Exploring the Throughput Boundaries of Randomized Schedulers in Wireless Networks

Transcription

1 Exporing the Throughput Boundaries of Randomized Scheduers in Wireess Networks Bin Li and Atia Eryimaz Abstract Randomization is a powerfu and pervasive strategy for deveoping efficient and practica transmission scheduing agorithms in interference-imited wireess networks. Yet, despite the presence of a variety of earier works on the design and anaysis of particuar randomized scheduers, there does not exist an extensive study of the imitations of randomization on the efficient scheduing in wireess networks. In this work, we aim to fi this gap by proposing a common modeing framework and three functiona forms of randomized scheduers that utiize queueength information to probabiisticay schedue non-conficting transmissions. This framework not ony modes many existing scheduers operating under a time-scae separation assumption as specia cases, but it aso contains a much wider cass of potentia scheduers that have not been anayzed. We identify some sufficient and some necessary conditions on the network topoogy and on the functiona forms used in the randomization for throughput-optimaity. Our anaysis reveas an exponentia and a sub-exponentia cass of functions that exhibit differences in the throughput-optimaity. Aso, we observe the significance of the network s scheduing diversity for throughputoptimaity as measured by the number of maxima schedues each ink beongs to. We further vaidate our theoretica resuts through numerica studies. Index Terms Randomized scheduing, throughput-optimaity, stochastic contro, distributed agorithm, network stabiity. I. INTRODUCTION One of the greatest chaenges in the efficient communication in wireess networks is the management of interference amongst simutaneous transmissions. A commony used mode, which we aso empoy in this paper, to capture such interference effects is through the use of a confict graph whereby transmissions that wi coide with each other are indicated as conficting. These confict graphs can represent a variety of interference modes of practica importance, incuding the primary interference mode (e.g., [23], [9]), secondary interference mode (e.g., [2], [3]), or SINR threshod-based interference mode (e.g., [0]). Such confict graphs can take on extremey compex forms, especiay with growing network sizes. Thus, a fundamenta question in the design of efficient wireess network protocos is the decision of which subset of non-conficting This work was supported in part by Qatar Nationa Research Fund (QNRF) under the Nationa Research Priorities Program (NPRP) grant number NPRP , DTRA Grant HDTRA , and NSF Awards: CAREER-CNS and CCF An earier version of this paper has appeared in the Proceedings of 30th IEEE Internationa Conference on Computer Communications (INFOCOM), Shanghai, China, Apri 20 [2]. Bin Li and Atia Eryimaz ({ib,eryimaz}@ece.osu.edu) are with the Department of Eectrica and Computer Engineering at The Ohio State University, Coumbus, Ohio 4320 USA. transmissions to activate, and when - an operation commony referred to as scheduing. Of particuar interest in the cass of scheduing protocos is the set of throughput-optima scheduing strategies (e.g., [26], [8]) that achieves any throughput (subject to network stabiity) that is achievabe by any other scheduing strategy. Thus, throughput-optima scheduers are critica especiay for resource-imited wireess networks as they achieve the argest possibe throughput region that is supportabe by the network. The semina works of Tassiuas and Ephremides [26], [27] and many subsequent works (e.g., [4], [8], [24]; see [5] for an overview) have estabished the throughput-optimaity of a variety of Queue-Length-Based (QLB) Scheduing strategies, which prioritize activation of inks with the greatest backog awaiting service, aso caed Maximum Weight Scheduing (MWS). These origina throughput-optima strategies require the maximum weight schedue to be determined repeatedy as the queue-ength eves change. This cas for computationay heavy (even NP-hard in certain interference modes) and typicay centraized operations, which is impractica. Such restrictions have motivated new research efforts to deveop more practica throughput-optima scheduers with reduced compexity. One such thread ed to the deveopment of a cass of evoutionary randomized agorithms (aso named pick and compare agorithms) with throughput-optimaity characteristics (see [25], [3], [22]). Another thread ed to the deveopment of distributed but suboptima randomized/greedy strategies (see [3], [8], []). More recenty, another exciting thread of resuts have emerged that can guarantee throughput-optimaity by cevery utiizing queue-ength information in the context of carrier sense mutipe access (CSMA) (see [4], [7], [20], [9]). In paper [7], the authors proposed an agorithm that adaptivey seects the CSMA parameters under a time-scae separation assumption, i.e., the Markov Chain underying the CSMA-based agorithm converges to steady-state quicky compared with the time-scae of updating parameters of the agorithm. In paper [2], the authors showed the throughput-optimaity of a CSMAbased agorithm in which the ink weights are chosen to be of the form og og(q + e) (where q is the queue ength) without the time-scae separation assumption. Ghaderia and Srikant [6] extended these resuts by showing that the throughputoptimaity of CSMA-based agorithm wi be preserved even if the ink weights have the form og(q)/g(q), where g(q) can be a function that increases to infinity arbitrariy sowy. Yet, to the best of our knowedge, there does not exist a genera framework in which a variety of randomized scheduers can be

2 2 studied in terms of their throughput-optimaity characteristics. Thus, in this work, we aim to fi this gap by deveoping a common framework for the modeing and anaysis of queueength-based randomized scheduers, and then by estabishing necessary and sufficient conditions on the throughputoptimaity of a arge functiona cass of such scheduers under the time-scae separation assumption. Our framework is buit upon the observation that a common characteristic to most of the deveoped scheduers is their randomized seection of transmission schedues from the set of a feasibe schedues. Specificay, given the existing queue-engths of the inks, each scheduing strategy can be viewed as a particuar probabiity distribution over the set of feasibe schedues. Whie the means with which this random assignment may vary in its distributiveness or compexity, this perspective aows us to mode a arge set of existing and an even wider set of potentia randomized scheduers within a common framework. This work buids on this origina point-of-view to expore the boundaries of randomization in the throughput-optima operation of wireess networks. Such an investigation is crucia in reveaing the necessary and sufficient characteristics of randomized scheduers and the network topoogies in which throughput-optimaity can be achieved. Next, we ist our main contributions aong with references on where they appear in the text. In Section II, we highight the pressing need for deveoping new randomized scheduers, for exampe, for operation under fading conditions and for serving deay-reated appication requirements. We aso note with a specific exampe that these new scheduers may possess fundamentay different probabiistic operation than existing distributed soutions with product form mappings. This motivates us to study the performance imitations of wide cass of randomization strategies. In Section II, we introduce three functiona forms of randomized queue-ength-based