Traffic Capacity of Multi-Cell WLANs

Transcription

1 Traffic Capacity of Multi-Cell WLANs Thomas Bonal, Ali Ibrahim, James Roberts Orange Labs 38- rue general Leclerc Issy-les-Moulineaux, France ABSTRACT Performance of WLANs has been extensively stuie uring the past few years While the focus has mostly been on isolate cells, the coverage of WLANs is in practice most often realise through several cells Cells using the same frequency channel typically interact through the exclusion region enforce by the RTS/CTS mechanism prior to the transmission of any packet In this paper, we investigate the impact of this interaction on the overall network capacity uner realistic ynamic traffic conitions Specifically, we represent each cell as a queue an erive the stability conition of the corresponing couple queuing system This conition is then use to calculate the network capacity To gain insight into the particular nature of interference in multi-cell WLANs, we apply our moel to a number of simple network topologies an explicitly erive the capacity in several cases The results notably show that the capacity gain obtaine by using M frequency channels can grow significantly faster than M, the rate one might intuitively expect In aition to stability results, we present an approximate moel to erive the impact of network loa on the mean transfer rate seen by the users Categories an Subject Descriptors C [Computer-Communication Networks]: Network Architecture an Design Wireless communication ; C [Computer Systems]: Performance of Systems Moeling techniques General Terms Performance Keywors Multi-cell WLAN, IEEE 8, flow-level moel, stability, capacity Permission to make igital or har copies of all or part of this work for personal or classroom use is grante without fee provie that copies are not mae or istribute for profit or commercial avantage an that copies bear this notice an the full citation on the first page To copy otherwise, to republish, to post on servers or to reistribute to lists, requires prior specific permission an/or a fee SIGMETRICS 8, June 6, 8, Annapolis, Marylan, USA Copyright 8 ACM /8/6 $5 INTRODUCTION Our objective in the present paper is to investigate the ownlink capacity of multiple interfering IEEE 8 Access Points (AP) using the Distribute Coorination Function (DCF) meium access control Use of this technology to access the Internet in homes, enterprises an WiFi hotspots is growing rapily leaing to an increasingly ense implantation of APs Interference between these APs an their respective users reuces overall traffic capacity We consier a generic multiple AP network that we refer to as a multi-cell WLAN The cell is the set of positions from which users will associate with a given AP While the performance of single cell networks is well unerstoo, there has as yet been relatively little evaluation of the impact of interference on the capacity of multi-cell networks We assume users situate within the coverage area of the WLAN ownloa files from the Internet via their nearest AP The rate at which ownloas are generate times the average flow size efines a traffic intensity in bits per secon The traffic capacity of the WLAN is the maximum intensity it can support without the number of simultaneous ownloas growing to infinity The capacity epens somewhat on the statistical characteristics of eman as well as its spatial istribution over the WLAN coverage area Our main focus, however, is on how capacity epens on the placement of the APs, their transmission range an the assignment of frequency channels to neighbouring cells The ultimate goal is to provie guielines for planning multicell WLANs, to maximize their capacity or to minimize their cost, for example However, our present ambition is limite to exploring the issues using relatively simple moels that capture the impact of inter-cell interference in toy network configurations Interference is manifeste in IEEE 8 networks by the CSMA collision avoiance protocol [9] Stations must listen to the channel an efer transmission when it is sense busy When the channel is ile, stations compete for access using the exponential backoff proceure In a multi-cell WLAN the efficiency of this proceure is compromise ue to the hien noe phenomenon necessitating the use of channel reservation This is performe through the exchange of short frames calle RTS (Request-To-Sen) an CTS (Clear-To- Sen) prior to the transmission of a ata packet Suppose, for example, that two users associate with ifferent APs are locate within their common transmission range while the two APs are not able to hear each other, as epicte in Figure (a) The transmission from AP to user u will not be etecte by AP leaing to a collision

2 AP AP u u (a) DATA-DATA collision AP AP u u present the analysis that leas to close form expressions for the traffic capacity an a proceure to evaluate mean flow ownloa times as a function of loa The results are applie in the next sections to simple network configurations, with one or several channels, in one an two imensional space We evaluate flow throughput for the simplest two-ap network configuration an valiate these results by means of packet-level simulations We first present a brief review of relate literature (b) DATA-ACK collision Figure : Impact of hien noes on reception if it begins to transmit to u The RTS/CTS mechanisms prevent this The CTS frame sent by u in response to an RTS from AP is hear by AP which refrains from transmitting until the channel is ile In Figure (b), where the two ownlink transmissions can coexist, collisions can still occur between ata packets in one cell an acknowlegements in the other With a prior RTS/CTS exchange, the secon transmission woul not take place since, after receiving the CTS from u, u woul not respon to an RTS sent later by AP The area where channel access is inhibite uring an ongoing transmission is thus extene to all user an AP positions within the transmission range of both sener an receiver We term this the exclusion region The process by which stations gain channel access is very complex Consier an AP with a packet to transmit to a particular user in its cell It must first wait for any ongoing transmission in its exclusion region to complete It may then continue to be blocke by some other concurrent transmission within