Optimal Distributed Scheduling under Time-varying Conditions: A Fast-CSMA Algorithm with Applications

Transcription

1 Optima Distributed Scheduing under Time-varying Conditions: A Fast-CSMA Agorithm with Appications Bin Li and Atia Eryimaz Abstract Recenty, ow-compexity and distributed Carrier Sense Mutipe Access (CSMA)-based scheduing agorithms have attracted extensive interest due to their throughput-optima characteristics in genera network topoogies. However, these agorithms are not we-suited for time-varying environments (i.e., serving rea-time traffic under time-varying channe conditions in wireess networks) 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; () since the dynamic CSMA parameters are infuenced by the arriva and channe state processes, the underying CSMA Markov Chain may not converge to a steady-state under strict deadine constraints and fading channe conditions. In this paper, we attack the probem of distributed scheduing for time-varying environments. Specificay, we propose a Fast- CSMA (FCSMA) poicy in fuy-connected topoogies, which converges much faster than the existing CSMA agorithms and thus yieds significant advantages for time-varying appications. Then, we design optima poicies based on FCSMA techniques in two chaenging and important scenarios in wireess networks for scheduing ineastic traffic with/without channe state information (CSI) over wireess fading channes. I. INTRODUCTION Efficient utiization of network resources requires carefu interference management among simutaneous transmissions. Of particuar interest in the efficient scheduing are Queue-Length- Based (QLB) scheduers (e.g., [], [3], [4], [5], [6]) due to their provaby optima performance guarantees. Randomization is usefu in aowing fexibiities in the design and impementation of such scheduers (e.g., [7]). However, it causes inaccurate operation and may be hurtfu if it is not performed within imits (see [8] for more detais). One of the most robust randomized scheduers is CSMA-based distributed scheduer (e.g., [9], [0], [], []), whose stationary distribution of the underying Markov chain has a product-form. It is we-known that CSMA-based scheduer can maximize ong-term average throughput for genera wireess topoogies. However, these resuts do not appy to time-varying environments (i.e., scheduing deadine-constrained traffic over wireess fading channes), since their throughput-optimaity This work is supported by the NSF grants: CAREER-CNS and CCF Aso, the work of A. Eryimaz was in part supported by the QNRF grant number NPRP An earier version of this paper has appeared in the workshop on Resource Aocation and Cooperation in Wireess Networks (RAWNET), Princeton, NJ, May, 0 []. 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 430 USA. reies: (i) on the convergence time of the underying Markov Chain to its steady-state, which grows with the size of the network; and (ii) on reativey stationary conditions in which the CSMA parameters do not change significanty over time so that the instantaneous service rate distribution can stay cose to the stationary distribution. Both of these conditions are vioated in time-varying environments. For exampe, packets of deadine-constrained traffic are ikey to be dropped before the CSMA-based agorithm converges to its steady-state, and the time-varying fading creates significant variations on the CSMA parameters, in which case the instantaneous service rate distribution cannot cosey track the stationary distribution. To the best of our knowedge, there does not exist a work that can achieve provaby good performance by using attractive CSMA principes under time-varying conditions. Whie achieving ow deay via distributed scheduing in genera topoogies is a difficut task (see [3]), in a reated work [4] that focuses on grid topoogies, the authors have designed an Unocking CSMA (UCSMA) agorithm with both maximum throughput and order optima average deay performance, which shows promise for ow-deay distributed scheduing in specia topoogies. However, UCSMA aso does not directy appy to deadine-constrained traffic since its measure of deay is on average. Moveover, it is not cear how existing CSMA or UCSMA wi perform under fading channe conditions. Thus, designing an optima distributed scheduing agorithm in timevarying environments remains an open question. With this motivation, in this work, we address the probem of distributed scheduing in fuy connected networks (e.g., Ceuar network, Wi-Fi network) for time-varying environments. We propose a Fast-CSMA (FCSMA) agorithm that, despite its simiarity of name, fundamentay differs from existing CSMA poicies in its design principe: rather than evoving over the set of schedues to reach a favorabe steady-state distribution, the FCSMA poicy aims to quicky reach one of a set of favorabe schedues and stick to it for a duration reated to time-varying scae of the appication. Whie the performance of the former strategy is tied to the mixing-time of a Markov Chain, the performance of our strategy is tied to the hitting time, and hence, yieds significant advantage for time-varying appications. In this work, we appy FCSMA techniques in two main scenarios: deadine-constrained scheduing with/without channe state information (CSI) over wireess fading channes. We aso consider the appication of FCSMA techniques in nondeadine-constrained scheduing over wireess fading channes in our technica report [5]. The two scenarios we considered