scheduing strategies that incude many existing strategies as specia cases (see Definitions 3, 4 and 5). These strategies differ in the manner in which they measure the weight of schedues, and hence are used to mode fundamentay different scheduing impementations. We categorize the set of a functions used by these strategies into functions of exponentia form and of sub-exponentia form (see Definition 6), coectivey covering amost a functions of interest. These two categories capture the steepness of the functions used in the scheduers, and hep revea a critica degree of steepness necessary for throughput-optimaity in arge networks. Then, we find some sufficient (in Section IV) and some necessary (in Section V) conditions on the topoogica characteristics of the confict graph for the throughput-optimaity of these scheduers as a function of the cass of functions used in their operation. Our resuts, graphicay summarized in Section III, revea the significance of the network s scheduing diversity that is measured by the number of schedues each ink beongs to. A. Basic Definitions II. SYSTEM MODEL We consider a fixed wireess network represented by a graph G = (N, L), where N is the set of nodes and L is the set of undirected inks. We assume a time-sotted system, where a nodes transmit at the beginning of each time sot. Due to the interference-imited nature of wireess transmissions, the success or faiure of a transmission over a ink depends on whether an interfering ink is aso active in the same sot. For ease of exposition, we assume that a successfu transmission over any ink in each sot transfers one packet. We use confict graphs to capture any such coision-based interference in the wireess networks. In a confict graph CG = (L, E) of G under a given interference mode, the set of inks L in G becomes the set of nodes, and E denotes the set of edges that connects inks that interfere with each other. In each time sot, we can successfuy transmit over nodes in a subset of L that form an independent set (i.e., that are not directy connected in CG). We ca each such independent set as a feasibe schedue, and denote it as S = (S ) L, where S = if ink is active and S = 0 is ink is inactive in the schedue. We aso treat S as a set of active inks and write S if S =. We use to denote the cardinaity of the set S. We further ca a feasibe schedue as maxima if no more nodes in CG can be added without vioating the interference constraint. As maxima schedues represent extreme points in the space of feasibe schedues, we coect them in the set S. Then, we can define the capacity region Λ as the convex hu of S and L-dimensiona a-zero vector, which wi give the upper bound on the achievabe ink rates in packets per sot that can be supported by the network under stabiity for the given interference mode. Given the topoogy and the interference mode of a wireess network, we define the scheduing diversity of ink L as the number of different maxima schedues m that ink beongs to. Since each ink L beongs to at east one maxima schedue, m shoud be the integer greater than or equa to. For a network topoogy with a compete N-partite confict graph 2, we have m =, L. As another exampe, a singe-hop wireess network where a inks interfere with each other, we have m = for a. Less triviay, a 2 2 switch has 2-partite confict graph in which each maxima schedue has ony 2 inks, and m = for each. Roughy speaking, the scheduing diversity increases as the network diameter 3 increases. Such a behavior can be observed directy in a inear network with L inks under the primary interference mode: for L 3, m = for a ; for L 6, m 2 for a. In its simpest form, a scheduer determines a maxima feasibe schedue S[t] S at each time sot t. This seection The convex hu of the set V is the minima convex set containing set V. 2 In a compete N-partite confict graph, the nodes are partitioned into N sets of nodes without a ink between them such that every node in each set is connected to a the nodes outside of that set. 3 Network diameter is the maximum of the shortest hop-count between any two nodes in the graph.

3 3 may be infuenced by the earier experiences of each transmitter, and may be performed through a variety of strategies. Here, we are not interested in the means of seecting schedues, but in the eventua seection modeed as a probabiistic function of the queue-ength state of the network. Before we define the cass of randomized scheduers we consider more expicity, we need to estabish the traffic and the queueing modes. For simpicity, we assume a per-ink traffic mode 4, where A [t] arrivas occur to ink in sot t that are independenty distributed over inks and identicay distributed over time with mean λ, and A [t] K for some K < 5. Accordingy, a queue is maintained for each ink L with Q [t] denoting its queue ength at the beginning of time sot t. Reca from above that S [t] denotes the number of potentia departures at time t. Further, we et U [t] denote the unused service for the queue in sot t. If the queue is empty and is schedued, then U [t] is equa to ; otherwise, it is equa to 0. Then, the evoution of the queue is described as foows: Q [t + ] = Q [t] + A [t] S [t] + U [t], L. () We define F := set of non-negative, nondecreasing and differentiabe functions f( ) : R + R + with im f(x) =. x We say that the queue is f-stabe for a function f F if it satisfies T im sup E[f(Q [t])] <. (2) T T t=0 We note that this is an extended form of the more traditiona strong stabiity condition (see [5]) that coincide when f(x) = x. Moreover, it is easy to show that f-stabiity impies strong stabiity when f is aso a convex function. We say that the network is f-stabe if a its queues are f-stabe. Accordingy, we say that a scheduer is f-throughput-optima if it achieves f- stabiity of the network for any arriva rate vector λ = (λ ) L that ies stricty inside the capacity region Λ. Again, in the specia case of f(x) = x, the notion of f-throughput-optimaity reduces to traditiona throughput-optimaity, and when f is convex, f-throughput-optimaity impies throughput-optimaity. B. Distributed Agorithms The operation of many existing scheduers are governed by probabiistic aws (e.g., [25], [3], [4], [7], [20], [3]). This is not ony because they mode possibe errors in the scheduing process, but aso because they aow significant fexibiities in the deveopment of ow-compexity and distributed impementations. Of particuar interest in this cass of probabiistic scheduers are distributed CSMA-based agorithms (e.g., [7], [5], [9]). We give the definition of continuous time CSMA agorithm for competeness. 4 This assumption can be reaxed by utiizing backpressure type routing strategy (see, for exampe, [26]), which is avoided for unnecessary compications. 