Asynchronous CSMA Policies in Multihop Wireless Networks with Primary Interference Constraints

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1 Asynchronous CSMA Polces n Multhop Wreless Networks wth Prmary Interference Constrants Peter Marbach, Atlla Erylmaz, and Asu Ozdaglar Abstract We analyze Asynchronous Carrer Sense Multple Access (CSMA) polces for schedulng packet transmssons n multhop wreless networks subject to collsons under prmary nterference constrants. Whle the (asymptotc) achevable rate regon of CSMA polces for sngle-hop networks has been wellknown, ther analyss for general multhop networks has been an open problem due to the complexty of complex nteractons among coupled nterference constrants. Our work resolves ths problem for networks wth prmary nterference constrants by ntroducng a novel fxed-pont formulaton that approxmates the lnk servce rates of CSMA polces. Ths formulaton allows us to derve an explct characterzaton of the achevable rate regon of CSMA polces for a lmtng regme of large networks wth a small sensng perod. Our analyss also reveals the rate at whch CSMA achevable rate regon approaches the asymptotc capacty regon of such networks. Moreover, our approach enables the computaton of approxmate CSMA lnk transmsson attempt probabltes to support any gven arrval vector wthn the achevable rate regon. As part of our analyss, we show that both of these approxmatons become (asymptotcally) accurate for large networks wth a small sensng perod. Our numercal case studes further suggest that these approxmatons are accurate even for moderately szed networks. Index Terms Asymptotc Capacty Regon of Wreless Networks, Carrer-Sense Multple Access, Fxed-Pont Approxmaton, Throughput-Optmal Schedulng. I. INTRODUCTION The desgn of effcent resource allocaton algorthms for wreless networks has been an actve area of research for decades. The semnal work [38] of Tassulas and Ephremdes has poneered n a new thread of resource allocaton mechansms that are throughput-optmal n the sense that the algorthm stablzes the network queues for flow rates that are stablzable by any other algorthm. Ths and subsequent works (e.g. [36], [], [0], [34], [32], [26], []) have proposed schemes that use queue-lengths to dynamcally perform varety of resource allocaton decsons, ncludng medum access, routng, power control, and schedulng. Schedulng (or medum access) has tradtonally been the most computatonally heavy and complex component of resource allocaton strateges due to the nterference-lmted P. Marbach s wth the Department of Computer Scence at the Unversty of Toronto, Toronto, Canada. Emal: marbach@cs.toronto.edu. A. Erylmaz s wth the Department of Electrcal and Computer Engneerng at the Oho State Unversty, Columbus, USA. Emal: erylmaz@ece.osu.edu. A. Ozdaglar s wth the Department of Electrcal Engneerng and Computer Scence at the Massachusetts Insttute of Technology, Cambrdge, USA. Emal: asuman@mt.edu. Ths work was supported n part by DTRA Grant HDTRA , and NSF Awards: CAREER-CNS and CCF nature of the wreless medum. The queue-length-based polces typcally have schedulng rules that use the queue-length nformaton to avod collsons whle prortzng the servce of more heavly loaded nodes. However, due to the couplng between the nterference constrants of nearby transmssons, such schedulng decsons can requre hghly complex and centralzed decsons. Ths observaton has motvated hgh research actvty n the recent years for the development of dstrbuted and low-complexty mplementatons of queuelength-based schemes (e.g. [37], [3], [7], [25], [8], [30], [4], [9], [42], [9]). Also, random access strateges have been nvestgated n a number of works (e.g. [22], [24], [39], [6], [6], [4], [35]) that acheve a fracton of the capacty regon. In the case of prmary nterference model and general network topology that we consder, ths fracton s /2 and s tght (.e. there exst networks for whch no rate outsde half of the capacty regon can be supported). These results have suggested that a sgnfcant porton of the capacty regon may need to be sacrfced to acheve dstrbuted mplementaton wth random access strateges. Besdes performance degradaton, the practcal mplementaton of exstng resource allocaton polces are also complcated by several factors: they usually rely on global synchronzaton of transmssons and requre a far amount of nformaton sharng (typcally n the form of queue-lengths) between nodes to perform decsons. In ths work, we consder an alternatve class of random access strateges wth favorable complexty and practcal mplementablty characterstcs. In partcular, we nvestgate Carrer Sense Multple Access (CSMA) polces n whch nodes operate asynchronously and sense the wreless channel before makng an attempt to transmt a packet, whch may result n collsons. We analyze such asynchronous CSMA polces for schedulng packet transmssons n multhop wreless networks subject to collsons under prmary nterference constrants. For a lmtng regme of large networks wth a small sensng perod, we derve an explct characterzaton of the achevable rate regon of CSMA polces. Whle an explct characterzaton of the (asymptotc) achevable rate regon of CSMA polces has been establshed n the specal case of snglehop networks, ther analyss for general multhop networks has been an open problem due to the complexty of the nteractons among coupled nterference constrants. Our work resolves ths problem for networks wth prmary nterference constrants through the ntroducton of a novel fxed pont formulaton that approxmates the lnk servce rates of CSMA polces. The man contrbutons of the paper are as follows. We provde an analytcal fxed-pont formulaton to approxmate the performance of asynchronous CSMA pol-

2 ces operatng n mult-hop networks subject to collsons wth prmary nterference constrants. Our formulaton makes nterestng connectons to work by Hajek and Krshna on the accuracy of the Erlang fxed pont for stochastc loss networks [7], [20]. Whle our techncal development focuses on the prmary nterference model, we note that t suggests a general approach that can be used to handle hgher-order nterference models. We rgorously show that our fxed pont formulaton to approxmate the performance of asynchronous CSMA polces s asymptotc accurate under an approprate lmtng regme where the network sze becomes large. We also demonstrate through smulaton results that such accuracy s acheved for moderately szed network. Ths s especally mportant snce t suggests that the approxmaton wll be useful even n realstc networks. We utlze the fxed-pont formulaton to characterze the achevable rate regon of our CSMA polces, and further provde a constructve method to fnd the transmsson attempt probabltes of a CSMA polcy that can stably support a gven network load n the achevable rate regon. To the best of our knowledge, ths consttutes the frst such characterzaton of CSMA achevable rate regon n mult-hop networks wth the explct ncorporaton of collsons. We show that for large networks wth a balanced traffc load, the CSMA achevable rate regon takes an extremely smple form that smply lmts the ndvdual load on each node to, whch s the maxmum supportable load. Ths result together wth the prevous shows that the capacty regon of large mult-hop wreless networks (asymptotcally) takes on a very smple form. The rest of the paper s organzed as