2 in this paper are most chaenging and important appication in practice, since wireess networks are expected to serve reatime traffic, such as video or voice appications, generated by a arge number of users over potentiay fading channes. These constraints and requirements, together with the imited shared resources, generate a strong need for distributed agorithms that can efficienty utiize the avaiabe resources whie maintaining high quaity-of-service for the rea-time appications. Yet, the strict short-term deadine constraints and ong-term throughput requirements associated with most rea-time appications compicate the deveopment of provaby good distributed soutions. A existing works in deadine-constrained scheduing (e.g., [6], [7], [8], [9]) assume centraized controers, and hence are not suitabe for distributed operation. To the best of our knowedge, this is the first work that proposes an optima and distributed agorithm under time-varying conditions caused by channe fading or time-sensitive appications. Our main contributions in this paper are: In Section II, we propose a FCSMA agorithm that aims to quicky reach one of a set of favorabe schedues. We design an optima distributed poicy based on FC- SMA techniques in scheduing deadine-constrained traffic with/without CSI over wireess fading channes in Section III and Section IV, respectivey. II. THE PRINCIPLE OF FAST-CSMA DESIGN We consider a fuy-connected network topoogy where L inks contend for data transmission over a singe channe. We assume a time-sotted system, where a inks start transmission at the beginning of each time sot. Due to the interference constraints, at most one ink can transmit in each sot. We ca a schedue where at most one ink is active in each sot as a feasibe schedue. Randomized scheduers (e.g., [7], [9], [0], [], [] and [3]) are widey studied due to their fexibiities in deveopment of ow-compexity and distributed impementations. The most promising and interesting randomized scheduers are distributed CSMA-based agorithms. We give the definition of continuoustime CSMA agorithm (see [9]) for competeness. In this paper, we adopt the same assumptions as in [9] that the sensing is instantaneous and the backoff time is continuous. Definition (CSMA Agorithm): Each ink independenty generates an exponentiay distributed random variabe with rate R [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 exponentiay distributed with mean. Figure a shows the state transition diagram of the underying Markov Chain for the CSMA Agorithm when there are 3 avaiabe inks at time t, where each state stands for a feasibe schedue. It is easy to see that the stationary distribution of this Markov Chain is P = R [t],, () + Z[t] R [ t ] ( 0,, 0 ) (, 0, 0 ) ( 0, 0, 0 ) (a) R [ t ] R 3 [ t ] ( 0, 0, ) ( 0,, 0 ) (, 0, 0 ) R [ t ] ( 0, 0, 0 ) R [ t ] R 3 [ t ] (b) ( 0, 0, ) Fig. : (a) Markov chain for a CSMA agorithm (b) Markov chain for a FCSMA agorithm where Z[t] = L = R [t]. Since R[t] = (R [t]) L = is chosen as a function of network state information (e.g., queue ength, channe state information, arrivas) in wireess networks, the underying Markov Chain for the CSMA Agorithm is inhomogeneous. Intuitivey, the CSMA parameters R[t] shoud change sowy such that the instantaneous service rate distribution can stay cose to the stationary distribution. Indeed, for the appication of scheduing over time-invariant channes (i.e., the transmission rate of each ink does not change over time), such mapping has been observed to be optima (e.g., [] and []) if the CSMA parameter R [t] of each ink can take certain functiona forms (e.g., og og( )) of its queue ength at time t. Note that the queue ength wi change sowy when it is arge enough. The purpose of choosing the sowy increasing function is further to make the CSMA parameters as a function of queue ength do not change significanty over time. However, for the appication of scheduing over wireess fading channes, the CSMA parameters R[t] need to be chosen as a function of channe sate information to yied good performance. In such case, no matter what function we choose for the channe state, R[t] wi change significanty as the fading state fuctuates and thus the instantaneous service distribution is not expected to track the stationary distribution. More generay, extending CSMA soutions to stochastic network dynamics or sophisticated appication requirements (e.g., serving traffic with strict deadine constraints over 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 deadineconstrained traffic; ) since the dynamic CSMA parameters R[t] are infuenced by the arriva and channe state process, the underying CSMA Markov chain may not converge to its steady-state under strict deadine constraints and wireess fading channe conditions. Thus, designing an optima and distributed scheduing agorithm for stochastic networks becomes quite chaenging. In this paper, we propose a Fast-CSMA strategy that provides provaby good performance under time varying conditions. Our approach fundamentay differs from existing CSMA soutions in that our FCSMA poicy expoits the fast convergence characteristics of hitting times instead of mixing times. Definition (Fast-CSMA (FCSMA) Agorithm): At the beginning of each time sot t, each ink independenty generates an exponentiay distributed random variabe with rate R [t],

