Achievable Rate Region of CSMA Schedulers in Wireless Networks with Primary Interference Constraints

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1 Achevable Rate Regon of CSMA Schedulers n Wreless Networks wth Prmary Interference Constrants Peter Marbach, Atlla Erylmaz, and Asuman Ozdaglar Abstract We consder Carrer Sense Multple Access (CSMA) schedulers for wreless networks. For networks where all nodes are wthn transmsson range of each other, t has been shown that such schedulers acheve the network capacty n the lmtng regon of large networks wth a small sensng delay. However the desgn and analyss of CSMA schedulers for general networks has been an open problem due to the complexty of the nteracton among coupled nterference constrants. For networks wth prmary nterference constrants, we ntroduce a tractable analyss of such CSMA schedulers based on a fxed pont approxmaton. We then use the approxmaton to characterze the achevable rate regon of statc CSMA schedulers. We show that the approxmaton s asymptotcally accurate for the lmtng regme of large networks wth a small sensng delay, and that n ths case the achevable rate regon of CSMA converges to the capacty regon. I. INTRODUCTION Recently, there has been a growng nterest n the development of dstrbuted transmsson polces for nterferencelmted wreless networks. A key objectve of these dstrbuted polces s to acheve any throughput wthn the capacty regon 1 of the network. In ths paper we focus on Carrer Sense Multple Access (CSMA) schedulers where nodes sense whether the channel s dle before makng an attempt to transmt a packet. For sngle-hop networks where all nodes are wthn transmsson range of each other, t s well-known that CSMA schedulers acheve network capacty n the lmtng regme of large networks wth a small sensng tme [2]. However the analyss of CSMA schedulers for general networks has been an open problem due to the complexty of the nteracton among coupled nterference constrants. In ths paper, we provde an analyss of CSMA schedulers for networks wth prmary nterference constrants. The man contrbutons of the paper are as follows. We provde a fxed pont approxmaton, called the CSMA fxed pont, to characterze the servce rates of CSMA schedulers n networks wth prmary nterference constrants, and show that (under some assumptons) the fxed pont approxmaton 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.2@osu.edu. A. Ozdaglar s wth the Department of Electrcal Engneerng and Computer Scence at the Massachussettes Insttute of Technology, Cambrdge, USA. Emal: asuman@mt.edu. The work of the second and thrd authors was supported by the Control- Based Moble Ad-Hoc Networkng (CBMANET) Program under DARPA subcontract no The capacty regon contans the set of arrval rates that are achevable by some feasble polcy. A more rgorous defnton wll be provded later. s asymptotcally accurate for large networks wth a small sensng tme. Usng the fxed pont approxmaton, we characterze the achevable rate regon of the statc CSMA schedulers for networks wth prmary nterference constrants. We show that the achevable rate regon converges to the capacty regon for large networks wth a small a sensng tme. Most recent work n ths area focused on dstrbuted mechansms that can mplement the throughput-optmal (or max-weght) polces (see for example [3], [7], [8], [10], [12]) that has been proposed n the semnal work of Tassulas and Ephremdes [11]. In vew of ther low-complexty nature, random access schemes provde an attractve alternatve class of dstrbuted transmsson strateges; however, the current understandng of such schemes s stll lmted. Random access protocols have been analyzed n [13] by Durvy and Thran usng a packng approach, and n [14] by Bordenave, McDonald, and Proutere, usng a mean-feld analyss approach. Related to [14], n ths paper we analyze CSMA random access schemes usng the CSMA fxed pont approxmaton. The paper s organzed as follows. We descrbe the wreless network and the statc CSMA schedulers that we consder for our analyss n Secton II. In Secton III, we revew well-know