follows. We start by notng several relevant works n the context of CSMA polces n Secton II. In Secton III, we defne our system model, and n Secton IV we descrbe the class of CSMA polces we consder n ths paper. In Secton V we provde a summary and dscusson of our man result, as well as an overvew of the analyss. We provde our fxed pont formulaton and prove ts asymptotc accuracy n Sectons VI and VIII, respectvely. Then, n Secton VII and IX, we provde a characterzaton of the achevable rate regon of the class of CSMA polces, and show that t s asymptotcally capacty achevng. We end wth concludng remarks n Secton X. II. RELATED WORK In ths secton, we provde a summary of the work on CSMA polces for sngle-hop and multhop networks that s most relevant to the analyss presented n ths paper, and note the key dfferences of our work n ths paper. For sngle-hop networks where all nodes are wthn transmsson range of each other, the performance of CSMA polces s well-understood [3]. Furthermore, the well-known nfnte node approxmatons provdes a smple characterzaton for the throughput of a gven CSMA polcy, as well as the achevable rate regon of CSMA polces, n the case of a sngle-hop networks [3]. Ths approxmaton has been nstrumental n the understandng of the performance of CSMA polces, as well as for the desgn of practcal protocols for wreless local area networks. For the case where nodes are saturated and always have a packet to sent, the achevable rate regon of CSMA polces s easly obtaned [5]. For the case where nodes only make a transmsson attempt when they have a packet to transmt has also recently been studed [5], [28]. For general multhop networks, results for CSMA polces are avalable for dealzed stuaton of nstantaneous channel feedback. Ths assumpton of nstantaneous channel feedback allows the elmnaton of collsons, whch sgnfcantly smplfes the analyss, and allows the use of Markov chans to model system operaton. Under such an nstant feedback assumpton, an early work [4] has shown that the statonary dstrbuton of the assocated Markov chan takes a product form. A more recent work [8] has utlzed such a productform to derve a dynamc CSMA polcy that, combned wth rate control, acheves throughput-optmalty whle satsfyng a gven farness crteron. Smlar results wth the same nstantaneous feedback assumpton have been ndependently derved n [33] n the context of optcal networks and later extend to wreless networks [29]. Another relevant recent work [27] suggests a way of handlng collsons under the synchronous CSMA operaton. Our approach dffers from much of ths lterature n that we do not assume nstantaneous feedback or tme synchronzaton, and explctly consder collsons, whch are unavodable n a real mplementaton. The ncorporaton of possble collsons requre the development of a completely dfferent modelng of the CSMA performance than the contnuous-tme Markov chan model used for the aforementoned dealzed setup. Instead, we develop a novel fxed-pont approxmaton for a specfc nterference model, and show ts asymptotc accuracy. An mportant byproduct of ths development s the quantfcaton of the proxmty of the CSMA achevable rate regon to the lmtng capacty regon as a functon of the sensng perod level. Such nformaton wll be extremely helpful n determnng how small the sensng perod should be to acheve a desred fracton of the capacty regon. Clearly, a non-zero sensng perod, however small, must be consdered n the CSMA operaton to account for the propagaton delay assocated wth transmssons. Yet, the ncluson of such a factor creates non-zero probabltes of collsons. Thus, n order to keep the collson level at a small level, the aggressveness of the CSMA polcy must depend on the partcular value of the sensng perod for the gven system. In our development, we explctly determne ths connecton and provde a constructve method to determne the CSMA parameters as a functon of the sensng perod. Moreover, n ths paper we consder a completely asynchronous CSMA operaton, whch relaxes any synchronsm assumptons amongst the nodes that wll facltate ts practcal mplementaton. Such a relaxaton creates many techncal challenges, whch are resolved n ths paper. III. SYSTEM MODEL Network Model: We consder a fxed wreless network composed of a set N of nodes wth cardnalty N, and a set L of 2

3 drected lnks wth cardnalty L. A drected lnk (, j) L ndcates that node s able to send data packets to node j. We assume that the rate of transmsson s the same for all lnks and all packets are of a fxed length. Throughout the paper we rescale tme such that the tme t takes to transmt one packet s equal to one tme unt. For a gven node N, let U := {j N : (j, ) L} be the set of upstream nodes,.e. the set contanng all nodes from whch can receve packets. Smlarly, let D := {j N : (, j) L} be set of downstream nodes,.e. the set contanng all nodes j whch can receve packets from. Collectvely, we denote the set of all the neghbors of node as N := U D. Also, we let L := {(, j) : j D } be the set of outgong lnks from node,.e. the set of all lnks from node to ts downstream nodes D (see Fg. for an example). Fg.. Example of a network where two routes f and g gven by R f = {(s f, ),(, j),(j, v), (v, w),(w, d f )} and R g = {(s g, k),(k, ),(, j),(j, n),(n, d g)}. In ths network: the set of upstream neghbors of node j s gven by U j = {, v}; the set of downstream neghbors of node j s gven by D j = {, s g, n, v}; the set of outgong lnks of node j s gven by L j = {(j, ), (j, s g), (j, v), (j, n)}; the set of lnks that nterfere wth (, j) s gven by I (,j) = {(j, ),(s f, ),(, k), (k, ), (j, s g), (j, v), (v, j),(j, n)}; the mean rate on lnk (, j) s gven by λ (,j) = λ f + λ g; and the load on node s Λ = 2λ f + 2λ g. Throughout the paper, we assume that U = D, for all N so that we have U = D = N, for each N. Ths assumpton smplfes the notaton as we can use a sngle set N to represent both D and U. Our analyss can be extended to the more general case requrng only notatonal changes. Thus, henceforth we wll descrbe a network by the tuple (N, L). Interference Model: We focus on networks under the wellknown prmary nterference, or node exclusve nterference, model [2], [40], defned next. Defnton (Prmary Interference Model). A packet transmsson over lnk (, j) L s successful f only f wthn the transmsson duraton there exsts no other actvty over any other lnk (m, n) L whch shares a node wth (, j). For each lnk l L, we use I l denote the set of lnks l L that nterfere wth lnk l,.e. the set of all lnks l L that have a node n common wth lnk l. The prmary nterference model apples, for example, to wreless systems where multple frequences/codes are aval- Notce that our defnton of nterference model does not requre a tme slotted operaton of the communcaton attempts, and hence apples to asynchronous network operaton. able (usng FDMA or CDMA) to avod nterference, but each node has only a sngle transcever and hence can only send to or receve from one other node at any tme (see [3], [7] for addtonal dscusson). Traffc Model: We characterze the network traffc by a rate vector λ := {λ r } r R where R s the set of routes used by the traffc, and λ r, λ r 0, s the mean rate n packets per unt tme along route r R. For a gven route r R, let s r be ts source node and d r be ts destnaton node, and let R r = {(s r, ), (, j),, (v, w), (w, d r )} L be the set of lnks