this exclusion region but compatible with the first transmission When it can eventually try to access the channel, it will be in competition with a variable number of other AP-user pairs Depening on the outcome of this competition, it may be necessary to make several successive attempts before successful transmission It seems impossible to precisely moel such a complex stochastic system where all cells are inter-epenent through their overlapping exclusion zones On the other han, simulation is harly useful to provie the general insights we seek We therefore make some quite bol assumptions an simplifications, as iscusse in the paper, in orer to erive analytical results In this simplifie setting, we are able to erive the stability conition an euce the traffic capacity of multi-cell WLANs When eman is less than capacity, we can estimate the mean time to perform a ownloa for any given AP an user position These general results are evaluate numerically for some toy network topologies The results show how capacity changes abruptly epening on whether APs interfere irectly or only through the users to which they transmit This iscontinuity has a significant impact on the potential gain in capacity brought by the use of ifferent frequency channels in neighbouring cells At best, this gain is significant, with capacity increasing up to four times as fast as the number of channels Arguably, the broa characteristics of this behaviour o not epen on the simplifications introuce in our moelling an therefore provie valuable insight into the performance of real multi-cell WLANs In the remainer of the paper we introuce our moel an RELATED WORK There is a huge amount of literature on the performance of IEEE 8 uner DCF access control However, much of this is focuse on the performance of isolate cells with a static user population In the seminal work of Bianchi [] an its generalizations, by Gupta an Kumar [8] an by Kumar et al [], for instance, the capacity of a single cell with N saturate stations uploaing packets to the AP is etermine as the solution to a fixe point equation This is not irectly relate to our work, however, where we focus on ownlink throughput an evaluate the impact of interference between multiple APs uner ynamic traffic The performance of multi-hop wireless networks calls for an evaluation of interference effects similar to those occurring in our multi-cell WLAN For example, Tassiulas an Ephremies [8] characterize the maximal capacity region of such networks an the scheuling algorithm that achieves this More recent work has focuse on etermining scheuling algorithms that realize given performance objectives, eg, [, 3] This work is useful in illustrating the inherent ifficulty of accounting for the impact of interference Our context is somewhat simpler in that connections are all single hop an we o not seek to efine a scheuling algorithm However, our focus is ifferent in that we evaluate capacity in a ynamic scenario an how it epens on the relative positions of the APs The work by Pana et al [6] is, at first sight, particularly relevant The authors consier the same configuration of interfering WLAN APs However, they consier uplink traffic only, generalizing the moel of Bianchi [] for a network of two APs with so-calle critical inter-ap spacing We consier ownlink traffic with two or more interfering APs Traffic capacity an throughput performance, in the flow level sense consiere here, was evaluate by Lebeugle an Proutière [] an Litjens et al [] for a single AP cell Their results are a useful justification for our assumption in the next sections that the DCF protocols leas to approximate fair sharing of a constant cell capacity The present analysis is also closely relate to previous work on the impact of inter-cell interference on the performance of cellular networks Bonal et al [3] evaluate bouns on ownloa throughput performance uner the flow level traffic moel consiere here Although the network is also represente as a couple queuing system, the nature of the interference phenomenon is ifferent A key feature of our moel of multi-cell WLANs is that the service rate of each queue oes not only epen on the activity state of the other queues, that is on the presence or absence of active users in the corresponing cells, but on the locations of these active users, that may collie or not with ongoing ata transfers in the consiere cell

3 3 MODEL In this section, we escribe the moel use to analyse the impact of interference on the traffic capacity of multi-cell WLANs Each cell is represente as a queue that interacts with other queues through couple service rates This coupling, that captures the interference between neighbouring cells, is erive from a simple packet-level moel representing the impact of RTS/CTS mechanisms 3 Network structure Consier a WLAN that consists of N access points (AP) We focus on the ownstream traffic from the APs to ranomly locate users in the coverage region of the network We assume this traffic consists of elastic ata transfers, typically uner the control of TCP There is an arbitrary finite set U of user classes, corresponing to homogeneous regions in terms of raio characteristics Specifically, two users of the same class are associate with the same AP an interact with other users an APs in the same way We enote by U i U the set of user classes associate with AP i This correspons to the cell serve by AP i Users interfere through the RTS/CTS mechanisms, as escribe in Section We moel this interference through a function χ from U U to {, }, such that χ(j, l) = if an only if the transmissions to a class-j user an a classl user cannot occur simultaneously In particular, we have χ(j, l) = if user classes j, l are associate with the same AP For all i =,, N an j U i, we refer to the set E i j = {l U : χ(j, l) = } as the exclusion region of user class j associate with AP i AP AP (a) (b) Figure : Notion of exclusion region Though we assume that both the number of APs an the set of classes are finite in the analysis, the results naturally exten to infinite networks with a continuous set of classes For D networks for instance, each class may represent a specific location in the network, so that U = R A simple interference moel in this context consists in consiering any transmission successful if