5 We note that the boundedness assumption on the arriva process simpifies the technica arguments, but can be reaxed (see, for exampe, [4]) to the ess strict assumption of E[A 2 (t)] <. Definition (CSMA Agorithm): Each ink independenty generates an exponentiay distributed random variabe with rate f(q [t]) and starts transmitting after this random duration uness it senses another transmission before. If ink senses the transmission, it suspends its backoff timer and resumes it after the competion of this transmission. The transmission time of each ink is exponentia distributed with mean. A common characteristics of these CSMA-based scheduers is the product form (see Definition 4) of the mapping of the tota queue-ength eves to the probabiity of the associated schedue. Such a mapping has been observed to cosey approximate the operation of the throughput-optima centraized MWS [26], and hence aso possesses throughput optimaity characteristics. However, these CSMA-based agorithm cannot be directy extended to operate under stochastic network dynamics or sophisticated appication requirements. As an important exampe, extending CSMA soutions to serving traffic with strict deadine constraints under wireess fading channes is difficut for two reasons: () the mixing time of the underying CSMA Markov Chain grows with the size of the network, which, for arge networks, generates unacceptabe deay for deadine-constrained traffic; (2) since the dynamic CSMA parameters are infuenced by the arriva and channe state process, the underying CSMA Markov Chain may not converge to a steady-state under strict deadine constrains and fading channe conditions. Thus, designing an optima distributed scheduing agorithm in deadine-constrained scheduing over fading channes becomes very chaenging. In a recent work [], we have found that, in some specia network topoogies, the foowing Fast carrier sensing mutipe access (FCSMA) agorithm can guarantee optima performance. Definition 2 (FCSMA Agorithm): At the beginning of each time sot t, each ink independenty generates an exponentiay distributed random variabe with rate f(q [t]), and starts transmitting after this random duration uness it senses another transmission before. The ink that captures the channe transmits its packets 6 unti the end of the sot and restarts in the next time sot. We note that FCSMA differs from CSMA in that it restarts every time sot, and hence increases the probabiity of meeting deadine requirements. Further remarks on FCSMA are shown as foows: Remarks: () Consider a compete N-partite confict graph, where each ink ony beongs to one schedue. If a queue engths are arge enough, then the ide duration in each sot wi quicky vanish, and FCSMA agorithm reaches one of the maxima schedue and stick to it for one time sot. Thus, the FCSMA agorithm serving a schedue S with probabiity P S = {S :S S} i S f(q i) j S f(q j) 6 If there are no packets awaiting in the ink, it transmits a dummy packet to occupy the channe. (3)

4 4 is f-throughput-optima, which is proven in Section IV. An interesting observation is that this scheduer does not approximate the MWS operation when queue-engths are arge, as CSMA does. For exampe, consider a 2 2 switch topoogy, where there are ony two maxima schedues S incuding two active inks and 2 and S 2 incuding two active inks 3 and 4. Suppose a queue engths are arge enough and Q + Q 2 = 0.5(Q 3 + Q 4 ), then the MWS chooses the schedue S 2 and CSMA agorithm seects the schedue S 2 with probabiity very cose to. However, FCSMA poicy chooses the schedue S 2 with probabiity cose to 2/3. This indicates the importance of understanding scheduers with fundamentay different behavior than MWS. (2) In a fuy connected network topoogy, due to its fast absorption time and quick adaptation to arriva and channe state processes, FCSMA poicy yieds significant advantages over traditiona CSMA poicies that evoves sowy to their steadystate, especiay in scheduing deadine constrained traffic over wireess fading channes. We refer the interested reader to [] for more detaied investigation of FCSMA operation. It is aso worth noting that for a given stationary distribution, it is possibe to construct a Markov Chain that converges to it by Metropois agorithm [6] or Gauber dynamics (e.g., [2], [6]). Yet, in this paper, we do not focus on the design of specific scheduing agorithms that can converge to the stationary distribution. Instead, we are interested in the throughput-optimaity characteristics of a wide cass of probabiistic mapping from the queue ength space to the feasibe schedues. In the foowing, we consider three casses of randomized scheduers which not ony mode many existing probabiistic scheduers as specia cases but aso contain a much wider casses of potentia scheduers that have not been anayzed. C. Randomized Scheduers In this subsection, we identify three casses of randomized scheduers that differ in the operation of the functiona forms used in them. Definition 3 (RSOF Scheduer): For a given f F and queue-ength vector Q at the beginning of a sot, the Ratio-of- Sum-of-Functions (RSOF) Scheduer picks a schedue S S in that sot such that i S P S (Q) := f(q i) j S f(q, for a S S. (4) j) {S :S S} Definition 4 (RMOF Scheduer): For a given f F and queue-ength vector Q at the beginning of a sot, the Ratio-of- Mutipication-of-Functions (RMOF) Scheduer picks a schedue S S in that sot such that υ S (Q) := {S :S S} i S f(q i) j S f(q, for a S S. (5) j) Definition 5 (RFOS Scheduer): For a given f F and queue-ength vector Q at the beginning of a sot, the Ratio-of- Function-of-Sums (RFOS) Scheduer picks a schedue S S in that sot such that f( i S π S (Q) := Q i) {S :S S} f( j S Q, for a S S. (6) j) Note that a the RSOF, RMOF and RFOS Scheduers are more ikey to pick a schedue with the arger queue ength, but with different distributions based on their form and the form of f F. In particuar, the steepness of the function f determines the weight given to the heaviy oaded ink in both RSOF and RMOF Scheduers and the heaviy oaded schedue in the RFOS Scheduer. Aso, note that the scheduers coincide in singe-hop network topoogies because each maxima schedue ony incudes one ink in such networks, and for the foowing choices of f: when f(x) = x, the RSOF and RFOS Scheduers coincide; when f(x) = e x, the RMOF and RFOS Scheduers coincide. These three casses cover a wide variety of scheduers incuding many of existing throughput-optima scheduers. For exampe, when f(x) = e x, the RMOS and RFOS Scheduers correspond to the throughput-optima CSMA poicy operating under time-scae separation assumption that attracted a ot of attention atey (see [7], [20], [9]); in a compete N-partite confict graph, the RSOF Scheduer corresponds to the FCSMA poicy when a the queue engths are arge enough. Yet, they aso contain a much wider set of scheduers, one for each f. The aim of this work is to identify the imitations of randomization for a wide cass of randomized dynamic scheduers that utiize functions of queue-engths to schedue transmissions. Even though randomization has significant advantage in owcompexity or distributed impementation, it causes inaccurate operation and may be hurtfu if not performed within imitations. In this work, we find that the performance of the randomized scheduers may especiay be sensitive to the topoogy of the confict graph and the functiona form used in the weighting. To see this, consider one maxima schedue S incuding three active inks, 2 and 3 in a 3 3 switch topoogy. We assume that arrivas ony happen to those 3 inks at rates λ, λ 2 and λ 3 with the constraints that λ i [0, ) for a i =, 2, 3, which ceary can be supported by a simpe poicy that aways serves the schedue S. Thus, by setting λ i arbitrariy cose to one for each i, this simpe poicy can achieve a sum rate of 3 λ i < 3. However, for a RFOS Scheduer with f(x) = x, we can easiy cacuate that 3 θ i = 2, where θ i (i =, 2, 3) is the probabiity of serving ink i. Thus, the RFOS Scheduer with f(x) = x cannot achieve fu capacity region in a 3 3 switch. Yet, in the same set up, if we use f(x) = e x instead of f(x) = x in the RFOS Scheduer, the mapping has the same probabiistic form as the CSMA poicy, and thus woud be throughput-optima. This shows the significant impact of the functiona form on the throughput performance of randomized scheduers. In addition, the RFOS Scheduer with f(x) = x is shown to be f-throughput-optima in a 2 2 switch (see Figure 3), which indicates that the network topoogy may aso affect the throughput performance of randomized scheduers.