traversed by the route. We allow several routes to be defned for a gven source and destnaton par (s, d), s, d N. Gven the rate vector λ = {λ r } r R, we let λ (,j) := λ r, (, j) L, () r:(,j) R r be the mean packet arrval rate to lnk (, j). Smlarly, we let Λ (λ) := [ λ(,j) + λ (j,) ], N. (2) be the mean packet arrval rate to node N (see Fgure for an example). To keep the notaton lght, we wll n the followng at tmes use the notaton Λ nstead of Λ (λ). IV. POLICY SPACE AND CSMA POLICY DESCRIPTION In ths secton, we ntroduce the space of schedulng polces that we are nterested n, and provde the descrpton of CSMA polces that we consder. We also defne the notons of stablty and achevable rate regon that we use for our analyss. A. Schedulng Polces and Capacty Regon Consder a fxed network (N, L) wth traffc vector λ = {λ r } r R. A schedulng polcy π then defnes the rules that are used to schedule packet transmssons on each lnk (, j) L. In the followng, we focus on polces π that have well-defned lnk servce rates as a functon of the rate vector λ = {λ r } r R. Defnton 2 (Servce Rate). For a gven network (N, L), the offered servce rate µ π (,j)(λ) for lnk l = (, j) L under polcy π and traffc vector λ = {λ r } r R s equal to the fracton of tme that polcy π allocates for successfully transmttng packets on lnk l = (, j) under the prmary nterference model,.e. the fracton of tme node can send packets on lnk l = (, j) that wll not experence nterference from any lnk l I l. Let P be the class of all polces π that have well-defned lnk servce rates. Note that ths class contans a broad range of schedulng polces, ncludng dynamc polces such as queuelength-based polces that are varatons of the MaxWeght polcy [38], as well as noncausal polces that know the future arrval of the flows. We then defne network stablty as follows. 3

4 Defnton 3 (Stablty). For a gven network (N, L), let µ π (λ) = {µ π (,j) (λ)} (,j) L be the vector of lnk servce rates of polcy π, π P, for the rate vector λ = {λ r } r R. We say that polcy π stablzes the network for λ f λ (,j) < µ π (,j) (λ), (, j) L. Ths commonly used stablty crtera [38] requres that for each lnk (, j) the lnk servce rate µ π (,j)(λ) s larger than the arrval rate λ (,j). The capacty regon of a network (N, L) s then defned as follows. Defnton 4 (Capacty Regon). For a gven network (N, L), the capacty regon C s equal to the set of all traffc vectors λ = {λ r } r R such that there exsts a polcy π P that stablzes the network for λ,.e. we have C = {λ 0 : π P wth λ (,j) < µ π (,j)(λ), (, j) L}. B. CSMA Polces In ths paper, we are nterested n characterzng the performance of CSMA polces that operate by actvely sensng the channel actvty and, when dle, performng random transmsson attempts accordng to the parameters of the partcular CSMA polcy. Before we descrbe the detals of CSMA polcy operaton n Defnton 6, we present our modelng of heterogeneous channel sensng delay that must exst n the real-world mplementaton of such polces. Defnton 5 (Sensng Delay { l (l )}). Consder a gven lnk l = (, j) L. When a lnk l n the nterference regon I l of a lnk l becomes dle (or busy), then transmttng node of lnk l wll not be able to detect ths nstantaneously, but only after some delay, to whch we refer to as the sensng delay 2 l (l ). We note that the sensng delay gven n the above defnton s lower-bounded by the propagaton delay between node and. The exact length of the sensng delay wll depend on the specfcs of the sensng mechansm deployed. In Appendx A, we descrbe two possble approaches to how channel sensng could be performed for networks wth prmary nterference constrants. Whle the sensng delay of dfferent node-lnk pars may dffer, throughout ths work, we make the assumpton that all sensng delays are bounded by a constant measured wth respect to the normalzed packet transmsson duraton. We refer to ths upper bound as the sensng (or dle) perod of a CSMA polcy. Assumpton. There exsts a constant to whch we refer to as the sensng (or dle) perod of a CSMA polcy such that for all lnks l L, we have that l (l ), l I l. Recall that throughout the paper we rescale the tme such that the tme t takes to transmt one packet s equal to one 2 In our subsequent dscusson, for ease of exposton we wll typcally refer to lnks as performng sensng or schedulng a packet transmsson. Ths must be understood as the transmttng node of the (drected) lnk performng the acton. tme unt. Hence, the duraton of an dle perod s measured relatve to the length of one packet transmsson,.e. f the length of an dle perod s L seconds and the length of a packet transmsson s L p seconds, then we have = L /L p. For a fxed L, the duraton of an dle perod wll become small f we ncrease the packet lengths. Hence, we can control the value of by modfyng L p for a fxed L. Defnton 6 (CSMA(p, ) Polcy). A CSMA polcy s gven by a transmsson attempt probablty vector p = (p (,j) ) (,j) L [0, ] L and a sensng perod (or dle perod) > 0, that satsfes Assumpton. Gven p and, the polcy works as follows: each node, say, senses the actvty on ts outgong lnks l L. We say that has sensed lnk (, j) L to be dle for a duraton of an dle perod f for the duraton of tme unts we have that (a) node has not sent or receved a packet and (b) node has sensed that node j has not sent or receved a packet. If node has sensed lnk (, j) L to be dle for a duraton of an dle perod, then starts a transmsson of a sngle packet on lnk (, j) wth probablty p (,j), ndependent of all other events n the network. If node does not start a packet transmsson, then lnk (, j) has to reman dle for another perod of tme unts before agan has the chance to start a packet transmsson. Thus, the epochs at whch node has the chance to transmt a packet on lnk (, j) are separated by perods of length durng whch lnk (, j) s dle, and the probablty that starts a transmsson on lnk (, j) after the lnk has been dle for tme unts s equal to p (,j). In the event that the dle perods of two lnks l and l that both orgnate at node end at the same tme, we use the followng mechansm to prevent the possblty that node starts n ths case a transmsson on both lnks l and l smultaneously (leadng to sure collson): lettng ˆL (t) denote the set of lnks n L for whch an dle perod ends at tme t, for each lnk l = (, j) ˆL (t) the probablty that node starts a transmsson on lnk ) l at tme t s gven, ndependently of all by ( ) ( p (,j) / {j :(,j ) ˆL p (t)} (,j ) other attempts by any node n the network. Fnally, we assume that packet transmsson attempts are made accordng to above descrpton regardless of the avalablty of packets at the transmtter. In the event of the absence of a data packet, the transmttng node transmts a dummy packet, whch s dscarded at the recevng end of the transmsson (see also our dscusson n Secton X), but s counted n the servce rate provded to that lnk. We note that whle all the nodes use the same sensng tme to detect whether a gven lnk s dle, the actual tme that t takes a node to detect that another node has stopped (or started) transmttng a packet s determned by ts ndvdual sensng delay as gven n Defnton 5, whch can be dfferent for dfferent nodes. Dfferent sensng delays wll lead to an asynchronous operaton of the network where the sensng and packet transmsson perods of dfferent nodes are not algned. Also note that, under our CSMA polcy, lnks make a transmsson attempt wth a fxed probablty after the channel has been sensed to be dle, ndependent of the current backlog 4