there is no other transmitting or receiving station within a istance R from the source an from the receiver, the istance R corresponing the transmission range of RTS/CTS signals for both users an APs The interference function between two users locate in u u u, u R is then given by: j (u, u χ(u, u ) = ) > R, (u, v ) > R (v, u ) > R, (v, v ) > R where v, v are the respective locations of the associate APs an (u, v) is the istance between u an v Thus the exclusion region of any transmission typically consists of two isks, one centere on the AP an the other on the user, as illustrate by Figure (a) When some APs are locate in the region forme by these two isks, the whole cells serve by these APs must be ae to the exclusion region This is illustrate by Figure (b) 3 Packet-level moel Packet scheuling is assume to be FIFO at each AP When the transmission of a packet is scheule, the AP performs DCF access control competing with other transmissions in its exclusion region until the packet is eventually elivere Thus the APs interact in a very complicate way that epens on the location of active users in the corresponing cells To get some insight into the impact of this interaction on the overall network behaviour, we consier the following simple packet-level moel All packets have the same size an the network operates synchronously in a slotte fashion: transmissions occur only at the beginning of a time slot an the transmission of each packet takes one slot The slot uration may be seen as the average transmission time of a packet for an isolate AP, incluing all overheas like the times to access the channel an to transmit control messages like RTS/CTS signals an MAC/TCP acknowlegements Thus the throughput of an isolate AP is equal to packet/slot, which is the throughput unit use in the rest of the paper Note that we neglect the impact of variable raio conitions, that may result in ifferent physical throughputs epening on the location in the cell In our moel, the throughput of a user epens on her/his location in a given cell through her/his interaction with other cells only Now let ξ i j be the probability that AP i attempts to serve a class-j user after any successful transmission We assume that the mean number of slots an AP remains ile before transmitting a packet is equal to the number of other active users in the corresponing exclusion region, when each AP i inepenently selects a class-j user with probability ξ i j Thus the mean time for a user to access the channel is proportional to the number of active transfers in her/his exclusion region, which is a reasonable assumption For all k i an j U i, the probability that the user selecte by AP k is in the exclusion region E j i of class-j users is given by: l U k ξ k l χ(j, l) We euce the average number of slots δj i neee by AP i to successfully transmit the packet of a class-j user: δj i = + ξl k χ(j, l) () k i l U k The average number of slots neee by AP i to successfully transmit any packet is then given by: δ i = j U i ξ i jδ i j ()

4 We conclue that the throughput of AP i is equal to /δ i This throughput is equal to for an isolate AP an is less than in the presence of inter-cell interference A fraction ξ i j of the throughput of AP i is allocate to class-j users Thus the throughput allocation is entirely etermine by the interference function χ an the probabilities ξ i j, for all i =,, N an j U i 33 Flow-level moel In the following, we refer to a class-j flow as a ata transfer to a class-j user We assume class-j flows arrive at AP i as a Poisson process of intensity λ i j Flow sizes are ii exponential with parameter µ Let x i j be the number of class-j flows at AP i We enote by x i the total number of flows at AP i: x i = j U i x i j We are intereste in the evolution of the network state x that escribes the number of ongoing flows of each class at each AP This epens on the throughput allocation in state x In orer to apply the above packet-level moel, it remains to etermine the probabilities ξ i j in state x, for all i =,, N an j U i Assume all flows associate with same AP get the same throughput This is a natural assumption uner the assume FIFO packet scheuling policy, since these flows experience the same packet elay an the same packet loss rate at the AP The probability that AP i selects the packet of a class-j flow after a successful transmission is then proportional to x i j in all states x such that x i > : ξ i j = xi j x i In view of () an (), the throughput of AP i is equal to /δ i, with an δ i j = + δ i = j U i x i j x i δ i j (3) k xl k i:x k > l U k x k χ(j, l) () In aition, this throughput is evenly share by all active flows serve by AP i Thus the moel correspons to a network of N multi-class processor-sharing queues with state-epenent service rates Class-j customers arrive at queue i as a Poisson process of intensity λ i j an have ii exponential service requirements with parameter µ In view of (3)-(), the service rate /δ i of queue i epens on the whole network state x that escribes the number of customers of each class in each queue It is this coupling of service rates that captures the impact of interference between neighbouring cells ANALYSIS We first etermine the stability region of the network, from which we erive the notion of traffic capacity We then propose a fixe-point approximation for the flow-level performance when the network is stable Stability region Let ρ i j = λ i j/µ be the traffic intensity of class-j flows at AP i We enote by α i j the proportion of class-j traffic at AP i: α i j = ρi j ρ i with ρ i = j U i ρ i j We say that the network is stable if the Markov process that escribes the evolution of the network state x is ergoic We have the following key results, prove in Appenix A Theorem The network is stable if for all i =,, N, ρ i j U i α i jβ i j <, where βj i = + α k l χ(j, l) (5) k i l U k Observe that, for isolate cells, the moel reuces to mutually inepenent processor-sharing queues so that the stability conition becomes ρ i < for all i =,, N The impact of inter-cell interference on stability is capture by the multiplying factor P j U i α i jβ i j on the traffic intensity at AP i In the following, we refer to the loa of AP i as the prouct: ρ i j U i α i jβ i j Theorem If the network is stable, there exists some i {,, N} such