5 5 F A x x e e x x (, 0) (0, 0) B C x,(og( x )) ( 0) Reca that since m denotes the number of different maxima schedues that ink beongs to, it may be viewed as a rough measure of the network diameter. Then, the main resuts for the RSOF and RFOS Scheduers are presented in Figures 2 and 3, respectivey. In these figures, we aso incude severa conjectures that are vaidated through simuations in Section VI. m Fig. : The reationship between casses A, B and C. Next, we identify the three casses of functions with varying forms that turn out to be crucia to our investigation. Definition 6: We consider the foowing subsets of F: (a) A := {f F : ɛ > 0, im x f(x + a) (b) B := {f F: im x f(x) f(( + ɛ)x) = 0}. =, for any a R}. f(x) (c) C := {f B: there exist K and K 2 satisfying 0 < K K 2 < such that K (f(x ) + f(x 2 )) f(x + x 2 ) K 2 (f(x ) + f(x 2 )), for a x, x 2 0}. We ca A as the cass of exponentia functions and C as the cass of sub-exponentia functions. The key exampes of functions with sets A, B, C and their interreationship are extensivey studied in Appendix A. Figure concisey demonstrates the most critica facts: that A and C are non-overapping casses; whie B has an intersection with A. Furthermore, the exampe functions are provided with a variety of forms that justify the names assigned to A and C : A contains rapidy increasing functions generay with exponentia forms; whie C contains sub-exponentiay increasing poynomia and ogarithmic functiona forms. In the study of necessary and sufficient conditions for throughputoptimaity, we sha find that most of the resuts depend on which of these three functiona casses the functions beong to. III. OVERVIEW OF MAIN RESULTS In this section, we present our main findings and resuting insights on the throughput-optimaity of the RSOF, RMOF and RFOS Scheduers (see Definitions 3, 4 and 5) with different functiona forms under different network topoogies. These resuts are rigorousy proven in Sections IV and V. To faciitate an accessibe figurative presentation, in the horizonta dimension, we conceptuay order the functions in F in increasing eve of steepness starting from f(x) = (og(x + )) α and f(x) = x α for any α > 0 that beong to C, foowed by f(x) = e xα x β for any 0 < α < and any β 0 that beongs to B A, and finishing with f(x) = e xα for any α and any x β β 0 that beongs to A. In the vertica dimension, we use the scheduing diversity (m ) L introduced in Section II to distinguish different topoogica and interference scenarios. m 2, m, unknown f-throughput-optima (og(x+)) x x x ( >0) (0< <) ( >) cass of sub-exponentia functions C non-throughput-optima unknown f-throughputoptima exp(x )/x (0< <, 0) unknown conjecture: f-throughput-optima throughput-optima (singe-hop network) exp(x )/x (, 0) cass of exponentia functions A network with high scheduing diversity network with ow scheduing diversity f Fig. 2: The throughput performance of the RSOF Scheduer. m m 2, m, non-throughput-optima unknown conjecture: f-throughput-optima f-throughput-optima (singe-hop network) (og(x+)) x x x ( >0) (0< <) ( >) cass of sub-exponentia functions C unknown Throughput-optima exp(x )/x exp(x )/x (0< <, 0) (, 0) cass of exponentia functions A network with high scheduing diversity network with ow scheduing diversity f Fig. 3: The throughput performance of the RFOS Scheduer. From Figure 2, we see that the RSOF Scheduer with the function f B is f-throughput-optima when m =, L. Aso, the RSOF Scheduer with the function f A \ B is throughput-optima in singe-hop network topoogies since the RSOF and RFOS Scheduers have the same probabiity distribution over schedues in such networks and the RFOS Scheduer with the function f A is throughput-optima (see Figure 3). However, if min L m 2, the RSOF Scheduer with any function f F cannot be throughput-optima. Thus, roughy speaking, the RSOF Scheduer is non-throughputoptima for the network with high scheduing diversity, whie the RSOF Scheduer with the function f B is f-throughputoptima for ow scheduing diversity. We note that athough the throughput performance of the RSOF Scheduer with some exponentia functions f A \ B (i.e. f(x) = x β e xα, α and β 0) is not yet expored in genera topoogies with m =, L, we conjecture that it is f-throughput-optima in this region, since the RSOF Scheduer with such functions reacts much more quicky to the queue ength difference

6 6 between schedues than that with sub-exponentia functions, especiay under asymmetric arriva patterns. We vaidate this conjecture through simuations in Section VI. Overa, the RSOF Scheduer is more sensitive to the network topoogy than the functiona form used in it. The horizonta unknown region corresponds to network topoogies where some inks have scheduing diversity and other inks have scheduing diversity at east 2. The vertica unknown region corresponds to randomized scheduers with functions that are not in the functiona casses A, B and C. In Figure 3, we observe that the RFOS Scheduer with the function f A is throughput-optima under any network topoogy. Aso, the RFOS Scheduer with the function f C is f-throughputoptima in singe-hop network topoogies, which foows from the fact that the RFOS and RSOF Scheduers have the same probabiity probabiistic forms in such networks, the resut that the RSOF Scheduer with the function f B is f-throughputoptima (see Figure 2) and the fact that C B. Aso, when the function f is inear, the RFOS Scheduer has the same probabiity form with the RSOF Scheduer and thus is f- throughput-optima when m =, L. However, the RFOS Scheduer with the function f C is not throughput-optima when min L m 2. Roughy speaking, the network with higher scheduing diversity requires much steeper functions (e.g., exponentia functions) for the throughput-optimaity of the RFOS Scheduer. Whie the throughput performance of the RFOS Scheduer with the function f C \ {inear functions} for genera network topoogies with m =, L is part of our ongoing work, we conjecture that it is f-throughput-optima in those topoogies since both RFOS and RSOF Scheduers with sub-exponentia functions have amost the same reaction speed to the queue ength difference between schedues. We aso vaidate this conjecture via simuations in Section VI. Overa, the RFOS Scheduer is more sensitive to the functiona form used in it than the network topoogy. The RMOF Scheduer with the function f satisfying og f B and f(0) is (og f)-throughput-optima under any network topoogy. This resut together with the RFOS Scheduer with the function f A extends the throughput-optimaity of CSMA scheduers (e.g. [7], [9]) to a wider cass of functiona forms. Whie this resut proves a weaker form of throughputoptimaity than f-throughput-optimaity for the RMOF Scheduer, we note that the RMOF Scheduer generay outperforms the RFOS and RSOF Scheduers in numerica investigations. Hence, we eave it to future research to