5 of the lnk. Ths may seem to be an unreasonable scenaro as t mples that a lnk mght make a transmsson attempt even f there s no packet to be transmtted. However, there are at least two reasons why ths stuaton s of nterest. Frst, such a polcy could ndeed be mplemented (where lnks send dummy packets once n a whle) Second, and more mportantly, beng able to characterze the throughput of such a polcy opens up the possblty of studyng more complex, dynamc CSMA polces where the attempt probabltes depend on the current backlog. In partcular, the results of our analyss can be used to formulate a flud-flow model for backlog-dependent polces, where the nstantaneous throughput at a gven state (backlog vector) s gven by the expected throughput obtaned n our analyss. Such polces are of nterest as they mght allow for dynamc adaptaton of the traffc load n the network (e.g. see [23]). Gven the length of an dle perod, n the followng we wll smply use p to refer to the CSMA(p, ) polcy. Next, we defne the achevable rate regon of a CSMA polcy. We wll provde a precse descrpton of the lmtng regme that we consder n Secton IX. The result that the achevable rate regon of CSMA polces s asymptotcally such that t can support any rate vector λ satsfyng (3) may seem very surprsng and counter-ntutve at frst. And ndeed, t s mportant to stress that our result does not state that the achevable rate regon of CSMA polces s always of the form as gven by (3), but only under the condtons that (a) becomes small and (b) the network resources are shared by many small flows. Let us brefly comment on these two condtons. The fact that needs to be small n order to obtan a large achevable rate regon s rather ntutve; clearly f s large (let s say close to ) then the above result wll not be true. The fact that we need the assumpton of many small flows n order to obtan our result s llustrated by the followng example. C. Achevable Rate Regon of CSMA Polces We show n Appendx C-F that a CSMA polcy p has a welldefned lnk servce rate vector to whch we refer as µ(p) = {µ (,j) (p)} (,j) L,.e. CSMA polces are contaned n the set P. Note that for a gven, the lnk servce rate under a CSMA polcy depends only on the transmsson attempt probablty vector p, and not on the arrval rates λ. The achevable rate regon of CSMA polces s then gven as follows. Defnton 7 (Achevable Rate Regon of CSMA Polces). For a gven network (N, L) and a gven sensng perod, the achevable rate regon of CSMA polces s gven by the set of rate vectors λ = {λ r } r R for whch there exsts a CSMA polcy p that stablzes the network for λ,.e. we have that λ (,j) < µ (,j) (p), (, j) L. V. OVERVIEW OF THE MAIN RESULTS AND ANALYSIS Ths secton provdes an overvew of the man results of ths work along wth an outlne of the analyss. In Secton IX, we derve an approxmaton Γ() for the achevable rate regon of CSMA polces for a gven network and a gven sensng perod, and show that n the lmt as the sensng perod approaches 0 we have that lm Γ() = {λ 0 : Λ (λ) <, for all N }. 0 Snce t s mpossble for any polcy to stablze the network f for a node we have that Λ (λ), ths result suggest that n the lmtng regme as becomes small, the capacty regon for schedulng polces n wreless networks wth prmary nterference constrants ncludes all rate vectors λ such that Λ (λ) <, N. (3) We verfy ths ntuton for large networks wth many small flows,.e. we show that asymptotc achevable rate regon of CSMA polces under the lmtng regme large networks wth many small flows and a small sensng s of the above form. Fg. 2. The pentagon network wth flows r,, r 5 on each lnk, and the fve possble smultaneous transmssons that can occur under the prmary nterference model. The rate λ r = ( ǫ)/2, =,,5, for any ǫ (0, 0.] s not achevable by any polcy for ths scenaro. Example. For the pentagon network of Fgure 2, let ǫ (0, 0.] and λ r = ( ǫ)/2 for each r =,, 5. Then, the load on each node s gven by Λ = ( ǫ) for each N. Although the resultng traffc vector λ satsfes Eq. (3), no schedulng polcy can stablze the network for λ. Ths can be seen by notng that at most two lnks out of fve can transmt successfully at a gven tme, as shown n the fgure. Hence, even an optmal centralzed controller cannot acheve a maxmum symmetrc node actvty of more than 2/5, and clearly, our result cannot hold for ths network. The reason that n the pentagon network a node cannot acheve a throughput of more than 2/5 s that under each maxmal schedule gven n Fgure 2, f one of the neghborng nodes of a gven node s busy transmttng, then node has to wat for a duraton of tme unt to get a chance to make a transmsson attempt. However, f we have a network where each node has many neghbors wth whch t exchanges data packets (many flows), then nodes wll typcally have to wat for much less than tme unt before they get the chance to start a packet transmssons. Intutvely, the larger the number of neghbors of a node, the shorter a node has to wat untl t gets a chance to start a packet transmsson. In addton to havng many flows, we need the assumpton that each flow s small n order to avod the stuaton where the dynamcs at each node s bascally determned by a small number of large flows, essentally leadng to a smlar behavor as n the case where each node has only a small numbers 5

6 of neghbors. Note however that these assumptons aren t suffcent n order to obtan our result; we also need to show that there exsts a CSMA polcy under whch nodes (a) do not wat too long before makng a transmsson attempt (and hence waste bandwdth), (b) are not too aggressve such that a large fracton of packet transmssons result n collsons, and (c) share the avalable network resources such that the resultng lnk servce rates ndeed support a gven traffc vector λ that satsfes Eq. (3). Below, we provde a bref descrpton of the dfferent steps taken n our analyss. Our frst step s to derve a tractable formulaton to characterze the lnk servce rates for a gven CSMA polcy. Specfcally, nspred by the reduced load approxmatons utlzed n the loss network analyss [20], n Secton VI-B we propose a novel fxed pont formulaton to model the performance of a CSMA polcy p. Smlar to the reduced load approxmaton n loss networks, the fxed pont equaton s based on an ndependence assumpton. We show that the fxed pont s well-defned,.e., there exsts a unque fxed pont. Our second step s to use the CSMA fxed pont to characterze the approxmate achevable rate regon n Secton VII, and show that ths characterzaton suggests that CSMA polces are throughput-optmal n the lmt as the sensng tme becomes small. In our thrd step, we show that the formulated CSMA fxed pont s asymptotcally accurate n the sense that t accurately characterzes the lnk servce rates of a CSMA polcy as becomes small for large networks wth many small flows. A techncal ssue that requres care n the proof s the scalng wth whch the sensng delay decays as a functon of the network sze N. We dentfy a proper scalng, as gven n Assumpton 2 of Secton VIII, that yelds the asymptotc accuracy result. Moreover, n the dervaton of the achevable rate regon usng the CSMA fxed pont, we obtan an algorthm that allows the constructve computaton of the CSMA polcy parameters that stablze the network for any gven rate vector