results for networks where all nodes share a sngle communcaton channel. Secton IV presents the CSMA fxed pont formulaton that we use to approxmate the performance of statc CSMA schedulers. We show n Secton IV that there always exsts a unque CSMA fxed pont and hence our fxed pont approxmaton s welldefned. In Secton V, we show that the fxed pont approxmaton s asymptotcally accurate for large networks wth a small sensng delay. Secton VI uses the CSMA fxed pont approxmaton to characterze the achevable rate regon of statc CSMA schedulers. Fnally, n Secton VII, we show that the achevable rate regon asymptotcally converges to the capacty regon for large networks wth a small sensng tme. Due to space constrants, we state our results wthout proofs. II. PROBLEM FORMULATION Consder a wreless network consstng of N nodes and L drected lnks. A lnk (, j) s gven by a node par and j, where s the sendng and j s the recevng node. Let L be the set of all lnks n the network. Let S be the set of all nodes j such that there exsts a lnk from node to node j, and let R be the set of all nodes j for whch there exsts a lnk from node j to node. Let N = S R denote the neghbourhood of node. Furthermore let L be the set of

2 lnks orgnatng at node,.e. we have that L = {(k, j) L k = }. In the followng we assume that S = R, = 1,..,N,.e. all nodes have the same transmsson range. For our analyss, we focus on networks under the prmary nterference, or node exclusve nterference, model as gven below (see also [8], [12]). Defnton 1 (Prmary Interference Model): An attempt by node to capture lnk (, j) s successful f t does not overlap wth any attempt by another node to capture a lnk (k,l) whch has a node n common wth lnk (, j). Ths nterference model apples, for example, to wreless systems where multple frequences/codes are avalable (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 [5] for addtonal dscusson). A. Statc CSMA Schedulers We consder a Carrer Sense Multple Access (CSMA) mechansm to schedule the tmes when nodes try to capture a lnk that s gven as follows. Before tryng to capture a lnk, nodes sense whether the lnk s dle. More precsely, we say that a node s dle f t currently does not capture any lnk (, j), j R, nor does any node j S capture the lnk ( j,). We say that the lnk (, j) s dle f both node and node j are dle. Nodes only try to capture lnks that have been dle for β tme unts. We refer to β as the sensng tme (or delay), and defne an dle slot for lnk (, j) as the event that lnk (, j) has been dle for β tme unts. In addton, we defne the end of an dle perod at lnk (, j) as the event that node and j have been jontly dle for exactly k dle slots, k {1,2,...}. Defnton 2 (Statc CSMA Polcy): A statc CSMA polcy s gven by a vector p = (p (, j) ) (, j) L [0,1] L. At a gven tme t 0, let L (t) be the set of lnks n L for whch an dle perod ends at tme t. Then node wll mark lnk (, j) L (t) at tme t wth probablty p (, j), ndependently of all other attempts by any node n the network. If at tme t, node marks a sngle lnk (, j) L (t), then t wll make an attempt at tme t to capture that lnk. If node marks two, or more, lnks n the set L (t) at tme t, then t wll choose an arbtrary lnk among the marked lnks, say (, j), and wll make an attempt at tme t to capture the lnk (, j). If a node tres to capture a lnk, t wll hold (occupy) the lnk for one packet transmsson duraton. In the followng, we assume that the transmsson tme of each packet s equal to 1 tme unt. If a node does not make an attempt to capture a lnk, then t wll wat untl the next tme that an dle perod of a lnk (, j), j N, ends. To smplfy the analyss, we assume n the rest of the paper that the propagaton delay between a node and a node j N, j (.e. the tme t takes a node to detect that node j stopped/started transmttng) s equal to the sensng tme β. Furthermore, we assume that all nodes are ntally dle and start sensng at tme t = 0. The analyss can be extended (but becomes more nvolved) to the case where the propagaton delay between neghbourng nodes s upper-bounded by β and the ntal sensng