strengthen this resut. Coectivey these resuts not ony highight the strengths and weaknesses of the three functiona randomized scheduers, they aso revea the interreation between the steepness of the functions and the scheduing diversity of the underying wireess networks. This extensive understanding of the imitations of randomization may motivate the network designers to use or avoid certain types of probabiistic scheduing strategies depending on the topoogica characteristics of the network. IV. SUFFICIENT CONDITIONS In this section, we study the sufficient conditions on the network s topoogica characteristics and the functions used in the RSOF, RMOF and RFOS Scheduers to achieve throughputoptimaity. A. f-throughput-optimaity of the RSOF Scheduer We study the throughput performance of the RSOF Scheduer for a network topoogy with m =, L. In such a network, each ink beongs to ony one maxima schedue. Lemma : If N λ i <, λ i > 0, and a i 0, for i =,..., N, then there exists a δ > 0 such that N a 2 i λ i ( N a i ) 2 ( + δ). (7) Proof: See Appendix B for the proof. Theorem : In a network topoogy with the scheduing diversity of each ink equa to, i.e., m =, L, the RSOF Scheduer with the function f B is f-throughput-optima. Proof: We assume that there are ony N avaiabe maxima schedues. Let S i (i =,..., N) denote the i th maxima schedue. In each maxima schedue S i, there are active inks. We use (S i, =,..., ) to denote the sequence of active inks in the maxima schedue S i. Note that we use i to index maxima schedue and to index ink. Since the schedue diversity of each ink is equa to, each ink beongs to ony one maxima schedue. Thus, we can denote the queues, arrivas, and scheduing statistics in terms of maxima schedues for easier exposition. To that end, we et Q i, λi and (i =,..., N, =,..., ) denote the queue-ength of ink S i, the average arriva rate for the ink S i and the probabiity of serving the ink S i, respectivey. In addition, A i [t], Si [t] and U i [t] denote the number of arrivas to ink S i at time sot t, the number of potentia departures of ink S i in sot t and the unused service for ink S i at time sot t, respectivey. Reca that each ink can ony beong to one maxima schedue and note that inks in different maxima schedues cannot be active at the same time. Thus, the capacity region for such network is P i C N := {λ : N λ i i <, i =,..., }. (8) Under the above notation, the RSOF Scheduer becomes : P S i = f(qi ) N k i =,..., N. (9) k= f(qk ), Note that P i = P S i, for =,...,. If λ i = 0 for some i and, then no arrivas occur in the ink S i. Thus, we don t need to consider such inks. In the rest of proof, we assume λ i > 0 (i =,..., N, =,..., ). Consider the Lyapunov function V (Q) := N h(q i ), where h (x) = f(x). By λ i using Lemma, it is shown in the Appendix C that there exist

7 7 positive constants γ and G such that V : = E [V (Q[t + ]) V (Q[t]) Q[t] = Q] γ N f(q i ) + G. (0) By using the Theorem 4. in [7], inequaity (0) impies the desired resut. B. Throughput-Optimaity of RMOF and RFOS Scheduers In this subsection, we investigate the sufficient condition for the throughput-optimaity of RMOF and RFOS Scheduers. Theorem 2: (i) The RMOF Scheduer with the function f F satisfying og f B and f(0) is (og f)-throughputoptima under any network topoogy; (ii) The RFOS Scheduer with the function f A is throughput-optima under any network topoogy. Proof: To prove this, we use a simiar approach as in [9] that uses the foowing resut from [4]: for a scheduing agorithm, given any 0 ɛ, δ <, there exists an M > 0 for which the scheduing agorithm satisfies the foowing condition: in any time sot t, with probabiity greater than δ, the scheduing agorithm chooses a schedue x[t] S that satisfies: x[t] w(q [t]) ( ɛ) max x S x[t] w(q [t]), whenever Q[t] > M, where Q[t] := (Q [t]) L, and w B. Then the scheduing agorithm is w-throughput-optima. (i) Given any ɛ and δ such that 0 ɛ, δ <. Let X := {x S : x og f(q [t]) < ( ɛ )W [t]}, () where W [t] := max x S x og f(q [t]). Then, we have υ(x ) = x υ x = f(q [t]) x X x X x S x f(q [t]) x X exp [ x og f(q [t]) ] = < x S exp [ x og f(q [t]) ] X exp [( ɛ )W [t]] x S exp [ x og f(q [t]) ]. Since x S exp [ x og f(q [t]) ] exp(w [t]), then we get υ(x ) < X exp [( ɛ )W [t]] exp(w [t]) = X exp(ɛ W (2) [t]). If some queue engths increase to infinity, then W [t] and thus we have υ(x ) 0. Hence, there exists a M > 0 such that Q[t] > M and the RMOF Scheduer with the function f F satisfying og f B and f(0) picks the schedue S[t] S \ X with probabiity δ and thus is og f-throughput-optima under any topoogy. (ii) Given any ɛ 2 and δ 2 such that 0 ɛ 2, δ 2 <. Let W 2 [t] := max x S x Q [t], and X 2 := {x S : x Q [t] < ( ɛ 2 )W 2 [t]}. Then, by using the same technique as in (i), we can prove that the RFOS Scheduer with f A is throughputoptima under any topoogy. V. NECESSARY CONDITIONS So far, we have shown that the RSOF Scheduer with the function f B is f-throughput-optima in the network topoogy with m =, L and the RFOS Scheduer with the function f A is throughput-optima under arbitrary network topoogies. However, the next resut estabishes that in network topoogies where each ink beongs to two or more schedues (i.e. when min L m 2), the RSOF Scheduer with any function f F and RFOS Scheduer with the function f C cannot be throughput-optima. Theorem 3: If the network is such that min L m 2, then (i) RSOF Scheduer is not throughput-optima for any f F; (ii) RFOS Scheduer is not throughput-optima for any f C. Proof: We prove these caims constructivey by considering an arriva process that is inside the capacity region, but is not supportabe by the randomized scheduers for the given functiona forms. To that end, et us consider any maxima schedue S 0 S and index its inks as {, 2,..., n} for convenience. We assume that arrivas ony happen to those n inks at rates λ,, λ n with the constraint that λ [0, ) for a =,, n, which is ceary supportabe by a simpe scheduing poicy that aways serves the schedue S 0. Thus, setting λ arbitrariy cose to one for each, this simpe poicy can achieve a sum rate of n λ < n. We define M = {S S : S S 0 }, K = S \ M, H = M \ {S 0 } and T = S \ {S 0 }. In the rest of the proof, we use AB to denote the intersection of A and B. Given this construction, we next prove the foowing statements for the RSOF and RFOS Scheduers respectivey: () If n λ > n 2, the RSOF Scheduer with any function f F is unstabe. (2) If n λ > n K 2K, where K 2 and K 2 are positive constants described in Appendix A, the RFOS Scheduer with the associated function f C is unstabe. Since the aforementioned simpe scheduer can stabiize the sum rate n λ < n, the RSOF Scheduer with any function f F and RFOS Scheduer with the associated function f C are not throughput-optima. We next prove these caims that compete the proof of Theorem 3. () Under the above mode, the RSOF Scheduer becomes P S = SS 0 f(q ) + \ S 0 f(0) S :S S ( S S 0 f(q ) + \ S 0 f(0)). Let P denote the probabiity that ink S 0 is served, then n n P = P S S M: SS 0 =:L {}}{ n ( f(q i ) + \ S 0 f(0)) S M: SS = 0 i SS 0. f(q ) + \ S 0 f(0) S:S S SS 0 S:S S }{{} =:L 2

