OPTIMAL QUEUE-SIZE SCALING IN SWITCHED NETWORKS

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1 The Annals of Appled Probablty 2014, Vol. 24, No. 6, DOI: /13-AAP970 Insttute of Mathematcal Statstcs, 2014 OPTIMAL QUEUE-SIZE SCALING IN SWITCHED NETWORKS BY D. SHAH 1,N.S.WALTON AND Y. ZHONG 1 Massachusetts Insttute of Technology, Unversty of Amsterdam and Columba Unversty We consder a swtched (queung) network n whch there are constrants on whch queues may be served smultaneously; such networks have been used to effectvely model nput-queued swtches and wreless networks. The schedulng polcy for such a network specfes whch queues to serve at any pont n tme, based on the current state or past hstory of the system. In the man result of ths paper, we provde a new class of onlne schedulng polces that acheve optmal queue-sze scalng for a class of swtched networks ncludng nput-queued swtches. In partcular, t establshes the valdty of a conjecture (documented n Shah, Tstskls and Zhong [Queueng Syst. 68 (2011) ]) about optmal queue-sze scalng for nput-queued swtches. 1. Introducton. A swtched network conssts of a collecton of, say, N queues, operatng n dscrete tme. At each tme slot, queues are offered servce accordng to a servce schedule chosen from a specfed fnte set, denoted by S. The rule for choosng a schedule from S at each tme slot s called the schedulng polcy. New work may arrve to each queue at each tme slot exogenously and work served from a queue may jon another queue or leave the network. We shall restrct our attenton, however, to the case where work arrves n the form of unt-szed packets, and once t s served from a queue, t leaves the network, that s, the network s sngle-hop. Swtched networks are specal cases of what Harrson [15, 16] calls stochastc processng networks. Swtched networks are general enough to model a varety of nterestng applcatons. For example, they have been used to effectvely model nput-queued swtches, the devces at the heart of hgh-end Internet routers, whose underlyng slcon archtecture mposes constrants on whch traffc streams can be transmtted smultaneously [8]. They have also been used to model multhop wreless networks n whch nterference lmts the amount of servce that can be gven Receved October 2011; revsed December Supported by NSF TF collaboratve project and NSF CNS CAREER project. When ths work was performed, the thrd author was afflated wth the Laboratory for Informaton and Decson Systems as well as the Operatons Research Center at MIT. The thrd author s now afflated wth the Department of Industral Engneerng and Operatons Research at Columba Unversty. MSC2010 subject classfcatons. 60K25, 60K30, 90B36. Key words and phrases. Swtched network, maxmum weght schedulng, flud models, state space collapse, heavy traffc, dffuson approxmaton. 2207

2 2208 D. SHAH, N. S. WALTON AND Y. ZHONG to each host [35]. Fnally, they can be nstrumental n fndng the rght operatonal pont n a data center [31]. In ths paper, we consder onlne schedulng polces, that s, polces that only utlze hstorcal nformaton (.e., past arrvals and schedulng decsons). The performance objectve of nterest s the total queue sze or total number of packets watng to be served n the network on average (approprately defned). The questons that we wsh to answer are: (a) what s the mnmal value of the performance objectve among the class of onlne schedulng polces, and (b) how does t depend on the network structure, S, as well as the effectve load. Consder a work-conservng M/D/1 queue wth a unt-rate server n whch unt-szed packets arrve as a Posson process wth rate ρ (0, 1). Then, the longrun average queue-sze scales 2 as 1/(1 ρ). Such scalng dependence of the average queue sze on 1/(1 ρ) (or the nverse of the gap, 1 ρ, from the load to the capacty) s a unversally observed behavor n a large class of queung networks. In a swtched network, the scalng of the average total queue sze ought to depend on the number of queues, N. For example, consder N parallel M/D/1 queues as descrbed above. Clearly, the average total queue sze wll scale as N/(1 ρ).on the other hand, consder a varaton where all of these queues pool ther resources nto a sngle server that works N tmes faster. Equvalently, by a tme change, let each of the N queues receve packets as an ndependent Posson process of rate