tmes are not algned. B. Performance Measures We defne the servce rate of a statc CSMA polcy for a lnk (, j) L as follows. Defnton 3 (Servce Rate): The servce rate T (, j) (p) of a statc CSMA polcy p for lnk (, j) L s the fracton of tme node successfully captures lnk (, j) under p. Note that the servce rate T (, j) (p) of the lnk (, j) s equal to the throughput for the lnk (, j) under the statc CSMA polcy p for the case where all lnks are saturated,.e. there s always a packet to be sent for all lnks (, j) L. Usng the servce rates T (, j) (p), (, j) L, we next defne the achevable rate regon of statc CSMA polces. Let R be a gven set of routes for the above wreless network, where route r R s characterzed by the lnks that t traverses. Let λ = {λ r } r R R R + be the mean packet arrval rate vector where λ r s the mean packet arrval rate for route r. Gven a rate vector λ, let λ (, j) = r:(, j) r λ r, (, j) L, (1) be the total mean packet arrval rate over all routes r R that traverse lnk (, j), and let Λ = j N [ λ(, j) + λ ( j,) ], = 1,..,N, (2) be the total mean packet arrval rate over all routes r R that pass through node. We assume that the ncomng packets for each lnk (, j) are stored n a separate nput-queue of nfnte sze. Defnton 4 (Stablty): Consder an arrval rate vector λ = {λ r } r R and a statc CSMA polcy p wth servce rates T (, j) (p), (, j) L. We say that p stablzes the network f λ (, j) < T (, j) (p), (, j) L,.e. the arrval rate s strctly less than the servce rate on all lnks (, j) L. The achevable rate regon of statc CSMA polces s then gven as follows. Defnton 5 (Achevable Rate Regon): The achevable rate regon of statc CSMA polces s the set of arrval rate vectors λ = {λ r } r R for whch there exsts a statc CSMA polcy that stablzes the network. In the followng we characterze the servce rates of a gven statc CSMA polcy, as well as the achevable rate regon of statc CSMA polces. III. SINGLE-HOP NETWORKS Before we present our analyss of statc CSMA polces for general networks, we revew n ths secton well-known results on CSMA schedulers for networks where N nodes share a sngle communcaton channel [2]. In ths case, an attempt by a node to capture the channel after an dle perod s only successful f t does not overlap wth an attempt by

3 any other node. A statc CSMA polcy s then gven by the vector p = (p 1,..., p N ) [0,1] N where p n s the probablty that node n makes an attempt to capture the channel. Under a gven statc CSMA polcy p, the servce rate T n (p) of node n s the fracton tme node n successfully captures the channel under p. The network servce rate T(p) s gven by T(p) = N n=1 T n (p). For a gven statc CSMA schedulng polcy p = (p 1,..., p N ), let G(p) = N n=1 p n denote the rate (also called the offered load) wth whch nodes make attempts to capture the channel at the end of an dle perod. For a gven statc CSMA polcy p, the followng approxmaton for the network servce rate s well-known (see for example [2]), τ(g(p)) = G(p)e G(p) β + 1 e G(p). (3) For β > 0, one can show that (see for example [2]) τ(g) < 1, G 0, (4) and for G + (β) = 2β, β > 0, we have that lmτ(g + (β)) = 1. (5) β 0 Usng Eq. (3), the servce rate τ n (p) of node n under a gven statc CSMA polcy p can be approxmated by τ n (p) = p n e G(p) 1+β e G(p), n = 1,...,N. (6) 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)) = β β + 1 e G(p), (7) where we have that lm β 0 ρ(g + (β)) = 0. Eq. (3)-(7) are obtaned by approxmatng the number of nodes that make an attempt to capture the channel after an dle perod by a Posson dstrbuton wth mean G(p). Ths approxmaton s asymptotcally accurate for large networks where each node makes an the attempt to capture the channel wth a small probablty. More precsely, consder a sequence of networks wth N nodes and let {p (N) } N 1, p (N) [0,1] N, be a sequence of CSMA polces such that N p (N) n = G, N 1 n=1 for some constant G 0, and lm N p (N) n = 0, n. Under the above scalng, the offered load stays constant, and equal to G, as the number of nodes N ncreases. Let T(p (N) ) be the network servce rate