3 3 and starts transmitting after this random duration uness it senses another transmission before. If a inks have their random duration greater than a sot, a inks wi keep sient in the current sot; otherwise, the ink that captures the channe transmits its data unti the end of the sot. The whoe process is repeated in the next time sot. Remarks: () The operation of the FCSMA Agorithm resembes that of the UCSMA Agorithm (see [4]). The difference ies in that the UCSMA agorithm restarts the CSMA Agorithm to achieve both maximum throughput and order optima average deay in grid network topoogies over time-invariant channes by carefuy choosing the running period. However, it is uncear whether the UCSMA agorithm can sti work we in timevarying appications. () By choosing the running period for the FCSMA Agorithm the same as the time scae of network dynamics (i.e., the bock ength for bock fading or maximum aowabe deadine), we can show in ater sections that the FCSMA Agorithm exhibits exceent performance in time-varying appications. (3) In genera muti-hop network topoogies, the FCSMA Agorithm can sti converge very fast to one feasibe schedue. Yet, the probabiity of serving each schedue may not have a product form and the performance of the FCSMA Agorithm is uncear. We eave it for future investigation. Figure b gives the state transition diagram of underying Markov Chain for the FCSMA Agorithm when there are 3 avaiabe inks, where each state represents a feasibe schedue. The convergence time of the FCSMA Agorithm is tied to the hitting time, whie the convergence time of the CSMA Agorithm is dominated by the mixing time of Markov chain, which generay is arge. The hitting time of the FCSMA Agorithm at sot t is exponentiay distributed with mean Z[t], which is generay sma in practice as we wi see in simuations. Due to its sma hitting time, the FCSMA Agorithm yieds significant advantages over existing CSMA poicies evoving sowy to the steady-state and may work we in more chaenging environments, i.e., scheduing reatime traffic over wireess fading channes. Because of the fast convergence property of the FCSMA Agorithm, we introduce the ideaized FCSMA agorithm for easier theoretica anaysis. The simuation resuts in the ater sections indicate that both FCSMA and Ideaized FCSMA Agorithm have the same system performance. Definition 3 (Ideaized FCSMA Agorithm): Ideaized FCSMA Agorithm is the FCSMA Agorithm with zero hitting time, which assumes that it can reach the favorabe state instantaneousy. For the Ideaized FCSMA Agorithm, the probabiity of If there is no data awaiting in the ink, it transmits dummy data to occupy the channe. The hitting time is an empty duration after which the Markov Chain stays in a non-zero feasibe schedue state (i.e., the channe is occupied by one of users) serving the ink in each sot t wi be: π [t] = R [t] Z[t]. () Let W [t] = og(r [t]) and W [t] = max W [t]. The foowing emma estabishes the fact that the Ideaized FCSMA Agorithm picks a ink with the weight cose to the maximum weight with high probabiity when the maximum weight W [t] is arge enough at each sot t. Lemma : Given ǫ > 0 and ζ > 0, W (0, ), such that if W [t] > W, then the Ideaized FCSMA Agorithm picks a ink satisfying Pr{W [t] ( ǫ)w [t]} ζ. (3) The proof is simiar to that in [0] and [8], and thus is omitted here for brevity. In the rest of paper, we mainy consider ineastic traffic. The ineastic traffic means that each arriva has a maximum deay requirement whie the eastic traffic does not have such a requirement. We appy the FCSMA technique in two chaenging scenarios: scheduing ineastic traffic with/without Channe State Information (CSI) over wireess fading channes. In our technica report [5], we aso consider scheduing eastic traffic over wireess fading channes. In each appication, we need to carefuy design the FCSMA parameters R[t] = (R [t]) L = at each sot t to yied optima performance. To faciitate the fexibiity in the design and impementation of the FCSMA agorithm, we define a set of functions (aso see [8]): F set of non-negative, nondecreasing and differentiabe functions f( ) : R + R + with im f(x) =. x f(x + a) B {f F: im =, for any a R}. x f(x) The exampes of functions that are in cass B are f(x) = og x, f(x) = x or f(x) = e x. Note that f(x) = e x does not beong to cass B. Now, we are ready to deveop optima FCSMA agorithms in two chaenging appications: scheduing ineastic traffic with/without CSI over wireess fading channes. III. SCHEDULING INELASTIC TRAFFIC WITH CSI A. Basic Setup We assume that the wireess channe is independenty bock fading at each ink. We capture the channe fading over ink via C [t], which measures the maximum amount of service avaiabe in sot t, if schedued. We assume that C[t] = (C [t]) L = are independenty distributed random variabes over inks and identicay distributed over time with C [t] C max,, t, for some C max <. We use a binary variabe S [t] to denote whether the ink is served at sot t, where S [t] = if the ink can be served at sot t and S [t] = 0, otherwise. Let S[t] = (S [t]) L = be a feasibe schedue. We use S to denote the coection of feasibe schedues. Reca that at most one ink can be active in a feasibe schedue, due to the fuy connected network topoogy we consider in this paper.