λ wthn the achevable rate regon. Fnally, n Secton IX, we derve the asymptotc achevable rate regon of CSMA polces for the lmtng regme of large networks wth many small flows and a small sensng perod. Ths result shows that n ths asymptote the CSMA achevable rate regon can be descrbed by a condton n the form of (3). VI. APPROXIMATE CSMA FIXED POINT FORMULATION In the frst part of our analyss, we ntroduce a fxed pont approxmaton, called the CSMA fxed pont, to characterze the lnk servce rates under a CSMA polcy p. The fxed pont approxmaton extends the well-known nfnte node approxmaton for sngle-hop networks (see for example [3]) to multhop networks whch we brefly revew below. In the followng we wll use τ to denote the servces rates obtaned under our analytcal formulatons that we use to approxmate the actual servce rates µ(p) under a CSMA polcy p as defned n Secton IV-C. A. Infnte Node Approxmaton for Sngle-Hop Networks Consder a sngle-hop network where N nodes share a sngle communcaton channel,.e. where nodes are all wthn transmsson range of each other. In ths case, a CSMA polcy s gven by the vector p = (p,, p N ) [0, ] N where p n s the probablty that node n starts a packet transmsson after an dle perod of length [3]. Suppose that the sngle-hop network s synchronzed,.e. the sensng delay s the same for all node pars n, n N and we have that (j) = k (l),, j, k, l N. Then the network throughput,.e. the fracton of tme the channel s used to transmt packets that do not experence a collson, can then be approxmated by (see for example [3]) τ(g(p)) = G(p)e G(p) + e G(p) (4) where G(p) = N n= p n. Note that G(p) captures the expected number of transmssons attempt after an dle perod under a CSMA polcy p. Ths well-known approxmaton s based on the assumpton that a large (nfnte) number of nodes share the communcaton channel. It s asymptotcally accurate as the number of nodes N becomes large and each node makes a transmsson attempt wth a probablty p n, n N that approaches zero whle the offered load G = N n= p n stays constant (see for example [3]). The followng results are well-known. For > 0, one can show that τ(g) <, G 0, (5) and for G + () = 2, > 0, we have that lm τ(g + ()) =. (6) 0 Usng (4), the servce rate µ n (p) of node n under a gven CSMA polcy p can be approxmated by p n e G(p) τ n (p) = + e G(p), n =,..., N. (7) In the above expresson, p n s the probablty that node n tres to capture the channel after an dle perod and e G(p) characterzes the probablty that ths attempt s successful,.e. the attempt does not collde wth an attempt by any other node. Smlarly, the fracton of tme that the channel s dle can be approxmated by ρ(p) = ρ(g(p)) = where we have that lm 0 ρ(g + ()) = 0. + e G(p), (8) B. CSMA Fxed Pont Approxmaton for Multhop Networks We extend the above approxmaton for sngle-hop networks to multhop networks that operate n an asynchronous manner as descrbed n Secton IV-B as follows. For a gven a sensng perod, we approxmate the fracton of tme ρ (p) that node s dle under the CSMA polcy p by the followng fxed pont equaton, ρ (p) =, =,, N, (9) ( + e G(p) ) 6

7 where G (p) = [ p(,j) + p (j,) ] ρj (p), =,, N. (0) Note that the fxed pont equaton can be gven both n terms of the fracton of dle tmes ρ by substtutng (0) n (9) or n terms of the transmsson attempt rates G by substtutng (9) n (0). Gven ths equvalence, we refer to ether one as the CSMA fxed pont equaton. We further let ρ(p) = (ρ (p),, ρ N (p)) and G(p) = (G (p),, G N (p)) denote partcular CSMA fxed ponts, and R(p) and G (p) denote the set of all fxed ponts of (9) and (0), respectvely. The ntuton behnd the CSMA fxed pont equaton (9) and (0) s as follows: suppose that the fracton of tme that node s dle under the CSMA polcy p s equal to ρ (p), and suppose that the tmes when node s dle are ndependent of the processes at all other nodes. If node has been dle for tme unts,.e. node has not receved or transmtted a packet for tme unts, then node can start a transmsson attempt on lnk (, j), j N, only f node j also has been dle for an dle perod of tme unts. Under the above ndependence assumpton, ths wll be (roughly) the case wth probablty ρ j (p), and the probablty that node start a packet transmsson on the lnk (, j), j N, gven that t has been dle for tme unts s (roughly) equal to p (,j) ρ j (p). Smlarly, the probablty that node j N starts a packet transmsson on the lnk (j, ) after node has been dle for tme unts s (roughly) equal to p (j,) ρ j (p). Hence, the expected number of transmsson attempts that node makes or receves, after t has been dle for tme unts s (roughly) gven by (0). Usng (8) of Secton VI-A, the fracton of tme that node s dle under p can then be approxmated by (9). There are two mportant questons regardng the CSMA fxed pont approxmaton. Frst, one needs to show that the CSMA fxed pont s well-defned,.e. that there always exsts a unque CSMA fxed pont. In the above notaton ths corresponds to provng that the sets R(p) and G (p) have a sngle element for any feasble p. To that end, the followng result, proven n Appendx B, establshes the unqueness of a fxed pont soluton for all such p. Theorem. For every CSMA polcy p (0, ) L, each of the set of fxed pont solutons R(p) and G (p) has a sngle element, denoted henceforth by ρ(p) and G(p), respectvely. Second, we need to check the accuracy of the above CSMA fxed pont approxmaton. Ths s postponed to Secton VIII, where we show that the CSMA fxed pont approxmaton s asymptotcally accurate for large networks wth a small sensng perod and approprately decreasng lnk attempt probabltes. In what follows, we focus on the CSMA achevable rate regon characterzaton based on the above fxed pont approxmaton. VII. APPROXIMATE CSMA ACHIEVABLE RATE REGION In ths secton, we use the CSMA fxed pont approxmaton (9) and (0) to characterze an approxmate achevable rate regon of CSMA polces. In Secton IX, we wll show that ths characterzaton s asymptotcally accurate for large networks wth many small flows and a small sensng tme,. We start by notng that, for a gven sensng perod, we can use the CSMA fxed pont G(p) for a polcy p to approxmate the actual lnk servce rate µ (,j) (p) under the CSMA polcy p by τ (,j) (p) that satsfes where τ (,j) (p) = p (,j)ρ j (p)e (G R (p)+gj(p)) () + e G(p) G R (p) p (j,) ρ j (p) represents the rate at whch node receves transmsson attempts by ts neghbors, and hence ts dfference from G (p). Note that the above equaton s smlar to (7) where p (,j) ρ j (p) captures the probablty that node makes an attempt to capture lnk (, j) f t has been dle for tme unts, and exp [ (G R (p) + G j(p)) ] s the probablty that ths attempt s successful,.e. the attempt does not overlap wth an attempt by another lnk that shares a node wth (, j). Note that τ (,j) (p) p (,j) e (G(p)+Gj(p)) ( ) ( + e G (p) + e Gj(p)) (2) as G (p) G R (p). The next result provdes an approxmate achevable rate regon of the CSMA polcy based on the CSMA fxed pont approxmaton and the approxmate servce rates (τ (,j) (p)) (,j) gven n (). Theorem 2. Gven a network (N, L) wth sensng perod > 0, let Γ() be gven by { } Γ() λ 0 Λ (λ) < τ(g + ())e (G+ ()), N, (3) where G + () 2, τ(g + ()) s as defned n (4), and Λ (λ) [ ] λ(,j) + λ (j,), for each N. Then, for every λ Γ(), we can explctly fnd (cf. Equaton (4)) a CSMA polcy parameter p for whch the correspondng CSMA fxed pont approxmaton yelds λ (,j) < τ (,j) (p), (, j) L, where τ (,j) (p) s as defned n (). In other words, by a proper selecton of p, the approxmate servce rates can be made to exceed the traffc load on each lnk as long as λ Γ(). Proof: For brevty, we wll denote Λ (λ) as Λ, whch, by defnton, satsfes Λ < τ(g + ())e G+ () for all N. For each node =,..., N, choose G [0, G + ()) such that and let e (G G+ ()) τ(g )e G+ () = Λ ρ = + e G. Such a G exsts snce the functon f(g ) = e (G G+ ()) τ(g )e G+ () 7