ρ/n, and each tme a common unt-rate server serves a packet from one of the nonempty queues. Then, the average total queue-sze scales as 1/(1 ρ). Indeed, these are nstances of swtched networks that dffer n ther schedulng set S,whch leads to dfferent queue-sze scalngs. Therefore, a natural queston s the determnaton of queue-sze scalng n terms of S and (1 ρ), whereρ s the effectve load. In the context of an n-port nput-queued swtch wth N = n 2 queues, the optmal scalng of average total queue sze has been conjectured to be n/(1 ρ),that s, N/(1 ρ) [29]. As the man result of ths paper, we propose a new onlne schedulng polcy for any sngle-hop swtched network. Ths polcy effectvely emulates an nsenstve bandwdth sharng network wth a product-form statonary dstrbuton wth each component of ths product-form behavng lke an M/M/1 queue. Ths crsp descrpton of statonary dstrbuton allows us to obtan precse bounds on the average queue szes under ths polcy. Ths leads to establshng, as a corollary of our result, the valdty of a conjecture stated n [29] for nput-queued swtches. In general, t provdes explct bounds on the average total queue sze for any swtched network. Furthermore, due to the explct bound on the statonary dstrbuton of queue szes under our polcy, we are able to establsh a form of large-devatons optmalty of the polcy for a large class of sngle-hop swtched networks, ncludng 2 In ths paper, by scalng of quantty we mean ts dependence (gnorng unversal constants) on 1 1 ρ and/or the number of queues, N, as these quanttes become large. Of partcular nterest s the scalng of ρ 1andN,nthatorder.

3 OPTIMAL SCHEDULING 2209 the nput-queued swtches, and the ndependent-set model of wreless networks, when the underlyng nterference graph s bpartte, for example, and more generally, perfect. The conjecture from [29] that we settle n ths paper, states that n the heavy-traffc regme (.e., ρ 1), the optmal average total queue-sze scales as N/(1 ρ). The valdty of ths conjecture s a sgnfcant mprovement over the best-known bounds of O(N/(1 ρ)) (due to the moment bounds of [24] forthe maxmum weght polcy) or O( N log N/(1 ρ) 2 ) (obtaned by usng a batchng polcy [25]). Our analyss conssts of two prncpal components. Frst, we propose and analyze a schedulng mechansm that s able to emulate, n dscrete tme, any contnuous-tme bandwdth allocaton wthn a bounded degree of error. Ths scheduler mantans a contnuous-tme queung process and tracks ts own queue sze process. If, valued under a certan decomposton, the gap between the dealzed contnuous-tme process and the real queung process becomes too large, then an approprate schedule s allocated. Second, we mplement specfc bandwdth allocaton named the store-and-forward allocaton polcy (SFA). Ths polcy was frst consdered by Massoulé, and was consequently dscussed n the thess of Proutère [26], Secton 3.4. It was shown to be nsenstve wth respect to phasetype servce dstrbutons n works by Bonald and Proutère [3, 4]. The nsenstvty of ths polcy for general servce dstrbutons was establshed by Zachary [41]. The store-and-forward polcy s closely related to the classcal product-form multclass queung network, whch have hghly desrable queue-sze scalngs. By emulatng these queung networks, we are able to translate results whch render optmal queue-sze bounds for a swtched network. An nterested reader s referred to [38] and [20] for an n-depth dscusson on the relaton between ths polcy, the proportonally far allocaton, and mult-class queung networks Organzaton. In Secton 2, we specfy a stochastc swtched network model. In Secton 3, we dscuss related works. Secton 4 detals the necessary background on the nsenstve store-and-forward bandwdth allocaton (SFA) polcy. The man result of the paper s presented and proved n Secton 5. Wefrst descrbe the polcy for sngle-hop swtched networks, and state our man result, Theorem 5.2. Ths s followed by a dscusson of the optmalty of the polcy. We then provde a proof of Theorem 5.2. A dscusson of drectons for future work s provded n Secton 6. Notaton. LetN be the set of natural numbers {1, 2,...},letZ + ={0, 1, 2,...}, let R be the set of real numbers and let R + ={x R : x 0}. LetI[A] be the ndcator functon of an event A, Letx y = mn(x, y), x y = max(x, y) and [x] + = x 0. When x s a vector, the maxmum s taken componentwse. We wll reserve bold letters for vectors n R N,whereN s the number of queues. For example, x =[x n ] 1 n N. Superscrpts on vectors are used to denote labels, not