4 4 We assume that a arrivas have the same deay bound of T time sots, which means that if the data cannot be served during T sots after it arrives, it wi be dropped. For convenience, we ca a set of T consecutive time sots a frame. In the context of fuy-connected networks, we associate each rea-time fow with a ink, and hence use these two terms interchangeaby. We assume that a data arrives at each ink at the beginning of each frame. Let A [kt] denote the amount of data arriving at ink in frame k that are independenty distributed over inks and identicay distributed over time with mean λ, and A [kt] A max for some A max <. A the remaining data is dropped at the end of a frame. Each ink has a maximum aowabe drop rate ρ λ, where ρ (0, ) is the maximum fraction of data that can be dropped at ink. For exampe, ρ = 0. means that at most 0% of data can be dropped at ink on average. Our goa is to find the schedue {S[t]} t under the scheduing constraint that at most one ink can be schedued at each time sot and dropping rate constraint that the average drop rate of each ink shoud not be greater than its maximum aowabe drop rate. To sove this optima contro probem, we use the inteigent technique in [4] to introduce a virtua queue X [kt] for each ink to track the amount of dropped data in frame k. Specificay, the amount of data arriving at virtua queue at the end of frame k is denoted as D [kt], which is equa to (k+)t A [kt] min C [t]s [t], A [kt]. We use I [kt] to denote the service for virtua queue at the end of the frame k with mean ρ λ, and I [kt] I max for some I max <. Further, we et U [kt] denote the unused service for queue at the end of frame k, which is upper-bounded by I max. Then, the evoution of virtua queue is described as foows: X [(k + )T] = X [kt] + D [kt] I [kt] + U [kt],. (4) In this and next section, we consider two main scenarios: known channe state and unknown channe state. For the known channe state case, we assume that the channe state is constant for the duration of a frame and each ink knows CSI at the beginning of each frame. For the unknown channe state case, we aow that the channe state changes from time sot to time sot and each ink does not know CSI before each transmission, but can determine how much data has been transmitted at each sot after we get feedback from the receiver. These assumptions are aso adopted in [8]. We consider the cass of stationary poicies P that seect S[t] as a function of (X[kT],A[kT],C[kT]) for the known channe state scenario and a function of (X[kT],A[kT]) for the unknown channe state scenario in frame k, which, then, form a Markov Chain, where X[kT] = (X [kt]) L = and A[kT] = (A [kt]) L =. If this Markov Chain is positive recurrent, then the average drop rate wi meet the required dropping rate constraint automaticay (see [5]). We define the maxima satisfiabe region as a maximum set of arriva processes for which this Markov Chain is positive recurrent under any poicy. We ca an agorithm optima if it makes Markov Chain positive recurrent for any arriva process within the maxima satisfiabe region. B. FCSMA agorithm impementation In this subsection, we first characterize the maxima satisfiabe region and then propose an optima FCSMA agorithm with CSI for scheduing ineastic traffic over fading channes. Consider the cass P of stationary poicies that base their scheduing decision on the observed vector (X[kT], A[kT], C[kT]) in frame k. The next emma estabishes a necessary condition for stabiizing the system. Lemma : If there is a poicy P 0 P that can stabiize the virtua queue X, then there exist non-negative numbers α(a,c;s 0,s,...,s T ) such that α(a,c;s 0,s,...,s T ) =, a,c, (5) s 0,s,...,s T S λ ( ρ ) < a P A (a) c α(a,c;s 0,s,...,s T )min P C (c) { T s 0,s,...,s T S } c s i, a,, (6) where s i = (s i )L =, P A(a) = P(A[t] = a) and P C (c) = P(C[t] = c). The proof is amost the same as in [6]. The main difference ies in that our proof deas with the necessary condition for stabiizing virtua queues instead of data queues as in [6]. We omit it for conciseness. Note that the right hand side of the inequaity (6) is the average service provided for ink during one frame; whie λ ( ρ ) is the average amount of data at ink that needs to be served. Thus, to meet the maximum aowabe drop rate requirement, (6) shoud be satisfied. We define the maxima satisfiabe region Λ (ρ,c) as foows: i=0 Λ (ρ,c) {A : α(a,c;s 0,s,...,s T ) 0, such that both (5) and (6) satisfy}. We are now ready to deveop an optima centraized agorithm with CSI for scheduing ineastic traffic over wireess fading channes. Centraized Agorithm with CSI: In each frame k, given (X[kT],A[kT],C[kT]), perform {S (k+)t [t]} argmax f(x [kt]) where f F. (k+)t {S[t]} min C [kt] (k+)t S [t], A [kt], (7) Remark: In [8], the authors proposed a centraized agorithm with f(x) = x. Our proposed centraized agorithm is more

5 5 genera, which aows more fexibiities in distributed impementations. Next, we estabish the optimaity of the centraized agorithm with CSI under certain conditions for function f. Theorem : If f B, the Centraized Agorithm with CSI for scheduing ineastic traffic is optima over wireess fading channes, i.e., for any arriva process A Λ (ρ,c), it makes the underying Markov Chain positive recurrent. The proof is a generaization of that in [8] and is a specia case of that in Theorem 3, where we use FCSMA techniques to mimic the Centraized Agorithm. Thus, we omit it for brevity. Even though the above centraized agorithm is optima, it cannot directy be appied in practice due to the need of centraized coordination. Next, we propose a greedy agorithm that is we suited for distributed impementation. To that end, we first give the key identity that faciitates the deveopment of greedy soutions. Lemma 3: Let a 0 and c[t] 0, t = 0,,..., T. If s[t] {0, }, t, then { T } + T t min c[t]s[t], a = min c[t], a c[j]s[j] s[t], (8) t=0 t=0 where (x) + = max{x, 0}. Proof: The proof directy foows by induction. Pease see our technica report [5] for detais. Based on Lemma 3, the objective function in (7) can be rewritten as (k+)t f(x [kt])min C [kt] S [t], A [kt] (k+)t t + = f(x [kt])min C [kt], A [kt] C [kt] S [j] S [t]. (9) We can observe that the equation (9) decoupes the scheduing decisions over a frame and hep deveop the greedy soutions that are easy to be impemented distributivey. Greedy