8 s contnuous n G wth f(0) = 0 and f(g + ()) = τ(g + ())e G+ () > Λ. Usng ρ for =,..., N as defned above, consder the CSMA polcy p gven by p (,j) = λ (,j) e 2G + (), (, j) L. (4) ρ ρ j By applyng the above defntons, at every node =,..., N we have that λ (,j) + λ (j,) [p (,j) + p (j,) ]ρ j = e 2G+ () ρ j ρ ρ j = e2g+ () ρ [λ (,j) + λ (j,) ] = e2g ρ + () Λ () = e2g+ e (G G+()) τ(g )e G+ () = eg τ(g ) ρ ρ = + G e G e G e G + e = G. G Ths mples that the above choces of G = (G,, G N ) and ρ = (ρ,, ρ N ) defne the CSMA fxed pont of the statc CSMA polcy gven by (4),.e. we have that ρ(p) = ρ and G(p) = G. Usng (2), the servce rate τ (,j) (p) on lnk (, j) under p s then gven by τ (,j) (p) p (,j)ρ j (p)e (G(p)+Gj(p)) + e G(p) ρ j e (G+Gj) = p (,j) + e = λ (,j) e 2G + () ρ je (G+Gj) G ρ ρ j + e G () (G = λ +G j) (,j) ρ ( + e G ) e2g+ = λ (,j) e 2G+ () (G +G j) > λ (,j), where we used n the last nequalty the fact that by constructon we have G, G j < G + (). The proposton then follows. The proof of Theorem 2 s constructve n the sense that gven a rate vector λ Γ(), we construct (cf. Equaton (4)) a CSMA polcy p such that λ (,j) < τ (,j) (p), (, j) L. We wll use ths constructon for our numercal results n Secton IX-C. Theorem 2 also leads to the followng nterestng corollary, whch ndcates the capacty achevng nature of CSMA polces n the small sensng delay regme. Corollary. In the small sensng delay regme,.e. as 0, the approxmate achevable rate regon Γ() converges to the followng smple set lm Γ() = {λ 0 Λ (λ) <, =,, N}. 0 Proof: The proof follows mmedately from the defnton of Γ() once we recall from Secton VI-A that lm 0 G + () = 0, and lm 0 τ(g + ()) =. Snce any rate vector λ for whch there exsts a node wth Λ cannot be stablzed by any polcy, Corollary establshes that for networks wth a small sensng tme, the approxmate achevable rate regon of statc CSMA polces get arbtrarly close to the above lmtng rate regon descrbed purely n terms of per node traffc load. As we noted n Example, such a rate regon s not achevable for all networks. In Secton IX, we show that the capacty regon does take on the above smple form for large networks wth many small flows and a small sensng perod. To that end, n the next secton, we frst establsh condtons on the network and CSMA parameters for whch CSMA fxed pont approxmaton becomes accurate. VIII. ASYMPTOTIC CSMA FIXED POINT ACCURACY In ths secton, we study the accuracy of the CSMA fxed pont approxmaton proposed n Secton VI (cf. Equatons (9) and (0)) n capturng the servce rate and dle fracton performance of the actual CSMA polcy (cf. Defnton 6). Our analyss establshes a large network and small sensng delay regme n whch the approxmaton becomes arbtrarly accurate. More precsely, we consder a sequence of networks for whch the number of nodes N ncreases to nfnty, and let L and N respectvely denote the set of all lnks and the set of neghbors of node for the network wth N nodes. Smlarly, as N ncreases, we consder a correspondng sequence of CSMA polces {p } N wth a sequence of sensng perods { } N, where (p, ) defnes the CSMA polcy for the network wth N nodes as descrbed n Defnton 6. We make the followng assumptons on the parameters of the CSMA polcy. Assumpton 2. For the sequences {p } N and { } N ntroduced above: (a) lm N = 0. (b) Lettng p max max p (,j) L (,j), we have lm p max = 0. (c) There exsts a postve constant χ and a fnte nteger N 0, such that for all N N 0 we have j N [p (,j) + p (j,) ] χ, =,, N. (5) These techncal assumptons have the followng nterpretaton: Assumpton 2(a) characterzes a small sensng delay regme by specfyng how fast decreases to zero as the network sze N ncreases; Assumpton 2(b) mples that the attempt probablty of each lnk becomes small as N becomes large, assurng that no sngle lnk domnates the servce provded by ts transmttng node; and Assumpton 2(c) states that the total rate (gven on the left of (5) by the expected number of transmsson attempts per sensng perod ) wth whch lnks ncdent to a gven node start a packet transmsson, s upper-bounded by a postve constant. Below we provde two examples of networks that satsfy Assumpton 2. 8

9 Example 2. Consder an N N swtch (depcted n Fgure 3) wth traffc flowng from the set, N S = {,, N}, of nput (or sender) ports to the set, N R = {N +,, 2N}, of output (or recever) ports. For ths setup where the degree of each node s N, we can select the CSMA polcy parameters as follows to satsfy the Assumpton 2: = /(Nlog), and p (,j) = χ /(2N), (, j) N S N R.(6) Example 3. Consder a network consstng of N nodes and assume that each node communcates wth log neghborng nodes. Ths setup resembles randomly generated dense network wthn a unt area, where the nodes wthn the communcaton range of each other are connected. Such a model s wdely studed n earler works (e.g. [5]) that establsh that f the communcaton radus s optmally selected for connectvty, the degree of each node scales as Θ(log) for a network wth N nodes. The followng parameters as a functon of the network sze N wll satsfy Assumpton 2: = /(Nlog), and p (,j) χ /(log) (, j) L. (7) Next, we analyze the accuracy of the CSMA fxed pont approxmaton for the lmtng regme gven by Assumpton 2,.e. we let ρ(p ) = (ρ (p ),, ρ N (p )) be the CSMA fxed pont for the network of sze N, and let σ (p ) be the actual fracton of tme that node s dle under the CSMA(p ) operaton. Then, we use the followng metrc to measure the dscrepancy of the two: δ ρ max =,,N ρ (p ) σ (p ), whch quantfes the maxmum approxmaton error of the CSMA fxed pont across the network. Smlarly, we let τ (,j) (p ) be the approxmate CSMA servce rate for lnk (, j) defned n (), and let µ (,j) (p ) be the actual CSMA servce rate for lnk (, j). Then, we defne the followng metrc to measure the dscrepancy between the two: δ τ max (,j) L τ (,j)(p ) µ (,j) (p ), whch quantfes the maxmum relatve approxmaton error of the lnk servce rates under the CSMA fxed pont. Note that under Assumpton 2 the lnk servce rate µ (,j) (p ) wll approach zero as N ncreases and the error term τ (,j) (p ) µ (,j) (p ) wll trvally vansh; ths s the reason why we consder the relatve error when studyng the accuracy of the CSMA fxed pont equaton for the lnk servce rates. The followng result, proven n Appendx C, establshes that n the lmt as N approaches nfnty, the fxed pont approxmaton for CSMA polces wth the above scalng becomes asymptotcally accurate. Theorem 3. Under the CSMA polcy scalng of Assumpton 2, we have that lm δ ρ = 0, and lm δ τ = 0,.e., the fxed pont approxmaton becomes asymptotcally accurate both n terms of dle fracton and servce rate approxmatons. A. Numercal Results In ths secton, we llustrate Theorem 3 usng numercal results obtaned for the N N swtch network dscussed n Example 3 and depcted n Fgure 3. The swtch topology s selected for numercal comparson snce such a topology s the