for the network wth N nodes, and let ρ(p (N) ) be the fracton of tme that the channel s dle. For the above scalng, t s then well-known (see for example [15]) that the number of nodes that make an attempt to capture the channel after an dle perod weakly converges to a Posson random varable wth mean G. Usng ths result, t then follows that lm N T(p(N) ) τ(g) = 0 and lm N ρ(p(n) ) ρ(g) = 0,.e. the approxmatons gven by Eq. (3) and Eq. (7) are asymptotcally accurate. IV. CSMA FIXED POINT ANALYSIS In ths secton, we approxmate the servce rates of a statc CSMA polcy p for a general network usng a fxed pont analyss. In the Secton V, we show that ths approxmaton s asymptotcally accurate for large networks wth a small sensng tme. A. Fxed Pont Formulaton Gven a statc CSMA polcy p, we approxmate the fracton of tme ρ (p) that node s dle under p by the followng fxed pont equaton, ρ (p) = where G (p) s gven by β (β + 1 e G(p), = 1,...,N, (8) ) G (p) = j N [ p(, j) + p ( j,) ] ρ j (p), = 1,...,N. (9) Let G(p) = (G 1 (p),,g N (p)). We refer to Eq. (8) as the CSMA fxed pont equaton and to ρ(p) = (ρ 1 (p),,ρ N (p)), as well as G(p) = (G 1 (p),...,g N (p)), as the CSMA fxed pont. The ntuton behnd the CSMA fxed pont can be explaned as follows. Suppose that the fracton of tme that node s dle under the statc 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 β tme unts then the probablty that node makes an attempt to capture the lnk (, j), j N, s equal to p (, j) ρ j (p), and the probablty that node j N makes an attempt to capture the lnk ( j,) s p ( j,) ρ j (p). Hence, the rate at whch node receves or makes attempts to capture a lnk after t has been dle for β tme unts s gven by Eq. (9). Usng Eq. (7) of Secton III, the fracton of tme that node s dle under p can be approxmated by Eq. (8). Let G R (p) = j N p ( j,) ρ j (p). Usng the above CSMA fxed pont formulaton, we approxmate the servce rate for lnk (, j) under a statc CSMA polcy p by τ (, j) (p) = p (, j)ρ j (p)e (GR (p)+g j(p)) 1+β e G (p) (10) Note that the above equaton s smlar to Eq. (6) where p (, j) ρ j (p) captures the probablty that node makes an attempt to capture lnk (, j) f t has been dle for β tme

4 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 any other node to capture a lnk that has an endpont n common wth lnk (, j). Note that τ (, j) (p) p (, j) β e (G (p)+g j (p)) ( 1+β e G (p) )( 1+β e G j(p)) (11) as G (p) G R (p). In the next subsecton we show that there always exsts a unque CSMA fxed pont, and hence the CSMA fxed pont approxmaton (8) s well-defned. B. Exstence of a Unque Fxed Pont The next proposton shows the exstence of a CSMA fxed pont for all p. Lemma 1: For every statc CSMA polcy p, there exsts a CSMA fxed pont ρ(p). The proof for Lemma 1 uses the contnuty propertes of the fxed pont equaton gven Eq. (8), and s a straghtforward applcaton of the Brouwer s fxed pont theorem. The next lemma states that for the CSMA polcy p wth p (, j) = 0, (, j) L, there exsts a sngle CSMA fxed pont. Lemma 2: Consder the CSMA polcy p wth p (, j) = 0, (, j) L. Then for any β > 0 the unque CSMA fxed pont ρ( p) s gven by ρ ( p) = 1 and G ( p) = 0, for = 1,...,N. Thes result of Lemma 2 s obtaned by applyng Eq. (8) and (9) to the CSMA polcy p wth p (, j) = 0. We next study the contnuty propertes of G(p). To do ths, we consder the mappng [ ] p(, j) + p ( j,) f (G,p) = G ( ), = 1,...,N. (12) 1+β e G j j N β Note that the mappng f(g,p) = [ f (G,p)] =1,...,N s contnuous and we have that f(g(p),p) = 0. We then obtan the followng result. Proposton 1: The correspondence G : [0,1] L R N + s upper-semcontnuous;.e., G(p) has a closed graph. Usng the mappng f(g,p) = [ f (G,p)] =1,...,N gven by Eq. (12), we obtan the next proposton whch establshes the local unqueness of the correspondence G(p). Proposton 2: For all statc CSMA polces p and all CSMA fxed ponts Ḡ G( p), there exst neghbourhoods U R N + of Ḡ and V [0,1]L of p such that for each p V the equaton f(g,p) = 0 has a unque soluton G U. Moreover, ths soluton can be gven by a functon G = φ(p) where φ s contnuously