Agorithm with CSI: At each time sot t {kt, kt +,..., (k+)t } in frame k, seect ink G [t] such that G [t] argmax where f F. t + f(x [kt])min C [kt], A [kt] C [kt] S [j], (0) Theorem : The Greedy Agorithm with CSI is an optima soution to the probem (7) and thus is optima for scheduing ineastic traffic over wireess fading channes if f B. The proof is a specia case of that in Theorem 5: the channe state is constant over a frame in the proof for Theorem, whie the channe state changes from sot to sot in a frame in that for Theorem 5, which makes it more chaenging to dea with. Next, we expand on the distributed impementation of the greedy soution by using the FCSMA technique. Ideaized FCSMA Agorithm with CSI: At each time sot t {kt, kt +,..., (k+)t } in frame k, choose the rates { ( R [t] = g(x [kt]) min C [kt], A [kt] C [kt] ) } t + S [j],, () where g F. Note that the Ideaized FCSMA Agorithm with CSI does not take the convergence time into consideration. For the FCSMA Agorithm with CSI, the rates can be seected as R [t] = g(x [kt]) min{c [kt],j [t]}, for any ink and any t {kt, kt +,..., (k + )T }, where J [t] is the remaining data at ink at the beginning of each time sot t. Next, we wi show that the Ideaized FCSMA Agorithm yieds the optima performance. Simuation resuts show that both FCSMA and Ideaized FCSMA Agorithm have the same performance. Theorem 3: If f(x) = og g(x) B and g(0), the Ideaized FCSMA Agorithm with CSI for scheduing ineastic traffic is optima over wireess fading channes, i.e., for any arriva process A Λ (ρ,c), it makes the underying Markov Chain positive recurrent. Proof: The proof foows from the Lyapunov drift argument. However, it is quite chaenging to argue that the Ideaized FCSMA Agorithm with CSI mimics the Centraized/Greedy Agorithm with CSI over a frame, which is an obvious case when T = (see []). By propery partitioning the space of weights chosen by the Greedy Agorithm with CSI within a frame, we tacke this difficuty and refer the reader to see Appendix A for the detais. C. Simuation Resuts In this subsection, we perform simuations to vaidate the optimaity of the proposed FCSMA poicy with CSI for scheduing ineastic traffic with deadine constraint of T sots over wireess fading channes. In the simuation, there are L = 0 inks and each frame has T = 5 sots. A inks require the maximum fraction of dropped data to not exceed ρ = 0.3. The amount of arrivas in each frame foows common Bernoui distribution that the amount of arrivas equa to T with probabiity λ. A inks suffer from the ON-OFF channe fading independenty with probabiity p = 0.8 that the channe is avaiabe in each frame. The service for virtua queue aso foows Bernoui distribution that the maximum avaiabe service equas to T with probabiity ρλ. Under this setup, we can use the same technique in paper [3] to get the maxima satisfiabe region: Λ (ρ,c) = {λ : L( ρ)λ < ( pλ) L }. Through numerica cacuations, we can get the maxima satisfiabe region: {λ : λ < 0.038}. In the simuations, we aso compare our proposed FCSMA poicy with the QCSMA agorithm (see [0]) with the og og function. From Figure, we can observe that the FCSMA Agorithm with both g(x) = e x and g(x) = x + can achieve maxima satisfiabe region. Aso, we see that the average virtua queue ength of the FCSMA Agorithm with exponentia function is

6 6 smaer than that with inear function. However, the meaning of smaer virtua queue ength is uncear in this setup. We wi expore it in our future research. In addition, we can observe that the FCSMA Agorithm has amost the same performance as that with the Ideaized FCSMA Agorithm, which indicates that the hitting time shoud be negigiby sma. Furthermore, the QCSMA agorithm with og og function cannot even support the arriva rate of λ = 0.00 (i.e., its corresponding virtua queues are unstabe for the arriva rate of 0.00). The reason for the poor performance of the QCSMA agorithm is that it does not have enough time to converge to the steady state under fast dynamics of the arriva and channe processes. Average virtua queue ength Ideaized FCSMA with g(x)=e x FCSMA with g(x)=e x Ideaized FCSMA with g(x)=x+ FCSMA with g(x)=x+ QCSMA with ogog function E-8 E-7 E-6 E-5 E-4 E Arriva rate Fig. : The performance of the FCSMA Agorithm with CSI IV. SCHEDULING INELASTIC TRAFFIC WITHOUT CSI In this section, we consider the ineastic traffic scheduing without CSI over wireess fading channes. We assume that each ink knows how much data has been transmitted at the end of each sot by using per-sot feedback information. The persot feedback compicates the design of distributed scheduing agorithm. But, we sti can find a simiar greedy soution as in Section III and design its distributed agorithm by using FCSMA techniques. A. FCSMA agorithm impementation Consider the cass P of stationary poicies that base their scheduing decision on the observed vector (X[kT], A[kT]) in frame k. The next emma estabishes a condition that is necessary for stabiizing the system. Lemma 4: If there is a poicy P 0 P that can stabiize the virtua queue X, then there exist non-negative numbers α 0 (a;s 0 ), α (a,s 0 ;s ),..., α T (a,s 0,...,s T ;s T ), such that α 0 (a;s 0 ) =, a, () s 0 S α i (a,s 0,...,s i ;s i ) =, a, i =,,..., T, (3) s i S λ ( ρ ) < a P A(a) α T (a,s 0,..., s T ;s T )E s 0,s,...,s T S [ min { T i=0 α 0(a;s 0 )α (a,s 0 ;s )... c s i, a }],. (4) The proof is amost the same as [6] and hence is omitted here. We define maxima satisfiabe region Λ (ρ,c) as foows: Λ (ρ,c) {A : α 0(a;s 0 ), α (a,s 0 ;s ),..., α T (a,s 0,...,s T ;s T ) 0, such that both (), (3), and (4) satisfy}. Next, we deveop