A Fast-CSMA Algorithm for Deadline-Constrained Scheduling over Wireless Fading Channels

Transcription

1 A Fst-CSMA Algorithm for Dedline-Constrined Scheduling over Wireless Fding Chnnels Bin Li nd Atill Eryilmz Abstrct Recently, low-complexity nd distributed Crrier Sense Multiple Access (CSMA)-bsed scheduling lgorithms hve ttrcted extensive interest due to their throughput-optiml chrcteristics in generl network topologies. However, these lgorithms re not well-suited for serving rel-time trffic under timevrying chnnel conditions for two resons: () the mixing time of the underlying CSMA Mrkov Chin grows with the size of the network, which, for lrge networks, genertes uncceptble dely for dedline-constrined trffic; (2) since the dynmic CSMA prmeters re influenced by the rrivl nd chnnel stte processes, the underlying CSMA Mrkov Chin my not converge to stedy-stte under strict dedline constrints nd fding chnnel conditions. In this pper, we ttck the problem of distributed scheduling for serving rel-time type trffic over time-vrying chnnels. Specificlly, we consider fully-connected topologies with independently fding chnnels (which cn model cellulr networks) in which flows with short-term dedline constrints nd longterm drop rte requirements re served. To tht end, we first chrcterize the mximl set of stisfible rrivl processes for this system nd, then, propose Fst-CSMA (FCSMA) policy tht is shown to be optiml in supporting ny rel-time trffic tht is within the mximl stisfible set. These theoreticl results re further vlidted through simultions to demonstrte the reltive efficiency of the FCSMA policy compred to some of the existing CSMA-bsed lgorithms. I. ITRODUCTIO Wireless networks re expected to serve rel-time trffic, such s video or voice pplictions, generted by lrge number of users over potentilly fding chnnels. These constrints nd requirements, together with the limited nture of shred resources, generte strong need for distributed lgorithms tht cn efficiently utilize the vilble resources while mintining high qulity-of-service gurntees for the rel-time pplictions. Yet, the strict short-term dedline constrints nd long-term drop rte requirements ssocited with most reltime pplictions complicte the development of provbly good distributed solutions. In the recent yers, there hs been n incresing understnding on the modeling nd service of such rel-time trffic in wireless networks (e.g., [4], [5], [6], [2]). However, existing works in this domin ssume centrlized controllers, nd hence re not suitble for distributed opertion in lrge-scle networks. In seprte line of work, it hs lso been shown tht CSMA-bsed distributed scheduling (e.g., [7], [2], [3], Bin Li nd Atill Eryilmz ({lib,eryilmz}@ece.osu.edu) re with the Deprtment of Electricl nd Computer Engineering t The Ohio Stte University, Columbus, Ohio 4320 USA. This reserch work is funded by Qtr tionl Reserch Fund (QRF) under the tionl Reserch Priorities Progrm (PRP) grnt number PRP [3]) cn mximize long-term verge throughput for generl wireless topologies. However, these results lso do not pply to strictly dedline-constrined trffic tht we trget, since their throughput-optimlity relies: (i) on the convergence time of the underlying Mrkov Chin to its stedy-stte, which grows with the size of the network; nd (ii) on reltively sttionry conditions in which the CSMA prmeters do not chnge significntly over time so tht the instntneous service rte distribution cn sty close to the sttionry distribution. Both of these conditions re violted in our context: (i) pckets of dedline constrined trffic re likely to be dropped before the CSMA-bsed lgorithm converges to its stedy-stte; nd (ii) the time-vrying fding cretes significnt vritions on the CSMA prmeters, in which cse the instntneous service rte distribution cnnot closely trck the sttionry distribution. While