8 8 S:S S Since SS 0 f(q ) = n f(q ) S M: SS 0, n S M: SS 0 \S 0 f(0) = S:S S S 0 \S 0 f(0), and n f(q i ) = f(q i ) S M: SS 0 i SS 0 S:S M SS 0 i SS 0 = n S 0 f(q i ) = f(q ) S 0, S:S M i SS 0 S M: SS 0 we can extend L and L 2 as foows: n L = f(q ) S 0 + S 0 \ S 0 f(0) S M: SS 0 S:S S n = f(q )(n + HS 0 ) + TS 0 T \ S 0 f(0), H H: HS 0 T:T T and n L 2 = f(q ) + \ S 0 f(0) S M: SS 0 S:S S n = f(q )( + ) + T \ S 0 f(0). H H: HS 0 T:T T Thus, we have n P = L L 2 = n Z Z 2, (3) where Z = n f(q ) H H: HS 0 (n HS 0 ) + T:T T (n TS 0 ) T \ S 0 f(0), and Z 2 = n f(q )( + H H: HS 0 ) + T:T T T \ S 0 f(0). Note that HS 0 n, for H H, and TS 0 n, for T T. Now, since m = S S: S 2, S 0, we have H H: HS 0, S 0. Then, we get n Z f(q ) H H: HS 0 + T:T T T \ S 0 f(0) Z 2 2 n f(q ) H H: HS T:T T T \ S 0 f(0) = 2. Thus, we have n P n 2. Hence, for topoogies where min L m 2, if n λ > n 2, in which case the tota arriva rate is greater than the tota service rate, then, the RSOF Scheduer is unstabe by foowing the Theorem 2.8 and Theorem 2.5 in [7]. Since n f( S M: SS 0 we have n P = i SS 0 Q i ) = S:S M S 0 f( S:S M S 0 f( SS 0 Q ) S:S M f( SS 0 Q ) + S:S K f(0) i SS 0 Q i ), nf( n = Q ) + H:H H HS 0 f( HS 0 Q ) f( n Q ) + H:H H f( HS 0 Q ) + S:S K f(0) H:H H =n (n HS 0 )f( HS 0 Q ) + n S:S K f(0) f( n Q ) + H:H H f( HS 0 Q ) + S:S K f(0). The fact that f C impies that there exist K and K 2 satisfying 0 < K K 2 < such that K m f(q i) f( m Q i) K 2 m f(q i), for m =,..., n, where Q i 0, i =,..., m, which foows from induction. Then, we have n P n K H:H H K 2 (n HS0 ) HS f(q ) + n 0 S:S K f(0) n f(q ) + H:H H HS f(q ) + 0 S:S K f(0) =n K K 2 n f(q ) H H: HS (n HS 0 0 ) + n S:S K f(0) n f(q ) + n f(q ) H H: HS + 0 S:S K f(0). Note that HS 0 n, for H H, and that m = S S: S 2, S 0, impies that H H: HS 0, S 0. Then, we get n P n K K 2 n f(q ) H H: HS 0 + S:S K f(0) 2 n f(q ) H H: HS S:S K f(0) n K. (5) 2K 2 Thus, by foowing the same argument as in the proof for statement (), we know that when min L m 2 and n λ > n K 2K, the RFOS Scheduer is unstabe. 2 VI. SIMULATION RESULTS In this section, we first perform numerica studies to vaidate the throughput performance of the proposed randomized scheduers with different functions in 2 2 and 3 3 switch topoogies. Then, we evauate the impact of functiona forms on the deay performance of proposed randomized scheduers in 2 2 switch topoogies. (2) With the same mode, the RFOS Scheduer becomes f( SS π S = 0 Q ) S :S M f( S S 0 Q ) + (4) S :S K f(0). Then, n P = n π s S M: SS 0 n S M: SS = 0 f( i SS 0 Q i ) S:S M f( SS 0 Q ) + S:S K f(0). A. Throughput Performance In a 2 2 switch, the scheduing diversity of each ink is and thus a proposed randomized scheduers are proven to be throughput-optima. In a 3 3 switch, the scheduing diversity of each ink is 2, for which the RFOS Scheduer needs to carefuy choose the functiona form to preserve the throughput optimaity whie the RSOF Scheduer is not f- throughput-optima with any function f F In a 2 2 switch, we consider arriva rate vector λ = ρh, where H = [H ij ] is a douby-stochastic matrix with H ij

9 9 Average Queue Length RSOF or RFOS with f(x)=x RSOF with f(x)=x 2 RSOF with f(x)=e x RFOS with f(x)=x 2 RFOS or RMOF with f(x)=e x RMOF with f(x)=x+ Average Queue Length RSOF or RFOS with f(x)=x RSOF with f(x)=x 2 RSOF with f(x)=e x RFOS with f(x)=x 2 RFOS or RMOF with f(x)=e x RMOF with f(x)=x+ Average Queue Length RSOF or RFOS with f(x)=x RSOF with f(x)=x 2 RSOF with f(x)=e x RFOS with f(x)=x 2 RMOF with f(x)=x+ RMOF or RFOS with f(x)=e x Time Step x 0 4 (a) Symmetric arrivas in a 2 2 switch Time Step x 0 4 (b) Asymmetric arrivas in a 2 2 switch Fig. 4: The throughput performance vaidation of the randomized scheduers Time Step (c) 3 3 switch Average Queue Length 0 0. RSOF with e x RSOF with x+ RSOF with og(x+e) RFOS with e x RFOS with x+ RFOS with og(x+e) RMOF with e x RMOF with x+ RMOF with og(x+e) MWS Load Factor Average Queue Length 00 0 RSOF RFOS RMOF MWS og(x+e) x Load Factor e x (a) Symmetric arrivas in a 2 2 switch (b) Asymmetric arrivas in a 2 2 switch Fig. 5: Deay performance comparison of the randomized scheduers with different functiona forms denoting the fraction of the tota rate from input port i that is destined to output port j. Then, ρ (0, ) represents the average arriva intensity, where the arger the ρ, the more heaviy oaded the switch is. We present two cases: symmetric arriva process (H = [ ; ]) and asymmetric arriva process (H 2 = [0. 0.9; ]) under high arriva intensity ρ = From Figures 4(a) and 4(b), we can observe that a randomized scheduers can stabiize the system under symmetric and asymmetric arriva traffics. So, there is a wide cass of choices under which the randomized scheduing can guarantee the throughput performance in the 2 2 switch. In addition, we can see that the RSOF Scheduer with the exponentia function and the RFOS Scheduer with the square function are aso stabe in both symmetric and asymmetric arriva processes, which support our conjecture in Section III that the RSOF Scheduer with the function f A and the RFOS Scheduer with the function f B are f-throughput optima in network topoogies with m =, L. In a 3 3 switch, we consider arriva rate vector λ = [ ; ; ], where the RSOF Scheduer with any function f F and the RFOS Scheduer with any function f C cannot stabiize. The evoution of average queue ength per ink over time for different scheduers with different functions are shown in figures 4(c). From Figure 4(c), we can observe that the average queue engths of the RSOF Scheduers with inear function, square function and even exponentia function increase very fast, which vaidates our theoretica resut that the RSOF Scheduer with any function f F cannot be throughput-optima in network topoogies with min L m 2. In addition, we can see that the average queue engths of the RFOS Scheduers with inear function and square function grow quicky whie the RFOS Scheduer with exponentia function aways keeps ow queue ength eve, which demonstrates that the steepness of functiona form needs to be high enough for the RFOS Scheduer to keep throughput optimaity in genera network topoogies. Even though our resut indicates that the RMOF Scheduer with any function