an optima centraized agorithm without CSI for scheduing ineastic traffic over fading channes. Centraized Agorithm without CSI: In each frame k, given (X[kT],A[kT]), sove the foowing optimization probem: max E (k+)t f(x [kt]) min C [t]s [t], A [kt], (k+)t {S[t]} (5) where f F, and the schedue at each sot is determined after knowing how much data has been transmitted in the previous sots in each frame. Remark: In [8], the authors designed a centraized agorithm with f(x) = x. Our proposed centraized agorithm generaizes this to a arge space of functions f, and aows for more fexibiities in distributed impementations. Next, we estabish the optimaity of the centraized agorithm without CSI under certain conditions for function f. Theorem 4: If f B, the Centraized Agorithm without CSI for scheduing ineastic traffic is optima over wireess fading channes, i.e., for any arriva process A Λ (ρ,c), it makes the underying Markov Chain positive recurrent. The proof is a generaization of that in [8] and foows the same argument as that in Theorem 3. Thus, we omit it here for brevity. The centraized agorithm without CSI is quite compicated, since it coupes the scheduing decisions in each frame. Under the per-sot feedback assumption, the optimization probem (5) can be soved by using dynamic programming. Based on Lemma 3, we have the foowing key identity: (k+)t f(x [kt])min C [t]s [t], A [kt] (k+)t t + = f(x [kt])min C [t], A [kt] C [j]s [j] S [t]. (6) By using (6), we can get the foowing backward equation (see [7]) for the optimization probem (5). Backward Equation for (5): At each sot t {kt, kt +,..., (k + )T } in frame k, given (X[kT],A[kT]) and {(C[j],S[j])} t, seect ink [t] such that

7 7 ( + + t [t] argmax f(x [kt])e min C T [t], A [kt] S [j]c [j] W d f(x d[i] [0])E min C i d[i][i], A d[i] [0] S d[i] [j]c d[i] [j] i=t (k+)t + L t + max f(x [kt]) =f(x d[t] [0])E min (k+)t {S[r]} r=t+ i=t+ C d[t][t], A d[t] [0] S d[t] [j]c d[t] [j] = + ) + i E min C T [i], A [kt] S [j]c [j] S [i], (7) + f(x [0])E min C i [i], A [0] S [j]c [j] S [i], i=t+ where f F. At first gance, the optima soution to probem (7) at each time sot depends on the future sots and thus is difficut to be impemented distributivey. However, it may sti be possibe to decoupe the scheduing decisions over a frame, since the channe states are i.i.d. across over time sots. Next, we wi show that this is the case in our setup. Greedy Agorithm without CSI: At each time sot t {kt, kt +,..., (k + )T } in frame k, given (X[kT],A[kT]) and {(C[j],S[j])} t, seect ink G [t] such that G [t] arg max where f F. t + f(x [kt])e min C [t], A [kt] C [j]s [j], (8) Theorem 5: The Greedy Agorithm without CSI is optima for probem (7) and thus is optima for scheduing ineastic traffic over wireess fading channes if f B. Proof: Without oss of generaity, we consider the frame k = 0. We wi show that if G [t] satisfies (8) at time t {0,,..., T }, then G [t] is an optima soution to the backward equation (7), that is, t + f(x G [0])E min [t] C G [t] [t], A G [0] S [t] G [j]c [t] G [j] [t] T i + + max {S[r]} T f(x [0])E min r=t+ i=t+ C [i], A [0] S [j]c [j] S [i] t + f(xm[0])e min Cm[t], Am[0] Sm[j]Cm[j] T i + + max {S[r]} T f(x [0])E min i=t+ r=t+ C [i], A [0] S [j]c [j] S [i], (9) hods for any m G [t]. Reca that at most one ink can be schedued at each sot. For ease of exposition, et d (d[t], d[t+],..., d[t ]) genericay denote the sequence of feasibe inks chosen from time sot t to the end of the frame by any agorithm, where the eement d[i] denotes the ink that is schedued at sot i. Note that the eements in d can be any possibe inks. The purpose of introducing d is to simpify the expression of (9). Let D be the coection of the sequence of seected inks from time sot t to the end of the frame. Let W d for a given d D be defined as where S d[i] [i] =, and S [i] = 0, d[i], for i = t, t +,..., T. Let F = {d D : d[t] = }. Then, (9) can be rewritten as max W d max W d, m G [t]. (0) d F G [t] d F m Given any m G [t], we have the foowing two cases: () If d F m incudes the eement G [t], then a permutation of d with the first eement being G [t] shoud be in F G [t]. Since the channe states are i.i.d. over time sots, any permutation of d does not change the vaue W d and thus W d max e F G [t] W e. () If d F m does not incude the eement G [t], then it is easy to see that W d W c, where c = ( G [t], d[t + ], d[t+],..., d[t ]). Since c F G [t], we have W d max e F G [t] W e. Thus, we have max e F G [t] W e W d, d F m, and hence we have the desired resut (0). Next, we iustrate the distributed impementation of greedy soutions by using FCSMA techniques. Ideaized FCSMA Agorithm without CSI: At each time sot t {kt, kt +,..., (k+)t } in frame k, given (X[kT],A[kT]) and {(C[j],S[j])} t, choose the rates [ { ( R [t] = g(x [kt]) E min C [t], A [kt] ) }] t + C [j]s [j],, () where g F. Note that the Ideaized FCSMA Agorithm without CSI does not consider the impact of the convergence time. For the FCSMA Agorithm without CSI, the rates can be chosen as R [t] = g(x [kt]) E[min{C [t],j [t]}], () for any ink and any time t {kt, kt +,..., (k+)t }, where J [t] is the remaining data at ink at the beginning of each time sot t. Next, we wi show that the Ideaized FCSMA Agorithm yieds the optima performance. Simuation resuts indicate that both FCSMA and Ideaized FCSMA Agorithm have the same performance. Theorem 6: If f(x) = og g(x) B and g(0), the Ideaized FCSMA Agorithm without CSI for scheduing ineastic traffic is optima over wireess fading channes, i.e., for any arriva process A Λ (ρ,c), it makes the underying Markov Chain positive recurrent.