chieving low dely vi distributed scheduling in generl topologies is difficult tsk (see [4]), in relted work [9] tht focuses on grid topologies, the uthors hve designed n Unlocking CSMA (UCSMA) lgorithm with both mximum throughput nd order optiml verge dely performnce, which shows promise for distributed scheduling in specil topologies. However, UCSMA lso does not directly pply to dedlineconstrined trffic since its mesure of dely is on verge. Moveover, it is not cler how existing CSMA or UCSMA implementtions will perform under fding chnnel conditions. With this motivtion, in this work, we ddress the problem of distributed scheduling in fully connected networks (e.g., Cellulr network, WLA) for serving rel-time trffic over independently fding chnnels. Our contributions re: In Section III-A, we chrcterize the mximl set of stisfible rel-time trffic chrcteristics s function of their drop rte requirements nd chnnel sttistics. In Section III-B, we propose n FCSMA lgorithm tht differs from existing CSMA policies in its design principle: rther thn evolving over the set of schedules to rech fvorble stedy-stte distribution, the FCSMA policy ims to quickly rech one of set of fvorble schedules nd stick to it for durtion relted to the ppliction dedline constrints. While the performnce of the former strtegy is tied to the mixing-time of Mrkov Chin, the performnce of our strtegy is tied to the bsorption time, nd hence, yields significnt dvntge for strictly dedline-constrined flows. In Theorem, we prove tht the FCSMA policy is optiml in the sense tht it cn stisfy the dedline nd drop rte requirements ny rel-time trffic within the chrcterized mximl stisfible set. In Section IV, we compre the performnce of FCSMA

2 2 with some of the existing CSMA policies under different scenrios, both to vlidte the theoreticl clims, nd to demonstrte the performnce gins due to our proposed strtegy. II. SYSTEM MODEL We consider fully-connected wireless network topology where users contend for dt trnsmission over single chnnel tht is independently block fding for ech user. We ssume tht the time scle of block fding is the sme s the durtion of the dedline constrint, nd thus uniformly clled s slot. We lso ssume tht ll links strt trnsmission t the beginning of ech time slot. We cpture the chnnel fding over link l vi C l [t], which mesures the mximum mount of service vilble over it in slot t, if scheduled. We ssume tht C[t] = (C l [t]) re independently distributed rndom vribles over links nd identiclly distributed over time. Yet, due to interference constrints, t most one link cn be scheduled for service in ech slot. We use binry vrible S l [t] to denote whether the link l is served t slot t, where S l [t] = if the link l cn be served t slot t nd S l [t] = 0, otherwise. Ech pcket hs dely bound of time slot, which mens tht if pcket cnnot be served during the slot it rrives, it will be dropped. In this context of fully-connected network, we ssocite ech rel-time flow with link, nd hence use these two terms interchngebly. Let A l [t] denote the number of pckets rriving t link l in slot t tht re independently distributed over links nd identiclly distributed over time with men λ l, nd A l [t] A mx for some A mx <. Ech link hs mximum llowble drop rte ρ l λ l, where ρ l (0, ) is the mximum frction of pckets tht cn be dropped t link l. For exmple, ρ l = 0. mens tht t most 0% of pckets cn be dropped t link l on verge. Under bove setup, we define our stochstic control problem (SCP) s follows: Definition : (SCP) where Mximize () {S[t]} t Subject to S l [t]c l [t] A l [t], l, t (2) λ l = limsup T µ l = liminf T λ l ( ρ l ) µ l, l (3) S l [t] (4) l S l [t] {0, }, l, t (5) T T T E[A l [t]] (6) t= T E[S l [t]c l [t]] (7) t= In the bove mximiztion problem: (2) indictes tht the number of served pckets cnnot be greter thn the number of rrivls