that ρ i j U i α i jβ i j The inequalities of Theorems an coincie, up to the critical case, when all APs have the same loa This allows us to calculate in Sections 5 an 6 the traffic capacity of a number of practically interesting symmetric networks, following the approach escribe below It proves very ifficult to erive the necessary an sufficient stability conition for heterogeneous loa istributions, as in most couple queuing systems with more than two queues [,, 6, 7] Traffic capacity We efine the traffic capacity of a network as the maximum traffic intensity such that the network is stable, assuming a fixe traffic istribution among user classes When all APs have the same loa, the necessary an sufficient stability conition is, up to the critical case, that for all i =,, N, ρ i j U i α i jβ i j < We euce the traffic capacity of cell i: C i = P j U i α i j βi j This traffic capacity is equal to if the cell is isolate (since β i j = for all j U i) an is less than in the presence of inter-cell interference

5 In the continuous setting introuce in Section 3, assuming cells of equal area A an uniform traffic istribution throughout the network, we obtain: C = RC A (6) β(u)u, where C is the consiere cell an, in view of (5), β(u) = Z χ(u, v)v (7) A U Note that β(u) is equal to the area of the exclusion region associate with a user locate in u normalize by the cell area It may be simply interprete as a measure of the average interference suffere at location u We apply formulas (6) an (7) to a number of symmetric network topologies in Sections 5 an 6 3 Mean transfer time Finally, it is interesting to analyse how the throughput of each user epens on network loa uner the stability conition In the following, we refer to the flow throughput of class j at AP i as the ratio of the mean flow size, /µ, to the mean uration W i j of class-j flows: γj i = µwj i By Little s law, the mean number of class-j flows at AP i is given by: i j = λ i jw i j (8) Unfortunately, there is no simple expression for the stationary istribution of the network state A useful approximation consists in ecoupling queues by replacing the stateepenent parameters δ i j by constants φ i j In view of (3), class-j flows are then serve at rate: P x i j l U i x i l φi l This is equivalent to a iscriminatory processor-sharing queue of unit service rate with weights φ i j, where class-j customers require exponential services of parameter µ/φ i j We assume that the queue is stable, that is: ρ i j U i α i jφ i j < We can then apply the results of Fayolle et al [7] to get the mean flow uration of each class: Wj i ρi α i lφ i l ρi α i lφ i lwl i = φi j µ (9) l U i l U i It remains to estimate the weights φ i j, for which we use the following fixe point equations: φ i j = P φ k l ρ k l U A k l k χ(j, l) l P () k i l U k l U k l k These equations follow from (3) by replacing the conition x k > by the probability that x k > an the ranom variables x k l by their mean k l This approximation turns out to be very accurate, as shown in Section 7 5 D NETWORKS In this section, we calculate the traffic capacity of D networks, assuming users are ranomly locate on a line The set of user classes is U = R The traffic istribution is assume to be uniform on the coverage region of the network We consier a simple raio moel where users associate with the nearest AP an any transmission is successful if an only if there is no other transmitting station within a istance R from the source an from the receiver We take R = in the rest of the paper so that the unit length correspons to the transmission range of both APs an users We first consier the simple case of two APs This provies useful insight into the impact of interference an constitutes a basic builing block for unerstaning the behaviour of more complex network topologies We then analyse the more practically interesting case of an infinite number of APs regularly locate on the line an sharing one or more frequency channels 5 Case of two APs Consier two APs separate by a istance : AP is locate at an AP at When >, each cell is a segment of length centere at the AP; when <, the cells are the segments [,] an [, + ] Now consier one user in each cell, locate at u an u, respectively The mutual interference region of these two users is shown by the shae areas of Figure, where r = u + an r = u correspon to the relative location of these users in the corresponing cells, as illustrate by Figure 3 r r AP u AP u Figure 3: D network with two APs Applying (6), we obtain the traffic capacity C of each cell as a function of We istinguish four cases: (a) Total coupling, (, ]: The two APs can never transmit simultaneously an thus behave as a single cell of unit service rate We have: C() = (b) Partial coupling, overlapping transmission areas, (, ]: The APs can only interfere inirectly through their estination We obtain: + C() = 5 + (c) Partial coupling, non-overlapping transmission areas, (, 3]: The two transmission areas no longer overlap, but the cells still interfere We have: 8 C() = () Inepenent cells, (3, ): Each cell behaves as if it were alone yieling: C() = The results are illustrate by Figure 5 We observe that capacity has irregular behaviour at the three critical istances =, an 3 At = capacity jumps iscontinuously from to 9 as the APs cease to interfere irectly

6 r (a) (, ] r r r (b) (, ] r r expresse as: β(u) = + e u + ««+ e, where e (s) is the total area of cells whose AP is locate on the segment (, s] This function is illustrate in Figure 7 The steps at s =,, occur as an AP is inclue in the segment (e (s) = s for s =,, ); e then remains constant uring the first half interval an increases linearly with slope uring the next half; it jumps again as the next AP is inclue r r (c) (, 3] () (3, ) Figure : Mutual interference region of two users Capacity 8 6 AP, D AP, D AP, D Figure 5: Traffic capacity vs inter-ap istance The form of the shae interference region changes abruptly as we move from case (a) to case (b) in Figure At = an = 3 capacity is continuous but its slope is iscontinuous The reason is again the change in form of the exclusion region as we move from (b) to (c) an from (c) to () 5 Infinite linear network in D space Consier an infinite D linear network where APs are locate respectively at n, n Z, as shown in Figure Figure 