10 0 f satisfying og f B and f(0) is (og f)-throughputoptima in genera network topoogies, we can see that the RMOF Scheduer is sti stabe even with inear function. This vaidates that our conjecture that the RMOF Scheduer with any function f F can be f-throughput-optima in genera network topoogies. B. Deay Performance In this subsection, we perform numerica studies to evauate the deay performance of proposed randomized scheduers with different functions in a 2 2 switch topoogy. From Figure 5(a), we can observe that, under symmetric arriva traffic, the deay performance is highy insensitive to the choice of the randomization and the functiona form being used in it especiay under high arriva oad. So, there is a wide cass of choices under which the randomized scheduing can yied good deay performance. On the other hand, Figure 5(b) demonstrates that, under asymmetric arriva traffic, the RMOF Scheduer is more robust to the choice of functions used in it than both the RSOF and RFOS Scheduers. In particuar, it appears that the steepness of f needs to be high enough for each randomization to yied good deay performance. Generay, the RMOF Scheduer outperforms the other two randomized scheduers especiay under asymmetric arriva traffic. In a cases, the RSOF and RFOS Scheduers have simiar performance and MWS has the best deay performance. Whie these numerica studies indicate a number of interesting facts on the mean deay performance of randomized scheduers, we eave a more carefu deay performance comparison to future research. There is ceary a need for a deeper investigation of deay performance of throughput-optima scheduers. This work forms the foundation to investigate these higherorder performance metrics in our future research. () In B, if im x f(x+a) f(x) exists for any a R, then this imit shoud be equa to. Indeed, et im x f(x+a) f(x) = b for any a R, where b > 0. Then b = im x f(x+2) f(x) = f(x+2) im x f(x+) f(x+) f(x) = b 2. Thus, b =. (2) If the definition of C is not constrained by the set B, then C is not necessariy a subset of B. In fact, we can construct a f(x+a) function f C for which im x f(x) does not exist and hence f B. (3) In C, if f F, then the ower bound of f(x + x 2 ) aways exists. Aso if there exists w > 0 such that f(2x) wf(x) for any x 0, then the upper bound of f(x + x 2 ) aways exists. Indeed, since f( ) is nondecreasing, f(x + x 2 ) f(x i ), for i = or 2. Hence f(x + x 2 ) 2 (f(x ) + f(x 2 )). Thus, et K = 2, then we aways have K (f(x )+f(x 2 )) f(x +x 2 ). On the other hand, f(x + x 2 ) max{f(2x ), f(2x 2 )} f(2x ) + f(2x 2 ) w(f(x ) + f(x 2 )). Thus, et K 2 = w, we have f(x + x 2 ) K 2 (f(x ) + f(x 2 )). (4) If f C, then given n N, there exist K and K 2 satisfying 0 < K K 2 < such that K m f(x i) f( m x i) K 2 m f(x i), for m =,..., n, where x i 0, i =,..., m. This directy foows from the induction. (5) A f(2x) C =. Indeed, if f A, then im x f(x) =. Thus, for any c > 0, M > 0 such that f(2x) > cf(x) for any x > M. Hence, f C. On the other hand, if f C, then d > 0 f(2x) such that f(2x) df(x). Hence, im sup x f(x) d and thus f A. APPENDIX B PROOF FOR LEMMA VII. CONCLUSIONS We expored the imitations of randomization in the throughput-optima scheduer design in a generic framework under the time-scae separation assumption. We identified three important functiona forms of queue-ength-based scheduers that covers a vast number of dynamic scheduers of interest. These forms differ fundamentay in whether they work with the queue-ength of individua inks or whoe schedues. For a of these functiona forms, we estabished some sufficient and some necessary conditions on the network topoogy and the functiona forms for their throughput-optimaity. We aso provided numerica resuts to vaidate our theoretica resuts and conjectures, which wi be further studied in our future work. APPENDIX A PROPERTIES OF FUNCTIONAL CLASSES The foowing remarks expore more properties of casses A, B and C. Proof: If n =, because λ (0, ), by assumption, there exists a 0 < δ < λ, such that a2 λ a 2 ( + δ ). Assume that n = k, it is true. That is, if k λ i < and λ i > 0 (i =,..., k), then there exists a δ k = δ(λ,..., λ k ) > 0 such that λ a λ k a 2 k (a a k ) 2 ( + δ k ). (6) Then for n = k + and λ λ k + λ k+ <, we have a a 2 k + a 2 k+ λ λ k λ k+ = a a 2 k λ λ k + ( λ k + λ k+ a 2 k + λ k + λ k+ a 2 λ k + λ k+ λ k λ k+) k+ [ λ k + λ k+ a a k + a 2 k λ + λ k + λ k+ a 2 k+ k λ k+ ( + δ k+ ) (by assumption). (7) ] 2

11 Since λ k + λ k+ λ k a 2 k + λ k + λ k+ λ k+ a 2 k+ (a k + a k+ ) 2 = λ k+ a 2 k + λ k a 2 k+ 2a k a k+ λ k λ k+ λ k+ 2 a 2 k λ λ k a 2 k+ k λ 2a ka k+ = 0, (8) k+ hence λ k + λ k+ λ k a 2 k + λ k + λ k+ λ k+ a 2 k+ (a k + a k+ ). (9) Thus, equation (7) becomes k+ k+ a 2 i ( a i ) 2 ( + δ k+ ). λ i APPENDIX C PROOF OF INEQUALITY (0) V : = E [V (Q[t + ]) V (Q[t]) Q[t] = Q] N [ ] = E λ i (h(q i [t + ]) h(q i [t])) Q[t] = Q. By the mean-vaue theorem, we have h(q i [t+]) h(qi [t]) = f(r i[t])(qi [t + ] Qi [t]) = f(ri [t])(ai [t] Si [t] + U i[t]), where R i[t] ies between Qi [t] and Qi [t + ]. Hence, we get N [ ] V = E λ i f(r[t])(a i [t] S i [t] + U i [t]) Q[t] = Q N [ ] = E λ i f(r[t])u i i [t] Q[t] = Q + } {{ } =: V N [ E λ i f(r[t])(a i i [t] S i [t]) Q[t] = Q } {{ } =: V 2 ]. For V, if Q i [t] = Qi > 0, then U i[t] = 0. If Qi [t] = Qi = 0, then U i[t] may be equa to. But in this case, Qi [t + ] K (since A i [t] K). Hence, f(ri [t]) f(k) <. Thus, N [ ] V = E λ i f(r[t])u i i [t] Q[t] = Q {Q i =0} N N f(k) D f(k), (20) λ i where D := min{λ i } < and { } is the indicator function. Next, et s focus on V 2. We know that f(r i[t]) = f(qi [t]+ a i ) ( ai K). According to the definition of function f B, given ɛ > 0, there exists M > 0, such that for any Q i [t] = Q i > M, we have f(r i [t]) f(q i ) < ɛ, that is, ( ɛ)f(q i ) < f(r i[t]) < ( + ɛ)f(qi ). Thus, we have f(r i [t])(a i [t] S i [t]) =f(r i [t]) [ (A i [t] S i [t]) + (A i [t] S i [t]) ] ( + ɛ)f(q i )(A i [t] S i [t]) + ( ɛ)f(q i )(A i [t] S i [t]) =f(q i )(A i [t] S i [t]) + ɛf(q i ) A i [t] S i [t] f(q i )(A i [t] S i [t]) + Kɛf(Q i ), (2) where (x) + = max{x, 0}, (x) = min{x, 0}, and A i [t] S i [t] Ai [t] K. Thus, we divide V 2 into two parts: V 2 = + N N [ E f(r[t])(a i i [t] S λ i i [t]) Q[t] = Q ] {Q i >M} } {{ } =: V 3 ] [ E f(r[t])(a i i [t] S λ i i [t]) Q[t] = Q {Q i M} } {{ } =: V 4 For V 3, by using (2), we have N V 3 λ i + DKɛ f(q i )(λ i P i ) {Q i >M} N f(q i ) {Q i >M}, (22) where P i = E [ S i[t] Q[t] = Q] = et s consider the term N can be expressed as: N λ i f(q i )(λ i P i ) N N = f(q i ) ( N = Since N ( λ i f(q i ) λ i f(qi ))2 N ( f(q i ) λ i f(qi ) N k= k f(qk f(q i )(λi P i f(qi ) N k= k f(qk ). Next, ). ), which f(q i ) )( λ i f(qi )) N f(qi ). )( f(q i )) N λ i ( f(q i )) 2, where λ i = max {,..., } λ i, and by Lemma, there exists a δ > 0 such that N λ i ( N f(q i )) 2 ( f(q i )) 2 ( + δ), (23)