4 2210 D. SHAH, N. S. WALTON AND Y. ZHONG exponents, except where otherwse noted; thus, for example, (x 0, x 1, x 2 ) refers to three arbtrary vectors. Let 0 be the vector of all 0s and 1 the vector of all 1s. The vector e s the th unt vector, wth all components beng 0 but the th component equal to 1. We use the norm x =max n x n. For vectors u and v, welet u v = N n=1 u n v n.leta T be the transpose of matrx A.ForasetS R N, denote ts convex hull by S. Forn N, letn! = n l=1 l be the factoral of n, and by conventon, 0!=1. 2. Swtched network model. We now ntroduce the swtched network model. Secton 2.1 descrbes the general system model, Secton 2.2 lsts the probablstc assumptons about the arrval process and Secton 2.3 ntroduces some useful defntons Queueng dynamcs. Consder a collecton of N queues. Let tme be dscrete, and ndexed by τ {0, 1,...}. LetQ (τ) be the amount of work n queue {1,...,N} at tme slot τ. Followng our general notaton for vectors, we wrte Q(τ) for [Q (τ)] 1 N. The ntal queue szes are Q(0). LetA (τ) be the total amount of work arrvng to queue,andb (τ) be the cumulatve potental servce to queue n, up to tme τ, wth A(0) = B(0) = 0. We frst defne the queung dynamcs for a sngle-hop swtched network. Defnng da(τ) = A(τ + 1) A(τ) and db(τ) = B(τ + 1) B(τ), the basc Lndley recurson that we wll consder s (1) Q(τ + 1) = [ Q(τ) db(τ) ] + + da(τ), where the operaton [ ] + s appled componentwse. The fundamental swtched network constrant s that there s some fnte set S R N + such that (2) db(τ) S for all τ. For the purpose of ths work, we shall focus on S {0, 1} N. We wll refer to σ S as a schedule and S as the set of allowed schedules. In the applcatons n ths paper, the schedule s chosen based on current queue szes, whch s why t s natural to wrte the basc Lndley recurson as (1) rather than the more standard [Q(τ) + da(τ) db(τ)] +. For the analyss n ths paper, t s useful to keep track of two other quanttes. Let Z (τ) be the cumulatve amount of dlng at queue n,defnedbyz(0) = 0 and (3) dz(τ) = [ db(τ) Q(τ) ] +, where dz(τ) = Z(τ + 1) Z(τ). Then, (1) can be rewrtten as (4) Q(τ) = Q(0) + A(τ) B(τ) + Z(τ). Also, let S σ (τ) be the cumulatve amount of tme that s spent on usng schedule σ up to tme τ,sothat (5) B(τ) = σ S S σ (τ)σ.

5 OPTIMAL SCHEDULING 2211 A polcy that decdes whch schedule to choose at each tme slot τ Z + s called a schedulng polcy. In ths paper, we wll be nterested n onlne schedulng polces. That s, the schedulng decson at tme τ wll be based on hstorcal nformaton, that s, the cumulatve arrval process A( ) tll tme τ Stochastc model. We shall assume that the exogenous arrval process for each queue s ndependent and Posson. Specfcally, unt-szed packets arrve to queue as a Posson process of rate λ.letλ =[λ ] N denote the vector of all arrval rates. The results presented n ths paper extend to more general arrval process wth..d. nterarrval tmes wth fnte means, usng a Possonzaton trck. We dscuss ths extenson n Secton Useful quanttes. We shall assume that the schedulng constrant set S s monotone. Ths s captured n the followng assumpton. ASSUMPTION 2.1 (Monotoncty). If S contans a schedule, then S also contans all of ts sub-schedules. Formally, for any σ S, fσ {0, 1} N and σ σ componentwse, then σ S. Wthout loss of generalty, we wll assume that each unt vector e belongs to S. Next, we defne some quanttes that wll be useful n the remander of the paper. DEFINITION 2.2 (Admssble regon). Let S {0, 1} N be the set of allowed schedules. Let S be the convex hull of S, thats, { S = α σ σ : } α σ = 1andα σ 0, for all σ. σ S σ S Defne the admssble regon C to be C = { λ R N + : λ σ componentwse, for some σ S }. Note that under Assumpton 2.1, the capacty regon C and the convex hull S of S concde. Gven that S s a polytope contaned n [0, 1] N, there exsts an nteger J 1, amatrxr R J + N and a vector C R J + such that (6) S = { x [0, 1] N : Rx C }. We call J the rank of S n the representaton (6). When t s clear from the context, we smply call J the rank of S. Note that ths rank may be dfferent from the rank of matrx R. Our results wll explot the fact that the rank J may be an order of magntude smaller than N.