6: Infinite linear network in D We first assume all APs use the same channel an thus use the subscript We istinguish two cases: (a) Overlapping transmission areas, (, ] The cell associate with AP i is the segment [(i /), (i + /)] Consier a particular user locate at u (, ] Its exclusion region is equal to the union of the segment [, u + ] an of all cells whose AP is locate on this segment In view of (7), the level of interference β(u) is equal to the area of this exclusion region normalize by the cell area, It may be em(s) 8 6 e Figure 7: Functions e M, M =,,3 We euce from (6) the capacity C of each cell as a function of the inter-ap istance: Z + C () = + e + e (s)s () (b) Non-overlapping transmission areas, (, 3] In this case, the transmission areas o not overlap as the separation istance is greater than A given cell interferes only with its immeiate neighbours since the separation istance with other cells is greater than The total interference seen from the cell is therefore equal to twice that of two interfering APs erive in the previous section an the cell capacity is: C () = s Figure 8 illustrates the cell capacity as a function of the inter-ap istance We observe that capacity presents exactly the same types of irregularity iscusse in the previous example but this time the set of irregular points is iscrete an infinite The graph can best be unerstoo by observing how cell capacity ecreases from to as the istance between access points shrinks from + to The first type of irregularity is iscontinuity occurring at points =, where capacity n n+ ecreases sharply to Each of these points correspons to the introuction of a new tier of interfering APs into the transmission range of the consiere AP, say The n-th interfering tier consisting of APs n an n enters the transmission range of AP when n = At this point e e n+, interference with tier n + is null an cell capacity is n + being the number of APs locate within the transmission range As the inter-ap istance ecreases further, interference with tier n + begins an procees as in the case of two cells The result is slope iscontinuity occurring when (n + /) = Capacity at this point can also be compute easily by observing that each cell of tier n +

7 Capacity Figure 8: Capacity, infinite network in D Maximal traffic ensity Figure 9: Density, infinite network in D increases the mean slot time by an all previous tiers are 8 within the transmission range The capacity is then 8n+5 Between each couple of irregular points, capacity behaves like that of two interfering cells Figure 9 shows the maximal traffic ensity normalise to the maximal traffic ensity of an isolate cell By efinition, the maximal traffic ensity is the ratio of cell capacity to the cell area: () = C() min(,) It emonstrates that a maximal ensity of is possible only when or > 3 In the latter case the network no longer has complete coverage The former is obtaine at the cost of very high AP ensity 53 Use of multiple channels We generalise the results of the previous section to a network with M > non-interfering channels The APs operating on channel k, k =,, M are place at (nm + k), n Z As in the case of a single channel, we istinguish two cases: overlapping ( ) an non-overlapping ( > ) transmission areas The case of non-overlapping is trivial, capacity C M is always equal to because the istance between any two interfering APs is greater than 3 The case of overlapping is obtaine from equation () by replacing e with e M: C M() = Z +, + e M + e M(s)s where e M(s) is the total area of cells whose AP is locate on the segment (, s] an uses the same channel as AP The C C C3 3 function is illustrate in Figure 7 for M =, 3 Note that e M steps by at sm, s Z as a new interfering AP is inclue in the segment; it remains constant for +(M ) an then increases linearly with slope until it reaches (s + )M; it jumps again at this point an so on Figure 8 illustrates the cell capacity for M =,3 Capacity increases as expecte with the number of channels while the qualitative behaviour remains the same We have jumps at points = an slope iscontinuity at = for nm nm n =,, The maximal traffic ensities M for M =,3 are shown in Figure 9 The figure shows that traffic ensity has an infinite number of (local maximum, local minimum) pairs This occurs because the capacity slope ecreases from to as we move from The local maximum is attaine in segment [ to (n )M nm, nm nm ] an the local minimum is attaine at The global maximum always occurs uring nm interference with the first tier only an is therefore locate between an with: M M M() = + + M The maximum of this expression is M M + +M occurring at M = / M M + For example, if only two channels are use, the optimal inter-ap istance is = / an the network can accept up to times as much traffic as a single channel network When M = 3 (as in 8b/g), capacity increases 3 fol when = / 5 6 D NETWORKS We now investigate the traffic capacity of D networks The set of user classes is U = R Again, the traffic istribution is assume to be uniform on the coverage region of the network We successively consier four network topologies We only outline results erive in a similar way to those for D networks To obtain analytic expressions we use the infinity norm istance, (u, v) = max( u v, u v ) for all u, v R, instea of the -norm istance, (u, v) = p (u v ) + (u v ) for all u, v R The transmission area of each AP is thus no longer a circle of raius but a square with sies of length Of course, the approach applies to the usual eucliian istance as well but leas to more complex expressions 6 Two an four APs APs in the two noe network are separate by a istance In the noe network they are at the vertices of a square with sies of length Formulas for the capacity can be erive by generalizing the metho use in 5 Figure 5 shows the respective network capacities The capacity of APs in D is slightly higher than in D This is because the fraction of user positions inclue in an exclusion zone is somewhat smaller in D On the other han, interference in the AP network is greater leaing to significantly reuce capacity 6 Infinite linear network in D space We generalise to D the network of Figure 6 Each cell is now a rectangle of height an with centre at the AP if an a square with sie length if > M frequency