dfferentable on V. Proposton 2 can be proved usng the mplct functon theorem [1]. Combnng Lemma 2 wth Propostons 1 and 2, we obtan the followng result. Theorem 1: For every statc CSMA polcy p, there exsts a unque CSMA fxed pont ρ(p). The unqueness result of Theorem 1 combned wth the upper-semcontnuty of Proposton 1 drectly mples the contnuty of G(p), and hence of ρ(p). Ths s stated n the followng corollary. Corollary 1: The fxed pont ρ(p) s contnuous n p. V. LARGE NETWORKS WITH A SMALL SENSING TIME In ths secton, we study the behavor of the CSMA fxed pont under the lmtng regme where the number of nodes N ncreases to nfnty and the sensng tme β decreases to zero. For ths case, we show that the CSMA fxed pont descrpton of the operatng pont s asymptotcally accurate. Ths result states that for large networks wth a small sensng tme β, the CSMA fxed pont approxmates well the actual performance of the statc CSMA polcy. Consder a sequence of networks for whch the number of nodes N ncreases to nfnty. Let L (N) be the set of all lnks n the network wth N nodes, and let N (N) be the set of neghbours of node. As the network sze ncreases, we assume that the sensng tme decreases as follows. Assumpton 1: The sensng tme β (N) for the network wth N nodes gven by β (N) = 1 K (N) where K (N) s a postve nteger such that N lm = 0. N K (N) For the above scalng, consder a sequence of statc CSMA polces {p (N) } N 1, where p (N) defnes a statc CSMA polces for the network wth N nodes. We make the followng assumptons for the sequence {p (N) } N 1. Assumpton 2: For the sequences {p (N) } N 1 and {β (N) } N 1 the followng s true. (a) For p (N) max we have that max = p (N) (, j) L (N) (, j) lm N p (N) max = 0. β (N) (b) There exsts a constant χ and an nteger N 0 such that for all N N 0 we have that [p (N) (, j) + p(n) ( j,) ] χβ (N), = 1,...,N. j N (N) Condton (a) mples that the attempt probablty of each lnk becomes small as N becomes large, and that the sensng delay does not decrease too fast as N ncreases. Condton (b) mples that the total rate wth whch lnks that orgnate or end at a gven node are captured, s upper-bounded by χ. For the above scalng, let ρ (N) (p (N) ) = (ρ (N) 1 (p (N) ),...,ρ (N) N (p(n) )) be the CSMA fxed pont for the network of sze N, and let ρ (N) (p (N) ) be the actual fracton of tme that node s dle. Furthermore, let δ (N) ρ = max =1,...,N ρ(n) (p (N) ) ρ (N) (p (N) ) be the maxmum approxmaton error of the CSMA fxed pont. Smlarly, let δ (N) τ = max (, j) L (N) 1 τ (N) (, j) (p(n) ) T (N) (, j) (p(n) )

5 be the maxmum relatve approxmaton error of the lnk servce rates under the CSMA fxed pont. We followng result states that n the lmt as N approaches nfnty the CSMA fxed pont approxmaton becomes asymptotcally accurate. Proposton 3: For the above defned scalng we have that lm N δ(n) ρ = 0, and lm N δ(n) τ = 0. To prove the above Proposton 3, we make a connecton wth the Erlang fxed pont for loss networks [6]. Specfcally, we show that for the above scalng the CSMA fxed pont s asymptotcally dentcal to the Erlang fxed pont of a loss network wth two-hop routes where each lnk can support at most one connecton, and use the analyss of Hajek and Krshna n [4]. VI. ACHIEVABLE RATE REGION In ths secton we use the CSMA fxed pont approxmaton to characterze the achevable rate regon of statc CSMA schedulers. In Secton VII we wll show that ths characterzaton s asymptotcally accurate for large networks wth a small sensng tme. Consder a wreless network wth sensng tme β > 0 as descrbed n Secton II. Note that R denotes the set of routes n the network. Also, recall the defntons of λ and Λ, N, as gven n (1) and (2), respectvely. Usng these defntons, let the set Γ(β) be gven by Γ(β)= { λ = {λ r } r R Λ < τ(g + (β))e (G+ (β)), = 1,...,N where G + (β) = 2β (see Secton III). The next proposton states that under the CSMA fxed pont approxmaton the achevable rate regon of statc CSMA polces s equal to Γ(β). Proposton 4: Gven β > 0, for every λ Γ(β) there exsts a statc CSMA polcy p such that λ (, j) < τ (, j) (p), (, j) L. Our proof