for ech slot; (3) indictes tht the provided verge service rtes stisfy the drop rte requirements of the rel-time trffic; (4) indictes tht t most one link cn be served t ech slot t. ormlly, it is difficult to solve SCP directly. Insted, we use the technique in [] to introduce virtul queue X l [t] for ech link l to trck the number of dropped pckets t slot t. Specificlly, the number of pckets rriving t virtul queue l t the end of slot t is denoted s R l [t], which is equl to A l [t] min{s l [t]c l [t], A l [t]}. We use I l [t] to denote the service for virtul queue l t the end of the slot t with men ρ l λ l, nd I l [t] I mx for some I mx <. Further, we let U l [t] denote the unused service for queue l t the end of slot t, which is upper-bounded by I mx. Then, the evolution of virtul queue is s follows: X l [t + ] = X l [t] + R l [t] I l [t] + U l [t], l =,,. In the rest of the pper, we consider the clss of sttionry policies G tht select S[t] s function of (X[t], A[t], C[t]), which, then, forms Mrkov Chin. If this Mrkov Chin is positive recurrent, then the verge drop rte will meet the required constrint utomticlly (see []). Accordingly, we cll n lgorithm optiml if it cn mke this Mrkov Chin positive recurrent for ny rrivl rte vector within the mximl stisfible region tht we will chrcterize in the next section. III. FCMSA ALGORITHM FOR THROUGHPUT OPTIMALITY In this section, we first study the mximl stisfible region given the drop rte nd chnnel sttistics. Then, we propose n optiml FCSMA lgorithm. A. Mximl Stisfible Region Consider the clss G of sttionry policies tht bse on their scheduling decision on the observed vector (X[t], A[t], C[t]) t slot t. The next lemm estblishes condition tht is necessry for stbilizing the system. Lemm : If there is policy G 0 G tht cn stbilize the virtul queue X[t], then there exist non-negtive numbers α(, c; s) such tht α(, c; s) = (8) s S P A () P C (c) α(, c; s) min{s c, } > λ ( ρ) (9) c s S where (A B) i = A i B i denotes Hdmrd product, P A () = P (A[t] = ) nd P C (c) = P (C[t] = c). The proof is lmost the sme s [5] nd hence is omitted here. ote tht the left hnd side of inequlity (9) is the totl verge service provided for ech link during one time slot; while λ ( ρ) is the totl verge mount of dt pckets t ech link tht need to be served within one slot. Thus, to the meet the constrint of drop rte, (9) should be stisfied. We define mximl stisfible region Λ(ρ) s follows: Λ(ρ) = {A : α(, c; s) 0, such tht both (8) nd (9)stisfy}

3 3 B. FCSMA lgorithm Before we present nd nlyze our proposed FCSMA lgorithm, we define set of functions (lso see [8]) tht llows some flexibility in the design nd implementtion of the lgorithm. F := set of non-negtive, nondecresing nd differentible functions f( ) : R + R + with lim f(x) =. x f(x + ) B := {f F: lim =, for ny R}. x f(x) The exmples of functions tht re in clss B re f(x) = (log x) α, f(x) = x α (α > 0) or f(x) = e x. ote tht the exponentil function f(x) = e x is not in clss B. Definition 2 (FCSMA Algorithm): At the beginning of ech time slot t, ech link l independently genertes n exponentilly distributed rndom vrible with men f(x l [t]) min(c l[t],a l [t]), nd strts trnsmitting fter this rndom durtion unless it senses nother trnsmission before. The link tht grbs the chnnel trnsmits its pckets until the end of the slot. It is esy to show tht the convergence time of FCMSA is nd the probbility tht link l successfully grbs j= H j H the chnnel is l. If we cn choose H = (H l ) j= Hj lrge enough, then the convergence time of FCSMA cn be rbitrrily smll. Follows, we consider FCSMA with H l = f(x l [t]) min{c l[t],a l [t]} for link l. The frction of serving the link l is: π l = f(x l[t]) min{c l[t],a l [t]} ( Z Z ) (0) where Z = j= f(x j[t]) min{c j[t],a j [t]}. Let W [t] = mx l log f(x l [t]) min{c l [t], A l [t]}. The following lemm