8 M M M M M M Table : Infinite linear network in D space Maximal traffic ensity linear, M = linear, M = gri,m = gri,m = channels are allocate to the APs as in 5 In this an the next sections, we evaluate the optimal traffic ensity M Proceeing similarly as in the previous sections, we erive the interference β at some fixe point (u, v) of the reference cell: β(u, v) = + e M + e M + e M u + e M «v The factor ( v )/ accounts for the fact that the receiver is now situate in D It is less than so that capacity in D is greater than in D We have: C M() = +e M Z + «+ e M(s)s e! 8 Capacity for M =, is illustrate in Figure Maximal traffic ensity is given by M() = CM() min(,) It is optimal when interference occurs only with tier for between In this range, M has the following expression: M an M M() = + 8 M + The optimal ensity can be shown to be greater than (M ), the value obtaine for = /(M ) Table gives the exact optimal ensity erive numerically an shows that the boun is a goo approximation for M The scaling behaviour is the same for the linear network in D an D Capacity 8 6 linear, M = linear, M = gri,m = gri,m = Figure : Capacity, infinite network in D Figure : Density, infinite network in D 63 Infinite gri network in D space In the D gri network, the APs are at the vertices of a regular gri where the istance between ajacent vertices is Each cell is a square with sies of length if < an if > The number of channels is M which we assume to be the square of an integer For the sake of completeness we give the expression of β(u, v) an C M: β(u, v) = + e M e M u + C M() = e M + v + e M e M + 3e M «+ «e M M «Z + +e e M (s)s u + e M ««v + «e M e M, + e M e M + Z + The maximal traffic ensity is now given by C M() M() = min(, ) e M (s)s! The capacity an maximal traffic ensity are plotte in Figures an, respectively, for M =, The optimal traffic ensity occurs in interval [ M, M ] where M is given by M() = + + M + M As above we can lower boun the optimal ensity by its value at = /( M ), ( M ) Table confirms that this boun is a goo approximation for M We observe that maximal traffic ensity increases times as fast as the number of frequency channels = ( M ) M M > M Again, for the purpose of comparison, we provie in Table the exact an approximate values for M an M The table shows that for large M, M scales approximately as ( M ) Thus, even if the cells are subject to more

9 M M M M M M Mean flow rate 8 6 =, u = 6 =, mean =, u = 8 6 =, u = 6 =, mean =, u = Table : Infinite gri network in D space interference in D, the maximal traffic ensity scales as four times the number of use frequency channels 7 FLOW-LEVEL PERFORMANCE In this section we evaluate the mean flow rate as a function of user position an inter-ap istance for the simple two AP network of 5 In the following is traffic ensity per unit of length 7 Impact of traffic intensity We take =, which correspons to the case of partial coupling an overlapping transmission areas, cf Figure The cell length is equal to + / = 6 In Figure we plot the mean flow rate as a function of traffic intensity per cell, ρ = 6, for two user positions One user position is at the ege of the right han cell (u = 6), the other miway between the two APs (u = ) Fig (a) is erive from the approximate analytical moel of 3 while Fig (b) plots results from simulation of the unerlying state-epenent processor sharing system of 33 The approximation agrees closely with the simulation results The figures show behaviour typical of a iscriminatory processor sharing system The mean rate (for all user positions) ecreases approximately linearly from at zero loa (any flow receives all the cell capacity) to as traffic attains the capacity C evaluate in 5 for = (C 7) Users at the far ege gain higher throughput on average than users situate between the APs but the ifference is slight an isappears at the extremes Users at u = 6 gain higher throughput since they o not interfere with the other cell while users at u = always interfere The ifference is not large, however, since all active users in a cell in any occupancy state receive exactly the same rate Figures (c) an () correspon to NS- simulations with UDP traffic The mean flow rate an traffic intensity in both scenarios are normalise to the capacity of an isolate AP The RTS/CTS signals are assume to be instantaneously exchange in the Fig (c) while () correspons to the actual IEEE 8 stanar The analytic results agree with simulations in Fig (c) but iffer somewhat in Fig () The flow rate goes to zero as traffic intensity approaches 6 which is less than the compute capacity C 7 This is because the non-negligible RTS/CTS transmission times reuce capacity through hien-noe collisions, as follows In 8b, the RTS transmission time is approximately equal to backoff slots while the minimal contention winow length is 3 slots For = the APs are not able to hear each other; if one AP transmits an RTS to a user locate miway between the two APs, the other AP will provoke a collision at the user station if it also attempts a Mean flow rate Traffic intensity (a) Analytic moel =, u = 6 =, mean =, u = 6 8 Traffic intensity (c) NS- with instant RTS/CTS Traffic intensity (b) PS simulation =, u = 6 =, mean =, u = 6 8 Traffic intensity () NS- simulation Figure : Impact of traffic ensity transmission uring the slot perio On the other han, if the APs can hear each other, the RTS/CTS scheme succees if an only if the backoff counters o not have the same value 7 Impact of istance Figure 3 plots the mean flow rate against the inter-ap istance for two traffic ensities, = 5 an = The flow rate variations are irregular since capacity evolves ifferently with in the four regions ientifie in 5 Note that flow rate is zero at loa = for [55, ] since traffic intensity excees capacity Flow rate increases between = an = 3 but, of course, this oes not take into account users situate out of range of either AP Simulation results in Fig 3(b) an Fig 3(c) confirm the accuracy the moel when RTS/CTS times are neglecte However, the plots in Fig 3() show some iscrepancies The flow rate is lower in the range < < This is ue to the hien-noe problem explaine above, the impact being particularly significant at high traffic loa when the flow rate goes to zero because of collisions On the other han, the flow rate is slightly larger when < < This is because the aggregate capacity of two APs that can hear each other is slightly larger than the capacity of a single isolate AP While the backoff overhea for an isolate AP is the mean of a ranom variable uniformly istribute over the contention winow, it is approximately equal to the mean of the minimum of two uniformly istribute ranom variables when two APs compete 73 Impact of user position Finally, we erive analytical results for the mean flow transfer time W(r) of a user as a function of her/his relative location in the cell, r By symmetry, we can focus on cell We enote by f(r) the position of that user of cell locate at the ege of the exclusion region of the user of relative location r in cell