for Proposton 4 s constructve n the sense that gven a rate vector λ Γ(β), we derve a statc CSMA polcy p such that λ (, j) < τ (, j) (p), (, j) L. It s also shown that the resultng polcy satsfes Assumpton 2, and hence s feasble. Usng the results of Secton III, we have that lmγ(β) = {λ = {λ r } r R Λ < 1 = 1,...,N}. β 0 Note that any rate vector λ for whch there exsts a node wth Λ 1 cannot be stablzed, as the servce rate at each node s upper-bounded by 1. Hence, the above result suggests that for network wth a small sensng tme the achevable rate regon of statc CSMA polces s equal to the capacty regon. In the next secton we show that ths result s true for the lmtng regme of large networks wth a small sensng tme. VII. ASYMPTOTIC OPTIMALITY OF CSMA In ths secton we consder the case of large networks wth a small sensng tme under the many flow asymptotc. }, A. Many Flow Asymptotc Recall that n Secton V we ntroduced a sequence of networks for whch the number of nodes N ncreases to nfnty, and let L (N) be the set of all lnks n the network wth N nodes, and N (N) be the set of neghbors of node n the network wth N nodes. In ths secton, we ntroduce a smlar scalng for the set of flows n addton to those. In partcular, we let R (N) be the set of gven routes for the network wth N nodes and let λ (N) = {λ r (N) } r R (N) be the mean arrval rate vector. For the mean rate vector λ (N), let λ (N) (, j) = r R (N) :(, j) r λ (N) r, (, j) L (N), be the total mean packet arrval rate over all routes r R (N) that traverse lnk (, j), and let Λ (N) [ (N) = λ (, j) + λ (N) ( j,)], = 1,...,N, j N (N) be the total mean packet arrval rate over all routes r R (N) that traverse node. Defnton 6 (Many Flow Asymptotc): Gven a sequence of networks {L (N),R (N) } N 1, we defne D as the set of all rate vector sequences {λ (N) } N 1 for whch the followng propertes hold. (a) There exsts a constant Λ and an nteger N such that for all N N we have that Λ (N) Λ, = 1,..,N. (b) For every ε > 0, there exsts an nteger N 0 such that for N N 0 we have that λ (N) (, j) < ε, (, j) L (N). A sequence {λ (N) } N 1 D defnes the lmtng regme where (a) the number of nodes and routes of the network ncreases, (b) the arrval on each route decreases, and (c) for each node we have that the total rate over all routes that pass through the node stays bounded. The defnton s closely related to the defnton of large networks wth dverse routng that has been used n the context of the Erlang fxed pont approxmaton [6], [4]. We defne the asymptotc achevable rate regon of statc CSMA polces for large networks as follows. Defnton 7: The asymptotc achevable rate regon of statc CSMA polces under the many flow lmt s the set of sequences {λ (N) } N 1 D for whch there exsts a sequence of statc CSMA schedulng polces {p (N) } N 1 such that lm nf N ( (N) mn 1 ) > 0, n 1. where (N) mn = mn (, j) L (N) T (N) (N) /λ (, j) (, j). The above defnton mples that every rate sequence {λ (N) } N 1 n the asymptotc rate regon can eventually be stablzed by a statc CSMA polcy. Note that a sequence {λ (N) } N 1 D for whch there exsts a node wth lm N Λ(N) 1 can not be stablzed as servce rate at each node s bounded by 1. Hence, the achevable regon under the many flow lmt

6 s contaned n the set { C = {λ (N) } N 1 D Λ < 1 and an nteger N such that } N N we have Λ (N) Λ, = 1,...,N. We refer to C as the capacty regon under the many flow lmt. B. Asymptotc Rate Regon In ths subsecton we characterze the asymptotc achevable rate regon of statc CSMA polces under the many flow lmt for networks wth a small sensng tme, and show that the asymptotc achevable rate regon of statc CSMA polces s equal to capacty regon C. Consder the same scalng of the sensng tme as n Secton V as descrbed n Assumpton 1. We have the followng result. Proposton 5: Under the scalng β (N), for every sequence λ (N) C there exsts a sequence of statc CSMA polces {p (N) } N 1 that asymptotcally stablzes the network,.e. where lm nf N ( (N) mn 1 ) > 0, n 1. (N) (N) mn = mn T(, j) (p(n) ) (, j) L (N) λ (N) (, j) We provde a sketch of the proof