estblishes the fct tht FCSMA policy picks link with the weight close to mximum weight with high probbility when the mximum weight W [t] is lrge enough. Lemm 2: Given ɛ > 0 nd ζ > 0, W <, such tht if W [t] > W, then FCSMA policy picks link k stisfying which lso implies P {W k [t] ( ɛ)w [t]} ζ E[W k [t] {W [t] W } A[t], C[t], X[t]] ( ɛ)( ζ)w [t] {W [t] W } () where W k [t] = log f(x k [t]) min{c k [t], A k [t]}. Proof: Define X = {l : log f(x l [t]) min{c l [t], Q l [t]} < ( ɛ)w [t]} If there re no pckets witing in the link l, it trnsmits dummy pcket to occupy the chnnel. Then, π(x ) := l X π l exp(log f(x l [t]) min{c l [t], Q l [t]}) n l X j= exp(log f(x j[t]) min{c j [t], Q j [t]}) X exp(( ɛ)w [t]) < n j= exp(log f(x j[t]) min{c j [t], Q j [t]}) exp(( ɛ)w [t]) exp(w [t]) = exp(ɛw [t]) (2) The first inequlity in (2) follows the fct tht Z. Thus, W < such tht W [t] > W implies π(x ) < δ. Since t most one link cn be ctivted t ech slot in single-hop network topologies, we cn lso write W k [t] = W l F [t], where Wl F [t] = log f(x l [t]) min{c l [t]sl F [t], A l[t]} nd S F [t] denotes the schedule chosen by FCSMA with Sk F [t] =. In the rest of the pper, we lso write W [t] = W l [t], where Wl [t] = log f(x l[t]) min{c l [t]sl [t], A l[t]} nd S [t] = rg mx S S W l [t]. Under certin conditions for the function f, we cn estblish the throughput-optimlity of FCSMA lgorithm. Theorem : FCSMA with R l = f(x l [t]) min{c l[t],a l [t]} for ech link l t every time slot t is optiml when log f B nd f(0). Proof: Let g(x) = log f(x). Consider the Lypunov function V (X) := h(x l), where h (x) = g(x). Then V : = E [V (X[t + ]) V (X[t]) = E [(h(x l [t + ]) h(x l [t])) By the men-vlue theorem, we hve h(x l [t+]) h(x l [t]) = g(x l )(X l[t + ] X l [t]) = g(x l )(R l[t] I l [t] + U l [t]), where X l lies between X l[t] nd X l [t + ]. Hence, we get V = = E [g(x l)(r l [t] I l [t] + U l [t]) E [g(x l)u l [t] =: V + E [g(x l)(r l [t] I l [t]) =: V 2 For V, if X l [t] = X l I mx, then U l [t] = 0. If X l [t] = X l < I mx, then U l [t] I mx. But in this cse, X l [t + ] (I mx +A mx ) (since R l [t] A mx ). Hence, g(x l ) g(i mx+

4 4 A mx ) <. Thus, V = E [ g(r l )U l [t] {Xl <I mx} X[t] = X ] I mx g(i mx + A mx ) (3) where { } is the indictor function. ext, let s focus on V 2. We know tht g(x l ) = g(x l[t] + l ) ( l A mx ). According to the definition of function g B, given β > 0, there exists M > 0, such tht for ny X l [t] = X l > M, we hve g(x l ) g(x l ) < β, tht is, Thus, we hve ( β)g(x l ) < g(x l) < ( + β)g(x l ) (4) g(x l)(r l [t] I l [t]) =g(x l) [(R l [t] I l [t]) + (R l [t] I l [t]) ] <( + β)g(x l )(R l [t] I l [t]) + ( β)g(x l )(R l [t] I l [t]) =g(x l )(R l [t] I l [t]) + βg(x l ) R l [t] I l [t] g(x l )(R l [t] I l [t]) + βa mx g(x l ) (5) where (x) + = mx{x, 0}, (x) = min{x, 0} nd R l [t] I l [t] A l [t] A mx. Thus, we divide V 2 into two prts: V 2 = + E [ g(x l)(r l [t] I l [t]) {Xl >M} X[k] = X ] =: V 3 E [ g(x l)(r l [t] I l [t]) {Xl M} X[t] = X ] =: V 4 For V 3, by using (5), we hve V 3 = E [ g(x l )(R l [t] I l [t]) {Xl >M} X[t] = X ] + βa mx g(x l ) {Xl >M} E[g(X l )(A l [t] I l [t]) {Xl >M} =:L E[ W F l [t] {Xl >M} X[t]] + =:L 2 βa mx g(x l ) {Xl >M} ext, let s focus on L. By Lemm, there exist non-negtive numbers α(, c; s) stisfying (8) nd for δ > 0 smll enough, we hve P A () P C (c) α(, c; s) min{s l c l, l } c s S λ l ( ρ l ) + δ (6) Let W l = g(x l ) min{s l c l, l }. By using (6), we hve L = g(x l )λ l ( ρ l ) {Xl >M} P A () P C (c) α(, c; s) W l {Xl >M} c s S δ g(x l ) {Xl >M} = P A () P C (c) α(, c; s) W l {Xl >M,W [t]>w } c s S + P A () P C (c) α(, c; s) W l {Xl >M,W [t] W } c s S δ g(x l ) {Xl >M} P A () P C (c) α(, c; s) W l {Xl >M,W [t]>w } c s S + W δ g(x l ) {Xl >M} (7) ext, let s consider L 2. Since ( ɛ)( ζ)e[ Wl [t] {Xl >M,W [t]>w } ( ɛ)( ζ)e[ Wl [t] {W [t]>w } E[ Wl F [t] {W [t]>w } (by Lemm 2) = E[ Wl F [t] {Xl >M,W [t]>w } + E[ Wl F [t] {Xl M,W [t]>w } E[ Wl F [t] {Xl >M,W [t]>w } + A mxg(m)