6 2212 D. SHAH, N. S. WALTON AND Y. ZHONG Defne the statc plan- DEFINITION 2.3 (Statc plannng problems and load). nng optmzaton problem PRIMAL(λ) for λ R N + to be (7) mnmze α σ, σ S (8) subject to λ α σ σ, σ S (9) α σ R + for all σ S. Defne the nduced load by λ, denoted by ρ(λ), as the value of the optmzaton problem PRIMAL(λ). Note that λ s admssble f and only f ρ(λ) 1. It also follows mmedately from Defnton 2.3 that (10) ρ(λ) = nf{γ 0:Rλ γ C}, and λ s admssble f and only f Rλ C, componentwse. In the sequel, we wll often consder the quanttes ρ j = R j λ /C j,forj {1, 2,...,J}, whch can be nterpreted as loads on ndvdual resources of the system (ths nterpretaton wll be made precse n Secton 4). They are closely related to the system load ρ(λ). We formalze ths relaton n the followng lemma, whose proof s straghtforward and omtted. LEMMA 2.4. Consder a nonnegatve matrx R R J + N and a vector C R J wth C j > 0 for all j. For a nonnegatve vector λ R N +, defne ρ(λ) by (10) and ρ j = ( R j λ )/C j. Then ρ(λ) = max j ρ j. (11) The followng s a smple and useful property of ρ( ):foranya, b R N +, ρ(a + b) ρ(a) + ρ(b) Motvatng example. An Internet router has several nput ports and output ports. A data transmsson cable s attached to each of these ports. Packets arrve at the nput ports. The functon of the router s to work out whch output port each packet should go to, and to transfer packets to the correct output ports. Ths last functon s called swtchng. There are a number of possble swtch archtectures; we wll consder the commercally popular nput-queued swtch archtecture. Fgure 1 llustrates an nput-queued swtch wth three nput ports and three output ports. Packets arrvng at nput k destned for output l are stored at nput port k, n queue Q k,l, thus there are N = 9 queues n total. (For ths example, t s more natural to use double ndexng, e.g., Q 3,2, whereas for general swtched networks t s more natural to use sngle ndexng, e.g., Q for 1 N.) The swtch operates n dscrete tme. At each tme slot, the swtch fabrc can transmt a number of packets from nput ports to output ports, subject to the two

7 OPTIMAL SCHEDULING 2213 FIG. 1. An nput-queued swtch, and two example matchngs of nputs to outputs. constrants that each nput can transmt at most one packet, and that each output can receve at most one packet. In other words, at each tme slot the swtch can choose a matchng from nputs to outputs. The schedule σ R s gven by σ k,l = 1 f nput port k s matched to output port l n a gven tme slot, and σ k,l = 0 otherwse. The matchng constrants requre that 3 m=1 σ k,m 1fork = 1, 2, 3, and 3 m=1 σ m,l 1forl = 1, 2, 3. Fgure 1 shows two possble matchngs. On the left-hand sde, the matchng allows a packet to be transmtted from nput port 3 to output port 2, but snce Q 3,2 s empty, no packet s actually transmtted. In general, for an n-port swtch, there are N = n 2 queues. The correspondng schedule set S s defned as { } n n (12) S = σ {0, 1} n n : σ k,m 1, σ m,l 1, 1 k,l n. m=1 It can be checked that S s monotone. Furthermore, due to Brkhoff von Neumann theorem, [2, 37], the convex hull of S s gven by { } n n (13) S = x [0, 1] n n : x k,m 1, x m,l 1, 1 k,l n. m=1 Thus, the rank of S s less than or equal to 2n = 2 N for an n-port swtch. Fnally, gven an arrval rate matrx 3 λ [0, 1] n n, ρ(λ) s gven by { n n ρ(λ) = max λ k,m, λ m,l }. 1 k,l n m=1 3. Related works. The queston of determnng the optmal scalng of queue szes n swtched networks, or more generally, stochastc processng networks, has m=1 m=1 m=1 3 Not a vector, for notatonal convenence, as dscussed earler.