8 8 The proof is simiar to that in Theorem 3 which considers the ineastic traffic with CSI over wireess fading channes. We skip it for conciseness. B. Simuation Resuts In this subsection, we perform simuations to vaidate the optimaity of the proposed FCSMA poicy without CSI for scheduing ineastic traffic with deadine constraint T sots over wireess fading channes. The simuation setup is the same as that in Section III-C. The main difference is that the fading channes change from sot to sot. The maxima satisfiabe region under this setup is Λ (ρ,c) = {λ : L( ρ)λ < p ( ( λ) L) }. Through numerica cacuations, we can get the maxima satisfiabe region: {λ : λ < 0.03}. As in Section III-C, we aso compare the Ideaized FCSMA Agorithm with the FCSMA Agorithm and the QCSMA agorithm with og og function. From Figure 3, we can observe the same phenomenon as in III-C. Average virtua queue ength Ideaized FCSMA with g(x)=e x FCSMA with g(x)=e x Ideazied FCSMA with g(x)=x+ FCSMA with g(x)=x+ QCSMA with ogog function E-8 E-7 E-6 E-5 E-4 E Arriva rate Fig. 3: The performance of the FCSMA Agorithm without CSI V. PRACTICAL IMPLEMENTATION SUGGESTIONS In the previous sections, we assume that the sensing is instantaneous and the backoff time is continuous, which excudes the possibe coisions. These key assumptions are important in aowing us to concentrate on the chaenging distributed scheduing probem in time-varying environments without considering the contention resoution procedure. Yet, in practice, the sensing time is non-zero and the backoff time is typicay a mutipe of mini-sots, where a mini-sot is equa to the time required to detect the data transmission from another ink (e.g., in IEEE 80.b, a mini-sot shoud be at east 8µs). In such cases, coisions happen, which reduces the system throughput. In this section, we expicity consider these practica chaenges and propose an easiy impementabe and efficient agorithm that is simiar to the one in [9]. The basic idea is to quantize the continuous rate R [t] into a set of discrete vaues, where each discrete vaue is assigned to a different contention window (CW) size. The smaer the quantized vaue is, the arger the corresponding CW size is. Thus, this can be easiy mapped to the service casses in IEEE 80.e. The suggested rate quantization procedure is as foows: (i) if R [t] R max, then et R [t] = R max. This corresponds to the first cass; (ii) if R i max R [t] < R i max for some i =, 3,..., N, then et R [t] = R i max, where N is the number of casses; (iii) if R [t] < R min N R max, then do not start transmissions. Thus, the probabiity of inks accessing the channe in cass i is roughy twice as arge as that in cass i +, which impies that the CW size of cass i + shoud be roughy twice that of cass i. Discrete-time version of the FCSMA Agorithm: At the beginning of each sot t, each ink generates a uniformy distributed random variabe r over {0,,..., CW[t] }, where CW[t] is chosen according to the quantized vaue of rate R [t] as described above. Each ink keeps sensing the channe for r mini-sots. If the channe is busy in any one of the first r mini-sots, then ink suspends its transmission; otherwise, ink starts its transmission 3 from the r th mini-sot to the end of this sot. If two or more inks have the same backoff time, then a coision happens in the current sot. The whoe process restarts in the next sot. We assume that the coherence time for scheduing ineastic traffic with and without CSI are 500ms and 00ms, respectivey. Since a mini-sot is typicay 0µs, without oss of generaity, we assume that a time sot contains 0000 mini-sots in both cases. The simuation setups for scheduing ineastic traffic with and without CSI are the same as that in Section III-C and Section IV-B, respectivey. In the simuations, we et R max = e 5, N = 6, and CW i = 3 i, i =,,..., N, where CW i is the CW of cass i. We aso compare the discretetime version of the FCSMA Agorithm with IEEE 80. Distributed Coordination Function (DCF). In IEEE 80. DCF, the contention window (CW) size depends on whether the transmission is successfu or not, rather than the current system state information. In particuar, the CW for a inks are initiaized to 3; if the transmission of ink is unsuccessfu, then its CW is doubed unti it reaches to the maximum vaue of 04; otherwise, its CW drops to the initia vaue. Average virtua queue ength FCSMA with g(x)=e x 00 Discrete-time version of FCSMA QCSMA with ogog function IEEE 80. DCF 0 E-5 E-4 E Arriva rate 0 E-5 E-4 E Arriva rate (a) Scheduing ineastic traffic with CSI (b) Scheduing ineastic traffic without CSI Average virtua queue ength FCSMA with g(x)=e x Discrete-time version of FCSMA QCSMA with ogog function IEEE 80. DCF Fig. 4: Performance comparison between FCSMA agorithm and its discrete-time version From Figure 4a and 4b, we can observe that the performance of the discrete-time version of the FCSMA Agorithm remains cose to that of the FCSMA Agorithm, and continue to perform much better than the QCSMA agorithm with og og function 3 If the number of inks is arge, each ink uses short packets, such as Request- To-Send (RTS) and Cear-To-Send (CTS) in IEEE 80.b, to contend for the wireess channe, which wi significanty reduce the cost of a coision.

9 9 and IEEE 80. DCF in both scheduing ineastic traffic with and without CSI. However, we note that if the coherent time is comparabe with the maximum CW size, then, the discrete-time version of the FCSMA Agorithm can perform poory, since a non-negigibe amount of resources is consumed by the backoff process instead of the data transmission. VI. CONCLUSIONS In this paper, we first proposed a Fast-CSMA (FCSMA) Agorithm that quicky reaches the favorabe state in fuy connected network topoogies. Due to the fast convergence time, the FCSMA Agorithm exhibits significant advantages over existing CSMA agorithms for time-varying appications, which are important and popuar in wireess networks. Then, we appy the FCSMA Agorithm to design optima poicies for scheduing ineastic traffic with/without CSI over wireess fading channes. In the future, we wi try to expore distributed scheduing agorithms for time-varying environments in genera wireess network topoogies. APPENDIX A PROOF FOR THEOREM 3 Consider the Lyapunov function V (X) L = h(x ), where h (x) = f(x). Then, by using a simiar argument to the proof of Lemma in [8] (aso see [9]), it is not hard to show that if for any process A Λ (ρ,c), there exists γ > 0 and H 0 such that L V (X) E[f(X )(D [kt] I [kt]) X[kT] = X] = γ L f(x ) + H. (3) = By the teescoping technique, we have im sup K K K k= = L E[f(X [kt])] H γ <, which impies the stabiity-in-the-mean and thus the Markov Chain is positive recurrent [30]. Next, we wi show inequaity (3) to compete the proof. By substituting the expression of D [kt] (see the discussion before (4)) into V (X), we have V (X) = L E [f(x )(A [kt] I [kt]) X[kT] = X] = }{{} V (X) L (k+)t E f(x ) min C [kt]s F [t], A [kt] = X[kT], }{{} V (X) where S F [t] = (S F[t])L = denotes the schedue chosen by FCSMA agorithm at time t. Let + W [t] = f(x [kt]) min C t [kt], A [kt] C [kt] S [j], for any t = kt, kt +,..., (k + )T, where S[j] = (S [j]) L = is a feasibe schedue at time sot j. Let W G [t] be the weight of ink picked by the Greedy Agorithm with CSI at time sot t. Reca that W G [t] = max W [t]. Next, we wi derive an upper bound for V (X) by using Lemma and give a ower bound for V (X). First, et s focus on V. By Lemma, there exist nonnegative numbers α(a,c;s 0,s,...,s T ) satisfying (5) and for a δ > 0 sma enough, we have λ ( ρ ) P A (a) P C (c) a c s 0,s,...,s T S T α(a,c;s 0,s,...,s T )min c s j, a δ. (4) By using (4), we have V = L f(x )λ ( ρ ) = P A (a) P C (c) α(a,c;s 0,s,...,s T ) a c s 0,s,...,s T S L T L f(x )min c s j, a δ f(x ) = a P A (a) c = (k+)t P C (c) W G [t] δ L f(x ), (5) where the ast step foows from Theorem that the Greedy L T Agorithm with CSI maximizes f(x )min c s j, a = = for any feasibe schedues s 0, s,...,s T, given virtua queue engths, channe state information and arrivas, and uses (5). Thus, we have (k+)t L V E W G [t] X[kT] = X δ f(x ) = (k+)t =E E W G [t] X[kT],A[kT],C[kT] X[kT] δ L f(x ). (6) = Note that W G [t] is non-increasing in t within each frame, since the number of remaining packets cannot increase as t increases.