10 Mean flow rate = 5 = 5 = = (a) Analytic moel (b) PS simulation Mean transfer time = 8, = 5 = 8, = 5 =, = 5 =, = Noe position (a) Analytic moel Noe position (b) PS simulation Mean flow rate = 5 = 5 = = (c) NS- with instant RTS/CTS () NS- simulation Mean transfer time = 8, = 5 = 8, = 5 =, = 5 =, = Noe position (c) NS- with instant RTS/CTS Noe position () NS- simulation Figure 3: Impact of normalise inter-ap istance Figure : Impact of user position Proposition Assuming inter-cell interference is constant, as in 3, the mean transfer time satisfies the ifferential equation: W (r) = θf (r)w( f(r)), () where θ is a constant that epens on inter-ap istance an traffic ensity only The proof is in Appenix B The function f is the upper limit of the shae exclusion regions illustrate in Figures (a) to () Equation () can thus be solve piecewise giving the following expressions that epen on the extent of cell coupling (a )- Total coupling, (,]: W(r) = µ ( + ) r [, ] (b )- Partial coupling, overlappe transmission areas, (, ]: 8 >< W( ) r [, ] W(r) = W( >: ) sin θ(r ) + π r [, ] W() r [, ] (c )- Partial coupling, non-overlappe transmission areas, (,3]: ( W( ) r [, ] W(r) = W( ) sin θ(r ) + π r [,] ( )- Inepenent cells, (3, ): W(r) = µ r [, ] Figure (a) plots W as a function of r for = 8 an =, corresponing to cases (b ) an (c ) above, respectively The results highlight the negative impact of interference for users situate close to the centre of the network an having a larger exclusion region Simulation results in Figures (b)-(c) confirm the accuracy of the analytical approximation However because of the hien noe problem mentione above, we observe a larger transfer time in Fig () 8 CONCLUSION We have propose a moel to evaluate the ownlink traffic capacity of a multi-cell WLAN The capacity is efine as the limiting traffic intensity (flow arrival rate mean flow size) beyon which realize flow throughput tens to zero The moel has allowe us to evaluate the capacity of some toy network configurations proviing insight into the impact of inter-cell interference We observe an abrupt increase in capacity as APs cease to interfere irectly The impact of interference via the users to whom they transmit is less significant an ecreases as the inter-ap istance increases For multi-channel networks with a regular pattern of frequency re-use, the variation in capacity as a function of istance prouces clear local maxima an minima in achievable traffic ensity Optimal AP placement correspons to spacing neighbouring APs using the same channel by slightly more than their transmission range Capacity rapily ecreases to a global minimum when the inter-ap spacing is small enough to bring APs into irect conflict With optimal AP placement, the capacity of an M-channel D gri network is M times that of a single channel network That the gain is amplifie by the factor of is ue to a contraction of the cell coverage area as APs (of all channels) are more closely space This phenomenon is not an artefact of the moel an is a significant observation for the esign of frequency re-use in real networks The moel allows an evaluation of the expecte throughput of a ownloa as a function of cell loa an user position Results for the simplest -AP network show that the system behaves broaly like a iscriminatory processor sharing system Users experiencing lower inter-cell interference gain higher throughput but the ifference between best an worst positions remains slight Performance is mainly governe by the traffic capacity that fixes the point where

11 throughput goes to zero The moel makes many simplifying assumptions whose significance we now briefly iscuss The traffic moel assumes Poisson flow arrivals an exponential flow sizes These assumptions are necessary for the proof of the key stability result but we expect a reasonable egree of insensitivity to carry over from the unerlying processor sharing system We have assume equal sharing of cell capacity between concurrent flows This woul not occur if users ha significantly ifferent roun trip times, ue to TCP RTT bias, for instance This iscrepancy is unlikely to affect our broa conclusions, however We ignore the impact of upstream traffic This is clearly a bol simplification since even ownloa traffic generates TCP ACK packets in the upstream We suppose these ACKs can be accounte for by extening the packet transmission time We have chosen a simple raio propagation moel an erive numerical results only for toy symmetric networks with uniform traffic These assumptions coul be remove at the cost of ae complexity The most contestable simplifications are in the way we moel the impact of interference As note in the introuction, the channel access process is extremely complex an clearly beyon any precise stochastic moelling approach Our assumptions, that time is slotte an that the time to transmit a packet is proportional to the number of programme transmissions within its exclusion zone, effectively e-couple the complex inter-cell interference allowing the analytical evelopments The assumption is clearly correct when there is no interference, an is reasonable when cells interfere completely In an average sense, it is also intuitively plausible that transmission times increase linearly with the amount of interference The moel ignores the impact of finite RTS/CTS transmission times We have observe that this introuces iscrepancies ue to the aitional collisions that occur leaing to an overestimate of network capacity In future work we inten to relax some of the above assumptions We also mean to verify an quantify the preicte phenomena