8 2214 D. SHAH, N. S. WALTON AND Y. ZHONG been an mportant ntellectual pursut for more than a decade. The complexty of the generc stochastc processng network makes ths task extremely challengng. Therefore, n search of tractable analyss, most of the pror work has been on tryng to understand optmal scalng and schedulng polces for scaled systems: prmarly, wth respect to flud and heavy-traffc scalng, that s, ρ 1. In heavy-traffc analyss, one studes the queue-sze behavor under a dffuson (or heavy-traffc) scalng. Ths regme was frst consdered by Kngman [21]; snce then, a substantal body of theory has developed, and modern treatments can be found n [5, 14, 39, 40]. Stolyar [33] has studed a class of myopc schedulng polces, known as the maxmum weght polcy, ntroduced by Tassulas and Ephremdes [35], for a generalzed swtch model n the dffuson scalng. In a general verson of the maxmum weght polcy, a schedule wth maxmum weght s chosen at each tme step, wth the weght of a schedule beng equal to the sum of the weghts of the queues chosen by that schedule. The weght of a queue s a functon of ts sze. In partcular, for the choce of one parameter class of functons parameterzed by α>0, f(x)= x α, the resultng class of polces are called the maxmum weght polces wth parameter α>0, and denoted as MW-α. In [33], a complete characterzaton of the dffuson approxmaton for the queue-sze process was obtaned, under a condton known as complete resource poolng, when the network s operatng under the MW-α polcy, for any α>0. Stolyar [33] showed the remarkable result that the lmtng queue-sze vector lves n a one-dmensonal state space. Operatonally, ths means that all one needs to keep track of s the one-dmensonal total amount of work n the system (called the rescaled workload), and at any pont n tme one can assume that the ndvdual queues have all been balanced. Furthermore, t was establshed that a maxweght polcy mnmzes the rescaled workload nduced by any polcy under the heavy-traffc scalng (wth complete resource poolng). Da and Ln [6, 7] have establshed that a smlar result holds (wth complete resource poolng) n the more general settng of a stochastc processng network. In summary, under the complete resource poolng condton, the results n [6, 7, 33] mply that the performance of the maxmum weght polcy n an nput-queued swtch, or more generally n a stochastc processng network, s always optmal (n the dffuson lmt, and when each queue sze s approprately weghted). These results suggest that the average total queue-sze scales as 1/(1 ρ) n the ρ 1 lmt. However, such analyses do not capture the dependence on the network schedulng structure S. Essentally, ths s because the complete resource poolng condton reduces the system to a onedmensonal space (whch may be hghly dependent on a network s structure), and optmalty results are then ntally expressed wth respect to ths one-dmensonal space. Motvated to capture the dependence of the queue szes on the network schedulng structure S, a heavy-traffc analyss of swtched networks wth multple bottlenecks (wthout resource poolng) was pursued by Shah and Wschk [32]. They

9 OPTIMAL SCHEDULING 2215 establshed the so-called multplcatve state space collapse, and dentfed a member, denoted by MW-0 + (obtaned by takng α 0), of the class of maxmumweght polces as optmal wth respect to a crtcal flud model. In a more recent work, Shah and Wschk [31] establshed the optmalty of MW-0 + wth respect to overloaded flud models as well. However, ths collecton of works stops short of establshng optmalty for dffuson scaled queue-sze processes. Fnally, we take note of the work by Meyn [23], whch establshes that a class of generalzed maxmum weght polces acheve logarthmc [n 1/(1 ρ)]regret wth respect to an optmal polcy under certan condtons. In a related model the bandwdth-sharng network model Kang et al. [18] have establshed a dffuson approxmaton for the proportonally far bandwdth allocaton polcy, assumng a techncal local traffc condton, but wthout assumng complete resource poolng. 4 They show that the resultng dffuson approxmaton has a product-form statonary dstrbuton. Shah, Tstskls and Zhong [30] have recently establshed that ths product-form statonary dstrbuton s ndeed the lmt of the statonary dstrbutons of the orgnal stochastc model (an nterchange-of-lmts result). As a consequence, f one could utlze a schedulng polcy n a swtched network that corresponds to the proportonally far polcy, then the resultng dffuson approxmaton wll have a product-form statonary dstrbuton, as long as the effectve network schedulng structure S (precsely S ) satsfes the local traffc condton. Now, proportonal farness s a contnuoustme rate allocaton polcy that usually requres rate allocatons that are a convex combnaton of multple schedules. In a swtched network, a polcy must operate n dscrete tme and has to choose one schedule at any gven tme from a fnte dscrete set S. For ths reason, proportonal farness cannot be mplemented drectly. However, a natural randomzed polcy nspred by proportonal farness s lkely to have the same dffuson approxmaton (snce the flud models would be dentcal, and the entre machnery of Kang et al. [18], buldng upon the work of Bramson [5] and Wllams [40], reles on a flud model). As a consequence, f S (more accurately, S ) satsfes the local traffc condton, then effectvely the dffuson-scaled queue szes would have a product-form statonary dstrbuton, and would result n bounds smlar to those mpled by our results. In comparson, our results are nonasymptotc, n the sense that they hold for any admssble load, have a product-form structure, and do not requre techncal assumptons such as the local traffc condton. Furthermore, such generalty s needed because there are popular examples, such as the nput-queued swtch, that do not satsfy the local traffc condton. Another lne of works so-called large-devatons analyss concerns exponentally decayng bounds on the tal probablty of the steady-state dstrbutons of 4 Kang et al. [18] assume that crtcally loaded traffc s such that all the constrants are saturated smultaneously.