smplest non-trval one that also leads to an analytcally tractable fxed pont soluton under symmetrc condtons. Yet, we emphasze that Theorem 3 apples to any large network as long as CSMA polcy satsfes Assumpton 2. Besdes confrmng the asymptotc accuracy of the approxmatons, our results also ndcate that the accuracy s observed even for relatvely small networks N N+ N+2... Fg. 3. Network topology for our numercal results conssts of a set of N sender nodes N S = {,..., N}, and a set of N recever nodes N R = {N +,...,2N}. The set of lnks L conssts of all drected lnks (, j) from a sender N S to a recever j N R. For ths network, we consder a sequence of CSMA polces p = (p (,j) ) (,j) L and the correspondng sequence of sensng perods as n (6) by settng χ = 0. Recall that a CSMA polcy wth parameters (p, ) determnes the lnk probabltes p (,j) wth whch sender N S starts a transmsson of a packet to recever j N R after lnk (, j) has been sensed to be dle for sensng perod of tme unts. Gven a sensng perod, the CSMA fxed pont for a polcy p s then gven by ρ (p ) = where G (p ) = G j (p ) = ( + e G(p ), =,, 2N, ) j N R N S p (,j) ρ j(p ), p (,j) ρ (p ), 2N N S, and j N R. Then, due the symmetry of the network topology as well as of the constructed CSMA polces p, the CSMA fxed pont ρ(p ) s unform and satsfes ρ (p ) = ρ j (p ),, j N N S N R. In Fgures 4 and 5, we evaluate the performance of the above sequence of CSMA polces for varyng sze N of the sender set N S. In partcular, Fgure 4 depcts the measured mean 9

10 Fracton of dle tme Throughput Performance Fxed Pont Analyss, ρ Smulated CSMA Performance, σ Number of sender nodes, N Fxed Pont Analyss, τ Smulated CSMA Performance, µ Number of sender nodes, N Fg. 4. Comparson of the actual fracton of dle tme under the CSMA polcy and the predcted values based on the fxed pont formulaton. Maxmum error n dle fracton, δ ρ Maxmum Error n rates, δ τ Number of sender nodes, N Number of sender nodes, N Fg. 5. Error terms of Theorem 3 for dfferent values of N. fracton of tmes that nodes are dle and mean node throughput under the actual CSMA polcy operaton, compared wth the performance predcted by the CSMA fxed pont. Fgure 5 shows the error terms of Theorem 3 for the approxmaton error n the fracton of tme that nodes are dle, and the lnk servce rates. Note that the above numercal results not only confrm the asymptotc clams of Theorem 3 but also ndcate that the CSMA fxed pont approxmaton s remarkably accurate even for smaller values of N. Ths suggests that the CSMA fxed pont approxmaton may be used to characterze the performance for moderate-sze networks where each nodes has a relatvely small number of neghbors. An extensve nvestgaton of ths mplcaton n more general network topologes s of practcal nterest and s left to future research. IX. ASYMPTOTIC CAPACITY REGION C In ths secton, we derve the asymptotc achevable rate regon for CSMA for a lmtng regme of large networks wth many small flows and a small sensng perod that s formally defned n Secton IX-A. A. Many Small Flows Asymptotc In Secton VIII, we ntroduced a sequence of networks for whch the number of nodes N ncreases to nfnty, and let L be the set of all lnks n the network wth N nodes, and be the set of neghbors of node n the network wth N nodes. In ths secton, we ntroduce a smlar scalng for the traffc arrval rate vectors to assure that the load on any lnk do not domnate the load n ts neghborhood. To that end, N we use the notaton λ = {λ r vector for the network wth N nodes. Furthermore, λ (,j) = Λ = r R :(,j) r j N } r R for the arrval rate λ r, (, j) L, and [ λ (,j) + λ (j,)], N, respectvely, denotes the mean packet arrval rate on lnk (, j) and the mean packet arrval rate at node. Defnton 8 (Many Small Flows Asymptotc). Gven a sequence of networks {N, L } N, we defne A as the set of all rate vector sequences {λ } N such that ( ) lm sup max λ (,j) L (,j) = 0. We say that {λ } N satsfes the many small flows asymptotc f t belongs to A. The above defnton characterzes the lmtng regme where the mean arrval of each flow becomes small as the network sze scales,.e. the network traffc conssts of many small flows. It s mportant to note that, whle the load on each lnk vanshes under the many small flows asymptote, the total load on a node may be non-vanshng f the number of neghbors also ncreases. We shall see that ths key characterstc of the many small flows regme wll allow CSMA polces to acheve maxmal per node loads under large and wellconnected network topologes. Before we establsh ths man result, we defne the asymptotc achevable rate regon of CSMA polces under the many small flows asymptotc as follows. Defnton 9 (Asymptotc CSMA Achevable Rate Regon). The asymptotc achevable rate regon of statc CSMA polces under the many flow lmt s the set of flow rate sequences {λ } N A for whch there exsts a sequence of CSMA schedulng polces (p, ) N such that lm nf µ (,j) (p ) mn >. (,j) L λ (,j) Thus, every flow rate sequence {λ } N n the asymptotc CSMA rate regon can be stablzed by the sequence of CSMA polces (p, ) N for large enough N. 0

11 Note that a sequence {λ } N A for whch there exsts a node wth lm Λ cannot be stablzed by any polcy as servce rate at each node s bounded by. Hence, the achevable regon under the many flow lmt s contaned n the set C { {λ } N A lmsup ( max =,...,N Λ ) } <. (8) We refer to C as the capacty regon under the many small flows asymptotc. B. Asymptotc CSMA Achevable Rate Regon In ths subsecton, we characterze the asymptotc achevable rate regon of CSMA polces under the many small flows asymptotc for networks wth a small sensng perod. To do ths, we agan consder a sequence of sensng perods { } N that satsfes Assumpton 2(a). The next theorem, proven n Appendx D, shows that n ths case the achevable rate regon of CSMA polces converges to the capacty regon under the many small flows asymptotc C. Theorem 4. Gven a sequence of networks {N, L } N, a sequence of sensng perods { } N satsfyng Assumpton 2(a), and a sequence of flow rates {λ } N C, we can explctly fnd a sequence of CSMA polcy attempt rates {p } N that asymptotcally stablzes the network,.e., that satsfes lm nf µ (,j) (p ) mn >. (,j) L λ (,j) It s nterestng to note that the proof of Theorem 4 n Appendx D s constructve n that sense that t provdes explct expressons for the lnk transmsson attempt probabltes that stablze the network for a gven rate vector sequence {λ } N n C. C. Numercal Results In ths secton, we verfy the statement of Theorem 4 usng the same swtch topology we used for the numercal results n Secton VIII-A (see also Fgure 3). As the network sze ncreases, we consder a sequence of dle perods { } N = 0./(Nlog) and traffc vectors {λ } N wth λ (,j) = 0.95 ( ) N e G+ τ(g + ( )), N S, j N R. Notce that {λ } N satsfes the many small flows asymptotc (cf. Defnton 8) and that the per node load satsfes Λ = 0.95 e G+ ( ) τ(g + ( )), N, whch s non-vanshng. Also note that the selected rate vector λ s wthn that approxmate CSMA achevable rate regon Γ( ) (cf. Equaton 3) for each N. In the proof for Theorem 2 we derve an explct constructon for obtanng a polcy p that supports a gven traffc vector λ Γ( ). Followng ths constructon for the above choce of flow rates, we choose G [0, G + ( )) such that e (G G + ( )) τ(g )e G+ ( ) = 0.95 e G+ ( ) τ(g + ( )), whch s shown to exst n the