for Proposton 5. By defnton, for each sequence {λ (N) } N 1 C there exsts a scalar Λ < 1 and an nteger N such that for N N we have Λ (N) Λ, = 1,...,N. As lm N β (N) = 0 and lm β 0 τ(g + (β)) = 1 (see Eq. (5)), there exsts a nteger N 0 and a constant ε 1 such that for N N 0 we have that Λ (N) < τ(g + (β (N) ))e (G+ (β (N) )) (1 ε 1 ), = 1,...,N. Usng the proof of Proposton 4, we can then construct a sequence of statc CSMA polces {p (N) } N 1 such that for N N 0 we have λ (N) (, j) < τ (, j)(p (N) )(1 ε 2 ),. (, j) L (N) where ε 2 > 0 s a constant that does not depend on N. Usng Proposton 3, the approxmaton τ (, j) (p (N) ) of the servce rate of lnk (, j) s then asymptotcally accurate as N ncreases, and the result follows. VIII. CONCLUSIONS In ths paper we ntroduced the CSMA fxed pont approxmaton to study statc CSMA schedulers n wreless networks wth prmary nterference, and showed that the approxmaton s asymptotcally accurate for large networks wth a small sensng tme. There are three mportant ssues that are not addressed n ths paper, but are beng nvestgated n our ongong work. Frst, whle we showed that the CSMA fxed pont approxmaton s asymptotcally accurate, we dd not nvestgate how large a network has to be n order to obtan a good approxmaton. We carred out several numercal case studes whch suggest that the CSMA fxed pont approxmaton s already remarkably accurate for networks where each nodes has more than 5-10 neghbours. Second, the constructon of a statc CSMA scheduler requres the a pror-knowledge of the arrval rate vector. We are nvestgatng an approach usng a queue-length based scheduler smlar to [9] n order to obtan CSMA schedulers that are able to dynamcally adapt to any rate vector n the achevable rate regon. Thrd, the current analyss s restrcted to wreless networks wth prmary nterference, and t s an open problem to extend the analyss to more general nterference models. REFERENCES [1] D. Bertsekas. Nonlnear Programmng. Athena Scentfc, Belmont, MA, [2] D. Bertsekas and R. Gallager. Data Networks. Prentce Hall, Englewood Clffs, NJ, [3] L. Bu, A. Erylmaz, R. Srkant, and X. Wu. Jont asynchronous congeston control and dstrbuted schedulng for wreless networks. Proceedngs of IEEE Infocom [4] B. Hajek and A. Krshna. Bounds on the accuracy of the reducedload blockng formula for some smple crcut-swtched networks. In Proceedngs of Internatonal Conference on New Trends n Communcaton, Control and Sgnal Processng, Ankara, Turkey, [5] S. Sarkar, and L. Tassulas. Maxmn Far Schedulng n Wreless Networks. In Proceedngs of IEEE INFOCOM, [6] F. P. Kelly. Loss networks. The Annals of Appled Probablty, 1(3): , August [7] X. Ln and S. Rasool. Constant-tme dstrbuted schedulng polces for ad hoc wreless networks. In Proceedngs of CDC, [8] X. Ln and N. Shroff. The mpact of mperfect schedulng on crosslayer rate control n multhop wreless networks. In Proceedngs of IEEE Infocom, Mam, FL, March [9] P. Marbach and Y. Lu. Actve queue management and schedulng for wreless networks: The sngle-cell case. In Proceedngs of Conference on Informaton Scences and System (CISS), Prnceton, NJ, [10] G. Sharma, R. Mazumdar, and N. Shroff. On the complexty of schedulng n multhop wreless systems. In Proceedngs of MobCom, Los Angeles, CA, September [11] L. Tassulas and A. Ephremdes. Stablty propertes of constraned queueng systems and schedulng polces for maxmum throughput n multhop rado networks. IEEE Transactons on Automatc Control, 36: , December [12] X. Wu and R. 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., [13] M. Durvy and P. Thran. A Packng Approach to Compare Slotted and Non-Slotted Medum Access Control. In Proceedngs of IEEE Infocom, [14] Ch. Bordenave, D. McDonald, A. Proutere, A Partcle System n Interacton wth a Rapdly Varyng Envornment: Mean Feld Lmts and Applcatons, submtted Stochastc Processes and ther Applcatons, [15] D. McDonald. Elements of Appled Probablty for Engneerng, Mathematcs, and Systems Scence. World Scentfc Publshng Co. Plc. Ltd., Rver Edge, NJ, 2003.

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