10 0 Pick any W > 0 and et F 0 {W G [kt] W,W G [kt + ] W,...,W G [(k + )T ] W }; F j {W G [kt + j ] > W,W G [kt + j] W }, j =,...,T ; F T {W G [(k + )T ] > W }, where F j corresponds to the event where the weight chosen by Greedy Agorithm is greater than W in the first j sots in frame k. Thus, (F j ) T forms a partition of a set {W G [kt], W G [kt +],..., W G [(k+)t ]}. Then, we have (k+)t E W G [t] X[kT],A[kT],C[kT] (k+)t T =E W G [t]½ Fj X[kT],A[kT],C[kT] [ T ( kt+j ] E W G [t]½ Fj + (T j)w) X[kT],A[kT],C[kT] [ T ] kt+j =E W G T(T + )W [t]½ Fj X[kT],A[kT],C[kT] +. j= Thus, V becomes T kt +j V E W G [t]½ Fj X[kT] = X + δ j= T(T + )W L f(x ). (7) = Second, et s consider V. Let + t W F [t] = f(x [kt])min C [kt], A [kt] C [kt] S F [j]. Then, by using Lemma 3 and switching the summations over and t, we have (k+)t L V = E W F [t]s F [t] X[kT] = X. (8) = Let ǫ > 0 and ζ > 0. For each event F j, j =,,..., T, we have W G [kt] > W,..., W G [kt + j ] > W. By using Lemma, we obtain that for any ζ > 0, choose W such that { L } Pr W F [t]s F [t] ( ǫ)w G [t] F j ζ, (9) = hods for any t = kt, kt +,..., kt + j. Hence, we have kt+j L kt+j Pr W F [t]sf [t] ( ǫ) W G [t] = F j { L } Pr W F [t]sf [t] ( ǫ)w G [t], t = kt,...,kt + j F j = jζ Tζ, (30) where we use the fact that given any two events E and E such that Pr{E } ǫ and Pr{E } ǫ, we have Pr{E E } ǫ ǫ. By picking ζ sma enough such that Tζ ζ, we have { kt+j } kt+j L Pr W F [t]s F [t] ( ǫ) W G [t] ζ, = for j =,..., T, which impies that [ kt+j ] L E W F [t]s F [t]½ Fj X[kT] = X = Pr{F j }E = [ kt+j L = Pr{F j }( ǫ)( ζ)e =( ǫ)( ζ)e [ kt+j Fj W F [t]s F [t] X[kT] = X, F j ] [ kt+j ] W G [t] X[kT] = X, F j ] W G [t]½ Fj X[kT] = X, (3) for j =,..., T. Thus, we have (k+)t T L V = E W F [t]s F [t]½ F j = X[kT] = X T kt +j L E W F [t]s F [t]½ F j j= = X[kT] = X T kt +j ( ǫ)( ζ)e W G [t]½ Fj X[kT] (3) j= Thus, by using (7) and (3), V becomes T kt +j V (ǫ + ζ ǫζ)e W G [t]½ Fj X[kT] Since T E E we have j= δ L f(x ) + = kt +j (k+)t V (ǫ + ζ ǫζ)a max T = γ j= T(T + )W W G [t]½ Fj X[kT] W G [t] X[kT] A max T =. (33) L f(x ), (34) = L L f(x ) δ f(x ) + = L f(x ) + H, (35) = where H = T(T+)W and γ = δ A max (ǫ + ζ ǫζ)t. We can choose β, ǫ, ζ sma enough such that γ > 0. T(T + )W

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Bin Li received his B.S. degree in Eectronic Engineering and M.S. degree in Communication Engineering, both from Xiamen University, China, in 005 and 008, respectivey. Between 008 and 009, 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 Boḡaziçi University, Istanbu, in 999, and the M.S. and Ph.D. degrees in Eectrica and Computer Engineering from the University of Iinois at Urbana- Champaign in 00 and 005, respectivey. Between 005 and 007, he worked as a Postdoctora Associate at the Laboratory for Information and Decision Systems at the Massachusetts Institute of Technoogy. He is currenty an Associate 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 00.