proof. Then, lettng ρ + e G, we construct a sequence of CSMA polcy parameters p satsfyng p (,j) λ (,j) (ρ ) 2 e 2G + ( ), (, j) L. Theorem 4 then states that for such constructed sequence of CSMA polces we have, for a large enough N, that µ (,j) > λ (,j), for all (, j) L. Also, notng that lm Λ = 0.95 for the above choce of flow rates, we have lm j N R lm N S µ (,j) > 0.95, N S, and µ (,j) > 0.95, j N R. To confrm these asymptotc clams and to nvestgate ther correctness for moderate values of N we smulate the above network to measure the true lnk servce rates for ncreasng N. Fgure 6 shows the average node throughput that we obtaned. Note that the average node throughput ndeed s above the value Λ for whch we desgned the CSMA polcy p. Furthermore, as N ncreases the average node throughput becomes larger then 0.95 as predcted by our theoretcal result. Moreover, these results ndcate that the results are qute accurate even for small network szes and that CSMA polces can be close to capacty achevng even f the number of neghbors of each node s relatvely small. Fgure 6 shows the dstrbuton of the rato of lnk servce rates to lnk loads. We know from Theorem 4 that ths rato wll eventually exceed for all lnks as N tends to nfnty. We observe n Fgure 6 that already at a moderate value of N = 20, more than 95% of the lnks exceed and the rest of the lnks acheve rates close to. X. CONCLUSIONS In ths work, we provded an extensve analyss of asynchronous CSMA polces operatng n mult-hop wreless networks subject to collsons wth prmary nterference constrants. To that end, we frst ntroduced a CSMA fxedpont formulaton to:(a) approxmate the performance of such CSMA polces; (b) approxmate ther achevable rate regon; and (c) provde a constructve method for determnng the transmsson attempt probabltes of the CSMA polcy that can support a gven rate vector n the achevable rate regon. We then showed that the CSMA fxed pont formulaton becomes asymptotcally accurate for an approprate lmtng

12 Throughput per port Smulated CSMA Performance, µ Predcted capacty per port, C( ) := e G+ ( ) τ(g + ( )) Traffc load, Λ = 0.95 C( ) Number of nput ports, N Number of lnks Rato of µ / λ(,j) for N=20 (,j) Fg. 6. Performance of the CSMA polcy for the network n Fgure 3 wth symmetrc load. The graph on the left shows that the polcy acheves rates close to the amed value of 0.95 per sender node even for moderate values of N. The graph on the rght shows the dstrbuton of the rato of acheved rates to load on each lnk amongst 400 exstng lnks n the network n Fgure 3 wth N = 20. regme where the network sze ncreases and the sensng delay decreases. Usng ths result we establshed that for large networks wth a balanced traffc load, the CSMA achevable rate regon takes an extremely smple form that smply lmts the ndvdual load on each node to, whch s the maxmum supportable load by any other schedulng polcy. 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Srkant. Regulated maxmal matchng: A dstrbuted schedulng algorthm for mult-hop wreless networks wth nodeexclusve spectrum sharng. In Proceedngs of IEEE Conference on Decson and Control., [4] X. Wu and R. Srkant. Bounds on the capacty regon of mult-hop wreless networks under dstrbutedgreedy schedulng. In Proceedngs of IEEE Infocom, [42] X. Wu, R. Srkant, and J. Perkns. Queue-length stablty of maxmal greedy schedules n wreless networks. IEEE Transactons on Moble Computng, pages , June APPENDIX A EXAMPLE CHANNEL SENSING MECHANISMS In ths secton, we dscuss two specfc channel sensng mechansms that operate under heterogeneous sensng delay characterstcs. We note that our model s flexble enough to allow other mechansm desgns. Mechansm : Suppose that each node N s assgned a channel c over whch t receves data packets, and suppose that the sensng radus and transmsson radus of the nodes are dfferent. The channel c could ether be a frequency range, or a code, f a FDMA-based, or a CDMA-based, approach respectvely s used to obtan a network wth prmary nterference constrants (see also our dscusson n Secton III). Nodes that are wthn the transmsson radus of a node can successfully receve ts packet transmsson f there are no collsons by another transmsson wthn the transmsson radus of the recever. Nodes that are wthn the sensng radus of the transmttng node can only detect the presence or absence of actvty together wth ts destnaton. The actvty wthn the sensng radus does not cause collsons, but t sgnals the presence of actvty. In ths settng, a node j N can sense whether node s currently sendng a packet by scannng the channels c k used by node for transmsson on ts outgong lnks (, k) L. Furthermore, f the sensng radus s at least twce the transmsson radus, then a node j N can sense whether node s currently recevng a packet by scannng channel c. Note that the tme (measured n seconds) that t takes a node to detect whether a neghborng node s busy, wll ncrease as the number of neghbors of a node ncreases; however, the sensng delay l (l ) measured relatve to the tme t takes to transmt a packet can stll kept low by ncreasng the sze of a packet, and hence ncrease the tme L p t takes to transmt a packet. Mechansm 2: Agan, suppose that each node N s assgned a communcaton channel c over whch t receves data Fg. 7. Nodes m,, j, and k are connected as shown on the left. Node starts a packet transmsson to node j at t 0, whch s overheard startng at t by node m. Thus, the sensng delay m(, j) s equal to (t t 0 ). Node j starts recepton of the packet at t 2 (hence ts sensng delay satsfes j (, j) = (t 2 t 0 )) and generates a sgnal over ts control channel c j to ndcate the actvty of lnk (, j). Node k senses the control sgnal of node j at tme t 3 (hence ts sensng delay s k (, j) = (t 3 t 0 ). The transmsson of the packet ends at tme t 4 whch equals (t 0 +) snce the packet transmsson duraton s normalzed to one. Nodes m, j, and k sense the end of the actvty at t 5, t 6, and t 7, respectvely. packets, and that n addton t s assgned a control channel c, where the bandwdth of the communcaton channel c s much larger than the one of the control channel c. Then, f node s currently recevng a packet transmsson on ts communcaton channel c, then t can send out a busy sgnal on the control channel c. In ths settng, a node j N can sense whether node s currently sendng a packet by scannng the channels c k used by node for transmsson on ts outgong lnks (, k) L. Furthermore, a node j N can sense whether node s currently recevng a packet by scannng the control channel c. Agan, the tme (measured n seconds) that t takes a node to detect whether a neghborng node s busy, wll ncrease as the number of neghbors of a node ncreases; but the sensng delay l (l ) measured relatve to the tme t takes to transmt a packet can stll kept low by ncreasng the sze of a packet. Fgure 7 gves a tmng-dagram for ths case. APPENDIX B EXISTENCE AND UNIQUENESS OF CSMA FIXED POINTS In ths secton, we prove Theorem whch states that for each choce of p (0, ) L there exsts a unque CSMA fxed pont. We frst establsh the exstence of a CSMA fxed pont. Lemma. For every CSMA polcy p [0, ] L, there exsts a CSMA fxed pont ρ(p) and G(p),.e., the sets R(p) and G (p) are non-empty. Proof: The proof uses the contnuty propertes of the fxed pont equaton gven (9), and s a straghtforward applcaton of the Brouwer s fxed pont theorem. 3