10 2216 D. SHAH, N. S. WALTON AND Y. ZHONG queue szes. Venkataramanan and Ln [36] establshed that the maxmum weght polcy wth weght parameter α>0, MW-α, optmzes the tal exponent of the 1 + α norm of the queue-sze vector. Stolyar [34] showed that a so-called exponental rule optmzes the tal exponent of the max norm of the queue-sze vector. However, these works do not characterze the tal exponent explctly. See [28] whch has the best-known explct bounds on the tal exponent. In the context of nput-queued swtches, the example that has prmarly motvated ths work, the polcy that we propose has the average total queue sze bounded wthn factor 2 of the same quantty nduced by any polcy, n the heavytraffc lmt. Furthermore, ths result does not requre condtons lke complete resource poolng. More generally, our polcy provdes nonasymptotc bounds on queue szes for every arrval rate and swtch sze. The polcy even admts exponental tal bounds wth respect to the statonary dstrbuton, and the exponent of these tal bounds s optmal. These results are sgnfcant mprovements to the state-ofthe-art bounds for best performng polces for nput-queued swtches. As noted n the Introducton, our bound on the average total queue sze s N tmes better than the exstng bound for the maxmum-weght polcy, and log N/(1 ρ) tmes better than that for the batchng polcy n [25]. (Here N s the number of queues, and ρ the system load.) For further detals of these results, see [29]. For a generc swtched network, our polcy nduces average total queue sze that scale lnearly wth the rank of S, under the dffuson scalng. Ths s n contrast to the best-known bounds, such as those for maxmum weght polcy, where the average queue-sze scales as N, under the dffuson scalng. Therefore, whenever the rank of S s smaller than N (the number of queues), our polcy provdes tghter bounds. Under our polcy, queue szes admt exponental tal bounds. The bound on the dstrbuton of queue szes under our polcy leads to an explct characterzaton of the tal exponent, whch s optmal for a wde range of sngle-hop swtched networks, ncludng nput-queued swtches and the ndependent-set model of wreless networks, when the underlyng nterference graph s perfect. 4. Insenstvty n stochastc networks. Ths secton recalls the background on nsenstve stochastc networks that underles the man results of ths work. We shall focus on descrptons of the nsenstve bandwdth allocaton n so-called bandwdth-sharng networks operatng n contnuous tme. Propertes of these nsenstve networks are provded n the Appendx. We consder a bandwdth-sharng network operatng n contnuous tme wth capacty constrants. The partcular bandwdth-sharng polcy of nterest s the store-and-forward allocaton (SFA) mentoned earler. We shall use the SFA as an dealzed polcy to desgn onlne schedulng polces for swtched networks. We now descrbe the precse model, the SFA polcy, and ts performance propertes. Model. Let tme be contnuous and ndexed by t R +. Consder a network wth J 1 resources ndexed from 1,...,J. Let there be N routes, and suppose

11 OPTIMAL SCHEDULING 2217 that each packet on route consumes an amount R j 0 of resource j, for each j {1, 2,...,J}.LetK be the set of all resource route pars (j, ) such that route uses resource j, thats,k ={(j, ) : R j > 0}. Wthout loss of generalty, we assume that for each {1, 2,...,N}, J j=1 R j > 0. Let R be the J N matrx wth entres R j.letc R J + be a postve capacty vector wth components C j. For each route, packets arrve as an ndependent Posson process of rate λ. Packets arrvng on route requre a unt amount of servce, determnstcally. We denote the number of packets on route at tme t by M (t), and defne the queue-sze vector at tme t by M(t) =[M (t)] N ZN +. Each packet gets servce from the network at a rate determned accordng to a bandwdth-sharng polcy. We also