5 5 L 2 becomes L 2 E[ Wl F [t] {Xl >M,W [t]>w } X[t]] ( ɛ)( ζ)e[ Wl [t] {Xl >M,W [t]>w } X[t]] A mx g(m) Thus, by using (7) nd (8), V 3 becomes V 3 P A () P C (c) α(, c; s) W l {Xl >M,W [t]>w } c s S E[ Wl [t] {Xl >M,W [t]>w } X[t]] + (ɛ + ζ ɛζ)e[ Wl [t] {Xl >M,W [t]>w } X[t]] + W + A mx g(m) δ g(x l ) {Xl >M} + βa mx g(x l ) {Xl >M} (8) Since P A () P C (c) α(, c; s) c s S E[ Wl [t] = P A () c P A () c P C (c) s S α(, c; s) P C (c) s S α(, c; s) W l W l W l 0 (9) Thus, we hve P A () P C (c) α(, c; s) W l {Xl >M,W [t]>w } c s S P A () P C (c) α(, c; s) W l c s S E[ Wl [t](by using (9)) = E[ Wl [t] {Xl >M} X[t]] + E[ E[ Wl [t] {Xl >M,W [t]>w } W l [t] {Xl M} X[t]] + E[ Wl [t] {Xl >M,W [t] W } + A mxg(m) E[ Wl [t] {Xl >M,W [t]>w } X[t]] + W + A mxg(m) In ddition, we hve E[ Wl [t] {Xl >M,W [t]>w } E[ Wl [t] A mx g(x l ) = A mx g(x l ) {Xl >M} + A mx A mx g(x l ) {Xl M} (20) g(x l ) {Xl >M} + A mx g(m) (2) then, by using (20) nd (2), we hve V 3 γ g(x l ) {Xl >M} + D (22) where D = 2W + (2 + ɛ + ζ ɛζ)a mx g(m) nd γ = δ βa mx A mx (ɛ + ζ ɛζ). We cn choose β, ɛ, ζ smll enough such tht γ > 0. For V 4, we hve V 4 E [g(x l)r l [t] {Xl M} E [g(x l)a l [t] {Xl M} A mx g(m + A mx )

6 6 Thus, we get V < γ γ g(x l ) {Xl >M} + D g(x l ) + E (23) where D := I mx g(i mx + A mx ) + D + A mx g(m + A mx ) < nd E := D + γg(m). Hence, by the Lypunov Drift theorem [], we hve T t=0 E[g(X l[t])] E γ <, which lim sup T T implies stbility-in-the men nd thus the Mrkov Chin is positive recurrent [0]. IV. SIMULATIO RESULTS In this section, we perform simultions to vlidte the throughput-optimlity of the proposed FCSMA policy with dedline constrint time slot in both fding nd non-fding chnnels. In the simultion, there re = 0 links. All links require tht the mximum frction of dropping pckets cnnot exceed ρ = 0.2. The number of rrivls in ech slot follows Bernoulli distribution. For the simultions of fding chnnel, ll links suffer from the O-OFF chnnel fding independently with probbility p = 0.9 tht the chnnel is vilble in ech time slot. Under this setup, we cn use the sme technique in pper [6] to get the cpcity region: Γ = {λ : ( ρ)λ < ( pλ) }. Through numericl clcultion, we cn get λ < 0.05 in non-fding chnnel nd λ < 0.03 in fding chnnel. We compre FCSMA with H l = exp(x l [t] min{c l [t], A l [t]}) with discrete time version of the clssicl CSMA lgorithm with the weight 2 X l [t] min{c l [t], A l [t]} [2]. To tht end, we divide ech time slot into M mini-slots. In FCSMA policy, if the link contends for the chnnel successfully, it will occupy tht chnnel in the rest of time slot; while in clssicl CSMA policy, ech link contends for the chnnel nd trnsmit the dt in mini-slot. Here, we don t consider the overhed tht the clssicl CSMA policy needs to contends for the chnnel. From Figure nd 2, we cn observe tht the verge virtul queue length grows very fst under the clssicl CSMA policy with M = while the verge queue length of FCSMA lwys keeps in low level. The reson for the poor performnce of clssicl CSMA scheme in dedline-constrined scheduling ppliction is tht the underlying Mrkov chin is controlled by the rrivl process. If the running time of CSMA policy hs the sme time scle with the dedline of the pcket, this Mrkov chin cnnot converge to the stedy stte. However, FCMSA policy cn quickly lock into one stte nd exhibits good performnce, which is shown to be throughput-optiml if we crefully choose the prmeters. In ddition, s M increses, the performnce of clssicl CSMA improves. The reson is tht the underlying Mrkov chin hs enough time to 2 In our setup, clssicl CSMA lgorithm with the weight log log(x l [t] min{c l [t], A l [t]} + e) hs much worse performnce thn tht with X l [t] min{c l [t], A l [t]}. converge