in a more realistic network configuration The ultimate objective is to erive practical guielines for the esign an operation of ense multi-cell, multi-channel WLAN access networks Finally, it woul be interesting to evaluate the impact on performance of possible moifications to network operation In particular, the assume FIFO queuing in APs might be replace by a more opportunistic scheme where a packets are chosen for transmission epening on how much interference they cause 9 REFERENCES [] M Armony an N Bambos Queueing networks with interacting service resources In Proc 37th Annual Allerton Conf Commun, Control, Comp, 999 [] G Bianchi Performance analysis of the ieee 8 istribute coorination function IEEE Journal on Selecte Areas in Communications, 8(3):535 57, [3] T Bonal, S Borst, N Hege, an A Proutiére Wireless ata performance in multi-cell scenarios In SIGMETRICS, pages ACM Press, [] S Borst, M Jonckheere, an L Leskela Stability of parallel queueing systems with couple rates to appear in Journal of Discrete Events an Dynamic Systems, 7 [5] J Dai On positive Harris recurrence of multiclass queueing networks: A unifie approach via flui limit moels Annals of Appl Probability, 5():9 77, 995 [6] G Fayolle an R Iasnogoroski Two couple processors: the reuction to a Riemann-Hilbert problem Z Wahr verw Ge, 7(3):35 35, 979 [7] G Fayolle, I Mitrani, an R Iasnogoroski Sharing a processor among many job classes J ACM, 7(3):59 53, 98 [8] N Gupta an P R Kumar A performance analysis of the 8 wireless LAN meium access control Communications in Information an Systems, 3():79 3, [9] IEEE 8 Stanar Wireless LAN Meium Access Control (MAC) an Physical Layer (PHY) Specifications, 999 [] C Joo an N B Shroff Performance of ranom access scheuling schemes in multi-hop wireless networks In INFOCOM 7, 7 [] A Kumar, E Altman, D Miorani, an M Goyal New insights from a fixe point analysis of single cell IEEE 8 WLANs In INFOCOM 5, 5 [] F Lebeugle an A Proutiere User-level performance in WLAN hotspots In ITC 9, 5 [3] Lin an S B Rasool Constant-time istribute scheuling policies for a hoc wireless networks In IEEE Conference on Decision an Control, 6, pages 58 63, 6 [] R Litjens, F Roijers, J Van en Berg, R Boucherie, an M Fleuren Performance analysis of wireless lans: An integrate packet/flow level approach In ITC 8, 3 [5] S P Meyn Transience of multiclass queueing networks an their flui moels Annals of Appl Probability, 5:96 957, 995 [6] M K Pana, A Kumar, an S H Srinivasan Saturation throughput analysis of a system of interfering IEEE 8 WLANs In WoWMoM 5, 5 [7] R R Rao an A Ephremies On the stability of interacting queues in a multiple-access system IEEE Trans on Information Theory, 3:98 93, 988 [8] L Tassiulas an A Ephremies Stability properties of constraine queueing systems an scheuling policies for maximum throughput in multihop raio networks IEEE Trans on Automatic Control, 37():936 98, 99 APPENDI A STABILITY Proof of Theorem The proof procees by applying the flui limit approach of Dai [5] Let x i j(t) be the class-j flui volume at AP i at time t This represents the number of class-j ongoing flows when the flow population an the time are scale by the

12 same factor, growing to infinity We enote by x i(t) the total flui volume at AP i at time t an by ξ i j(t) the proportion of class-j flui volume at AP i at time t, when x i(t) > : ξ i j(t) = xi j(t) x i(t) with x i(t) = j U i x i j(t) It follows from the strong law of large numbers that at any time t such that x i(t) > for all i =,, N: with an x i j t = λi j µ x i j(t) δ i(t) x, i(t) δ i(t) = j Ui ξ i j(t) δ i j(t) δ j(t) i = + ξk l (t)χ(j, l) k i l U k We first prove that ξ i j(t) tens to α i j, the proportion of class-j traffic at AP i, for all j U i Note that: x i j t = λi j > if x j i (t) = an xi(t) > Now for all j, l U i, we have at any time t such that x i j(t) > an x i j(t) > : x i «t ln j (t) x i l (t) = xi j t x i j (t) xi l(t) t x i l (t) = λi j x i j (t) λi l x i l (t) Thus the ratio x i j(t)/ x i l(t) ecreases if an only if it is larger than λ i j/λ i l Since ξ j(t)/ ξ i l(t) i = x i j(t)/ x i l(t) an α i j/α i l = λ i j/λ i l, the ratio ξ j(t)/ ξ i l(t) i ecreases if an only if it is larger than α i j/α i l Using the fact that: ξi j (t) = α i j =, j U i j U i we euce that ξ i j(t) tens to α i j for all j U i when t tens to infinity In view of () an (5), this in turn implies that δ i j(t) tens to β i j for all j U i an that δ i(t) tens to P j U i α i jβ i j when t tens to infinity Thus for all ε >, we have for sufficiently large t such that x i(t) > for all i =,, N: x i j t λi j µ P α i j l U i α i l βi l ( ε) Note that this inequality hols even if x k (t) = for some k i, since this can only increase the service rate of AP i Define the flui workloa of AP i at time t as: w i(t) = j U i x i j(t) µ βi j Proof of Theorem P We show that if i j U i α i jβj i > for all i =,, N, the flui moel introuce in the proof of Theorem is unstable We prove in a similar way that ξ j(t) i tens to α i j an δ i(t) tens to P j U i α i jβj i when t tens to infinity Thus for all ε >, we have for sufficiently large t such that x i(t) > for all i =,, N: an x i j t λi j µ P w i t α i j l U i α i l βi l ( + ε) i j U i α i jβ i j ε, which is positive for all i =,, N for sufficiently small ε Thus w i(t) increases at least linearly for all i =,, N: the flui moel is unstable, which implies the transience of the unerlying Markov process [5] B PERFORMANCE Proof of Proposition Cell correspons to the interval [a, b] with a = an b = min( +,) Equations (8), (9) an () applie to cell have the following continuous counterparts: W(r) Z b a (r) = λw(r), φ(r)r Z b a φ(r)w(r)r = φ(r) µ, Z b R b f(r) φ(r) = + φ(r)r (r)r R b (r)r, a a where λ = µ is the flow arrival ensity an the last equality follows by symmetry Let an φ be the integrals of an φ over [a, b] Differentiating the above equations with respect to r, we obtain: We euce (), with (r) = λw (r), φ W (r) = φ (r) µ, φ (r) = φ f (r)( f(r)) φ θ = ( φ ) We have for sufficiently large t such that w i(t) > : w i i α i t jβj i + ε, j U i which is negative for sufficiently small ε Thus w i(t) = an x i j(t) = for all j U i for sufficiently large t This property hols for all APs: the flui moel is stable, which implies the ergoicity of the unerlying Markov process [5]