12 2 we have N ( Thus, we get f(q i ) λ i N λ i Hence, we have N λ i )( N f(q i )) ( f(q i )) 2 ( + δ). f(q i )(λ i P i ) δ f(q i )(λ i P i ) {Q i >M} N f(q i ). (24) N N δ f(q i ) {Q i >M} δ f(q i ) {Q i M} N λ f(q i )(λ i P i ) {Q i M} N N δ f(q i ) {Q i >M} + λ i f(q i )P i {Q i M} N N δ f(q i ) {Q i >M} + D f(m). (25) Thus, we can choose ɛ sma enough such that γ = δ DKɛ > 0, and thus we have N N V 3 γ f(q i ) {Q i >M} + D f(m) N N γ f(q i ) + (D + γ) f(m) For V 4, we have V 4 N λ i [ ] E λ i f(r[t]) A i i [t] S i [t] Q[t] = Q N N Kf(M + K) DK f(m + K). Thus, we get where G := D N K) + (D + γ) N V γ {Q i M} N f(q i ) + G, (26) f(k) + DK N f(m) <. REFERENCES f(m + [] L. Bui, A. Eryimaz, R. Srikant, and X. Wu. Joint asynchronous congestion contro and distributed scheduing for wireess networks. In Proc. IEEE Internationa Conference on Computer Communications. (INFOCOM), Barceona, Spain, Apri [2] P. Chaporkar, K. Kar, and S. Sarkar. Throughput guarantees through maxima scheduing in wireess networks. In Proc. Aerton Conference on Communication, Contro, and Computing (Aerton), Monticeo, Iinois, September [3] A. Eryimaz, A. Ozdagar, D. Shah, and E. Modiano. Distributed crossayer agorithms for the optima contro of mutihop wireess networks. IEEE/ACM Transactions on Networking, 8(2):638 65, 200. [4] A. Eryimaz, R. Srikant, and J. Perkins. Stabe scheduing poicies for fading wireess channes. IEEE/ACM Transactions on Networking, 3(2):4 424, [5] L. Georgiadis, M. Neey, and L. Tassiuas. Resource aocation and cross-ayer contro in wireess networks. Foundations and Trends in Networking, (): 44, [6] J. Ghaderi and R. Srikant. On the design of efficient csma agorithms for wireess networks, 200. Avaiabe onine at cache/ arxiv/pdf/003/ v.pdf. [7] L. Jiang and J. Warand. A csma distributed agorithm for throughput and utiity maximization in wireess networks. In Proc. Aerton Conference on Communication, Contro, and Computing (Aerton), Monticeo, Iinois, September [8] C. Joo, X. Lin, and N. Shroff. Understanding the capacity region of the greedy maxima scheduing agorithm in muti-hop wireess networks. In Proc. IEEE Internationa Conference on Computer Communications. (INFOCOM), Phoenix, Arizona, Apri [9] K. Kar, S. Sarkar, and L. Tassiuas. Achieving proportiona fairness using oca information in aoha networks. IEEE Transactions on Automatic Contro, 49(0): , [0] H. Lee, E. Modiano, and L. Le. Distributed throughput maximization in wireess networks via random power aocation. In Proc. IEEE Modeing and Optimization in Mobie, Ad Hoc, and Wireess Networks (WiOPT), Seou, Korea, June [] B. Li and A. Eryimaz. A fast-csma agorithm for deadine-constrained scheduing over wireess fading channes. In Proc. Internationa Workshop on Resource Aocation and Cooperation in Wireess Networks (RAWNET/WNC3), Princeton, New Jersey, May 20. [2] B. Li and A. Eryimaz. On the imitations of randomization for queue-ength-based scheduing in wireess networks. In Proc. IEEE Internationa Conference on Computer Communications. (INFOCOM), Shanghai, China, Apri 20. [3] X. Lin and N. Shroff. The impact of imperfect scheduing on cross-ayer rate contro in mutihop wireess networks. In Proc. IEEE Internationa Conference on Computer Communications. (INFOCOM), Miami, Forida, March [4] P. Marbach, A. Eryimaz, and A. Ozdagar. Achievabe rate region of csma scheduers in wireess networks with primary interference constraints. In Proc. IEEE Conference on Decision and Contro (CDC), New Oreans, Louisiana, December [5] P. Marbach, A. Eryimaz, and A. Ozdagar. On the throughput-optimaity of csma poicies in mutihop wireess networks, avaiabe at eryimaz/csmatechreport.pdf. [6] M. Mitzenmacher and E. Upfa. Probabiity and Computing: Randomized Agorithms and Probabiistic Anaysis. Cambridge University Press, New York, NY, [7] M. Neey. Stochastic Network Optimization with Appication to Communication and Queueing Systems. Morgan & Caypoo, 200. [8] M. Neey, E. Modiano, and C. Rohrs. Dynamic power aocation and routing for time varying wireess networks. In Proc. IEEE Internationa Conference on Computer Communications. (INFOCOM), San Francisco, Caifornia, Apri [9] J. Ni and R. Srikant. Distributed csma/ca agorithms for achieving maximum throughput in wireess networks. In Proc. IEEE Internationa Workshop on Information Theory and Appications (ITA), San Diego, Caifornia, February [20] S. Rajagopaan and D. Shah. Distributed agorithm and reversibe network. In Proc. IEEE Conference on Information Sciences and Systems (CISS), Princeton, New Jersey, March [2] S. Rajagopaan, D. Shah, and J. Shin. Network adiabatic theorem: an efficient randomized protoco for contention resoution. In Proc. ACM Internationa Conference on Measurement and Modeing of Computer Systems(SIGMETRICS), Seatte, Washington, June [22] S. Sanghavi, L. Bui, and R. Srikant. Distributed ink scheduing with constant overhead. In Proc. ACM Internationa Conference on Measure-

13 3 [23] [24] [25] [26] [27] ment and Modeing of Computer Systems. (SIGMETRICS ), San Diego, Caifornia, June G. Sasaki and B. Hajek. Link scheduing in poynomia time. IEEE Transactions on Information Theory, 34(5):90 97, 988. S. Shakkottai and A. Stoyar. Scheduing for mutipe fows sharing a time-varying channe: The exponentia rue. Transations of the AMS, Series 2, A voume in memory of F. Karpeevich, 207:85 202, L. Tassiuas. Linear compexity agorithms for maximum throughput in radio networks and input queued switches. In Proc. IEEE Internationa Conference on Computer Communications. (INFOCOM), San Francisco, Caifornia, Apri 998. L. Tassiuas and A. Ephremides. Stabiity properties of constrained queueing systems and scheduing poicies for maximum throughput in mutihop radio networks. IEEE Transactions on Automatic Contro, 36(2): , 992. L. Tassiuas and A. Ephremides. Dynamic server aocation to parae queues with randomy varying connectivity. IEEE Transactions on Information Theory, 39(2): , 993. Bin Li received his B.S. degree in Eectronic Engineering and M.S. degree in Communication Engineering, both from Xiamen University, China, in 2005 and 2008, respectivey. Between 2008 and 2009, he was with the University of Texas at Arington. He is currenty pursuing the Ph.D. degree in the Department of Eectrica and Computer Engineering at The Ohio State University. His research interests incude wireess communication and networks, optima network contro, distributed agorithms, and optimization theory. Atia Eryimaz (S 00-M 06) received his B.S. degree in Eectrica and Eectronics Engineering from Bog azic i University, Istanbu, in 999, and the M.S. and Ph.D. degrees in Eectrica and Computer Engineering from the University of Iinois at UrbanaChampaign in 200 and 2005, respectivey. Between 2005 and 2007, he worked as a Postdoctora Associate at the Laboratory for Information and Decision Systems at the Massachusetts Institute of Technoogy. He is currenty an Assistant Professor of Eectrica and Computer Engineering at The Ohio State University. Dr. Eryimaz s research interests incude design and anaysis for communication networks, optima contro of stochastic networks, optimization theory, distributed agorithms, pricing in networked systems, and information theory. He received the NSF-CAREER and the Lumey Research Awards in 200.