to the stedy stte nd thus yields better performnce. Furthermore, we cn see tht FCSMA policy hs lmost the sme performnce s tht in stedy stte. Recll tht FCSMA policy wits for rndom time before ccessing the chnnel, this rndom time cn be rbitrrily smll when the number of links increses nd the virtul queue length is high. Averge Queue Length Averge Queue Length FCSMA in Stedy Stte FCSMA CSMA with M= CSMA with M=0 3 CSMA with M= Arrivl Rte Fig.. on-fding chnnel FCSMA in Stedy Stte FCSMA CSMA with M= CSMA with M=0 3 CSMA with M= Arrivl Rte Fig. 2. Fding chnnel V. COCLUSIOS In this pper, we first chrcterized the stisfible rte region given the drop rte nd chnnel sttistics nd then proposed n optiml distributed FCSMA policy for scheduling rel-time trffic over fding chnnel. We vlidted the performnce of FCSMA policy by compring it with existing CSMA policies

7 7 in simultions. We ssumed tht the dedline of pcket nd the fding of chnnel hve the sme time scle, which is not lwys the cse in relity. We will relx this ssumption in our future work. REFERECES [] J. Di. A fluid-limit model criterion for instbility of multiclss queueing networks. Annls of Applied Probbility, 6:75 757, 996. [2] H. Gngmmnvr nd A. Eryilmz. Dynmic coding nd rte-control for serving dedline-constrined trffic over fding chnnels. In Proc. IEEE Interntionl Symposium on Informtion Theory. (ISIT), Austin, TX, June 200. [3] J. Ghderi nd R. Sriknt. On the design of efficient csm lgorithms for wireless networks. In Proc. IEEE Interntionl Conference on Decision nd Control. (CDC), Atlnt, GA, December 200. [4] I. Hou, V. Borkr, nd P. R. Kumr. A theory of qos for wireless. In Proc. IEEE Interntionl Conference on Computer Communictions. (IFOCOM), Rio de Jneiro, Brzil, April [5] I. Hou nd P. R. Kumr. Scheduling heterogeneous rel-time trffic over fding wireless chnnels. In Proc. IEEE Interntionl Conference on Computer Communictions. (IFOCOM), Sn Diego, CA, Mrch 200. [6] J. Jrmillo nd R. Sriknt. Optiml scheduling for fir resource lloction in d hoc networks with elstic nd inelstic trffic. In Proc. IEEE Interntionl Conference on Computer Communictions. (IFOCOM), Sn Diego, CA, Mrch 200. [7] L. Jing nd J. Wlrnd. A csm distributed lgorithm for throughput nd utility mximiztion in wireless networks. In Proc. Allerton Conference on Communiction, Control, nd Computing (Allerton 2008), Monticello, Illinois, September [8] B. Li nd A. Eryilmz. On the limittion of rndomiztion for queue-length-bsed scheduling in wireless networks. In Proc. IEEE Interntionl Conference on Computer Communictions. (IFOCOM), Shnghi, Chin, April 20. [9] M. Lotfinezhd nd P. Mrbch. Throughput-optiml rndom ccess with order-optiml dely. Submitted to IEEE IFOCOM 20. [0] S. Meyn nd R. Tweedie. Criteri for stbility of mrkovin processes i: Discrete time chins. Advnces in Applied Probbility, 24: , 992. [] M. eely. Stochstic etwork Optimiztion with Appliction to Communiction nd Queueing Systems. Morgn & Clypool, 200. [2] J. i nd R. Sriknt. Distributed csm/c lgorithms for chieving mximum throughput in wireless networks. In Proc. IEEE Interntionl Workshop on Informtion Theory nd Applictions (ITA 2009), Sn Diego, Cliforni, Februry [3] S. Rjgopln, D. Shh, nd J. Shin. etwork dibtic theorem: n efficient rndomized protocol for contention resolution. In Proc. IEEE Interntionl Joint Conference on Mesurement nd Modeling of Computer Systems. (SIGMETRICS), Settle, WA, June [4] D. Shh, D. Tse, nd J. Tsitsiklis. Hrdness of low dely network scheduling. Submitted to IEEE Trnsctions on Informtion Theory, [5] L. Tssiuls. Scheduling nd performnce limits of networks with constntly vrying topology. IEEE Trnsctions on Informtion Theory, 43: , My 997. [6] L. Tssiuls nd A. Ephremides. Dynmic server lloction to prllel queues with rndomly vrying connectivity. IEEE Trnsctions on Informtion Theory, 39(2): , 993.