Ergin, K. Ramachandran, and M. Gruteser, “An ExperimentalStudy of Inter-cell Interference Effects on System Performance inUnplanned Wireless Deployments,”Computer Networks, 2008, acceptedmanuscript.

[4] P. Belanger, “Enterprise Wireless LAN Scale Testing,” Novarum Inc.,Tech. Rep., 2007.

[5] G. Bianchi, “Performance Analysis of the IEEE 802.11 DistributedCoordination Function,”IEEE Journal on Selected Areas in Commu-nications, vol. 18, no. 3, pp. 535–547, March 2000.

[6] F. Cali, M. Conti, and E. Gregori, “Dynamic Tuning of the IEEE802.11 Protocol to Achieve a Theoretical Throughput Limit,” IEEE/ACMTransactions on Networking, vol. 8, no. 6, pp. 785–799, December 2000.

[7] A. Kumar, E. Altman, D. Miorandi, and M. Goyal, “New insights froma fixed point analysis of single cell IEEE 802.11 WLANs,”IEEE/ACMTransactions on Networking, vol. 15, no. 3, pp. 588–601, June 2007,also appeared in INFOCOM 2005.

[8] R. Bruno, M. Conti, and E. Gregori, “An accurate closed-form formulafor the throughput of long-lived TCP connections in IEEE 802.11WLANs,” Computer Networks, vol. 52, pp. 199–212, 2008.

[9] R. Boorstyn, A. Kershenbaum, B. Maglaris, and V. Sahin, “ThroughputAnalysis in Multihop CSMA Packet Radio Networks,”IEEE Transac-tions on Communications, vol. 35, no. 3, pp. 267–274, March 1987.

[10] M. Garetto, T. Salonidis, and E. W. Knightly, “ModelingPer-flowThroughput and Capturing Starvation in CSMA Multi-hop Networks,”IEEE/ACM Transactions on Networking, to appear, Also appeared inINFOCOM 2006.

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[13] S. Roy, H. Ma, and R. Vijayakumar, “Optimizing 802.11 Wireless MeshNetwork Performance Using Physical Carrier Sensing,” University ofWashington, Tech. Rep. UWEETR-2006-0005, 2006.

[14] H. Q. Nguyen, F. Baccelli, and D. Kofman, “A Stochastic GeometryAnalysis of Dense IEEE 802.11 Networks,” inIEEE INFOCOM’07.

[15] T. Bonald, A. Ibrahim, and J. Roberts, “Traffic Capacityof Multi-CellWLANs,” in ACM SIGMETRICS’08.

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APPENDIX ADERIVATION OF EQUATION 7

Consider a tagged node in Cell-i. Let

Ai(T ) : number of attempts made by the tagged node up to timeT

Ci(T ) : number of collisions as seen by the tagged node up to timeT

AAi (T ) : number of attempts made by the tagged node in State-A up to timeT

CAi (T ) : number of collisions as seen by the tagged node in State-A up to timeT

BAi (T ) : number of back-off slots elapsed in Cell-i in State-A up to timeT

Then, we have

γi = limT−→∞

Ci(T )

Ai(T )=

limT−→∞

1

T

∑

A∈A : i∈UA

CAi (T )

limT−→∞

1

T

∑

A∈A : i∈UA

AAi (T )

=

limT−→∞

1

T

∑

A∈A : i∈UA

BAi (T )×

CAi (T )

BAi (T )

limT−→∞

1

T

∑

A∈A : i∈UA

BAi (T )×

AAi (T )

BAi (T )

=

∑

A∈A : i∈UA

(

limT−→∞

BAi (T )

T

)

×

(

limT−→∞

CAi (T )

BAi (T )

)

∑

A∈A : i∈UA

(

limT−→∞

BAi (T )

T

)

×

(

limT−→∞

AAi (T )

BAi (T )

)

=

∑

A∈A : i∈UA

(

limT−→∞

BAi (T )

T

)

×

(

limT−→∞

AAi (T )

BAi (T )

)

×

(

limT−→∞

CAi (T )

AAi (T )

)

∑

A∈A : i∈UA

(

limT−→∞

BAi (T )

T

)

×

(

limT−→∞

AAi (T )

BAi (T )

)

=

∑

A∈A : i∈UA

(

π(A)

σ

)

× βi ×

1− (1 − βi)ni−1

∏

j∈Ni : j∈UA

(1 − βj)nj

∑

A∈A : i∈UA

(

π(A)

σ

)

× βi

=

∑

A∈A : i∈UA

π(A)

1− (1− βi)ni−1

∏

j∈Ni : j∈UA

(1 − βj)nj

∑

A∈A : i∈UA

π(A)(15)

where the last but one step follows from the facts that: (i) the time spent in State-A up to timeT is equal toπ(A)T andif Cell-i is in back-off in State-A, thenBA

i (T ) = π(A)Tσ

, (ii) the second limit within the brackets in the numerator is thelong-run fraction of back-off slots in State-A in which the tagged node attempts which is equal toβi irrespective of State-Agiven that Cell-i is in back-off in State-A, and (iii) the third limit within brackets in the numerator is the long-run fractionof attempts made by the tagged node in State-A that result in collisions and depends on the set of neighborsthat are also inback-off in State-A.

APPENDIX BPROOF OFTHEOREM 5.1

Solving the balance equations for the CTMC{A(t), t ≥ 0} (Equation 4) together with the normalization equation, we obtain

π(Φ) =1

∑

A∈A

(

∏

i∈A

ρi

) =1

∆. (16)

Equation 8 can now be expanded as

xi =∑

A∈A : i∈A

π(A) +∑

A∈A : i∈UA

π(A) (sinceA ∩ UA = Φ)

=∑

A∈A : i∈A

∏

j∈A

ρj

π(Φ) +∑

A∈A : i∈UA

∏

j∈A

ρj

π(Φ) (by Equation 4)

=1

∆

∑

A∈A : i∈A

∏

j∈A

ρj

+∑

A∈A : i∈UA

∏

j∈A

ρj

(by Equation 16)

=1

∆

∑

A∈A : i∈A, A∩Ni=Φ

∏

j∈A

ρj

+∑

A∈A : A∩({i}∪Ni)=Φ

∏

j∈A

ρj

=1

∆

ρi∑

A∈A : A∩{i}=Φ, A∩Ni=Φ

∏

j∈A

ρj

+∑

A∈A : A∩({i}∪Ni)=Φ

∏

j∈A

ρj

, (17)

where the fourth step follows from the facts that: (i) if Cell-i is active, then none of its neighbors can be active, and (i) ifCell-i is in back-off, then neither Cell-i nor any of its neighbors can be active. Observe now that, the independent sets ofGi

are also independent sets ofG. Thus,∆i can be obtained by restricting the summation in Equation 9 toonly those independentsets ofG that do not contain Cell-i or any cell inNi, i.e.,

∆i :=∑

A∈A : A∩({i}∪Ni)=Φ

(

∏

i∈A

ρi

)

=∑

A∈A : A∩{i}=Φ, A∩Ni=Φ

(

∏

i∈A

ρi

)

, (18)

where the second equality follows from the fact that union oftwo sets will be equal to the empty set iff each of the sets is anempty set. Applying Equation 18 in the last step of Equation 17, we obtain Equation 10.

modeling multi-cell ieee 802.11 wlans with …arxiv:0903.0096v2 [cs.ni] 14 mar 2009 modeling...

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