denote the total resdual workload on route at tme t by W (t), andlet the vector of resdual workload at tme t be W(t) =[W (t)] N. Once a packet receves ts total (unt) amount of servce, t departs the network. We consder onlne, myopc bandwdth allocatons. That s, the bandwdth allocaton at tme t only depends on the queue-sze vector M(t). When there are m packets on route, that s, f the vector of packets s m =[m ] N, let the total bandwdth allocated to route be φ (m) R +. We consder a processor-sharng polcy, so that each packet on route s served at rate φ (m)/m,fm > 0. If m = 0, let φ (m) = 0. If the bandwdth vector φ(m) =[φ (m)] N satsfes the capacty constrants (14) Rφ(m) C componentwse for all m Z N +, then, n lght of Defnton 2.2, we say that φ( ) s an admssble bandwdth allocaton. A Markovan descrpton of the system s gven by a process Y(t) whch contans the queue-sze vector M(t) along wth the resdual workloads of the set of packets on each route. Now, on average, λ unts of work arrve to route per unt tme. Therefore, n order for the Markov process Y( ) to be postve (Harrs) recurrent, t s necessary that (15) Rλ < C componentwse. All such λ =[λ ] N RN + wll be called strctly admssble, n the same sprt as strctly admssble vectors for a swtched network. Smlarly to the correspondng swtched network, gven λ R N +, we can defne ρ(λ), the load nduced by λ, usng (10), as well as ρ j = ( R j λ )/C j. Then by Lemma 2.4, ρ(λ) = max j ρ j, where ρ j can be nterpreted as the load nduced by λ on resource j. Store-and-forward allocaton (SFA) polcy. We descrbe the store-and-forward allocaton polcy that was frst consdered by Massoulé and later analyzed n the thess of Proutère [26]. Bonald and Proutère [4] establshed that ths polcy nduces product-form statonary dstrbutons and s nsenstve wth respect to phase-type dstrbutons. Ths polcy s shown to be nsenstve for general servce tme dstrbutons, ncludng the determnstc servce consdered here, by

12 2218 D. SHAH, N. S. WALTON AND Y. ZHONG Zachary [41]. The relaton between ths polcy, the proportonally far allocaton, and mult-class queung networks s dscussed n depth by Walton [38] and Kelly, Massoulé and Walton [20]. The nsenstvty property mples that the nvarant measure of the process M(t) only depends on the parameters λ =[λ ] N RN +, and no other aspects of the stochastc descrpton of the system. We frst gve an nformal motvaton for SFA. SFA s closely related to quasreversble queung networks. Consder a contnuous-tme mult-class queung network (wthout schedulng constrants) consstng of processor sharng queues ndexed by j {1,...,J} and job types ndexed by the routes {1,...,N}. Each route job has a servce requrement R j at each queue j, and a fxed servce capacty C j s shared between jobs at the queue. Here each job wll sequentally vst all the queues (so-called store-and-forward) and wll vst each queue a fxed number of tmes. If we assume that jobs on each route arrve as a Posson process, then the resultng queung network wll be stable for all strctly admssble arrval rates. Moreover, each statonary queue wll be ndependent wth a queue sze that scales, wth ts load ρ,asρ/(1 ρ). For further detals, see Kelly [19]. So, assumng each queue has equal load, the total number of jobs wthn the network s of the order Jρ/(1 ρ). In other words, these networks have the stablty and queue-sze scalng that we requre, but do not obey the necessary schedulng constrants (14). However, these networks do produce an admssble schedule on average. For ths reason, we consder an SFA polcy whch, gven the number of jobs on each route, allocates the average rate wth whch jobs are transferred through ths mult-class network. Next, we descrbe ths polcy (usng notaton smlar to those used n [20, 38]). Gven m Z N +,defne { U(m) = m = ( m j : (j, ) K ) Z K + : j : j For L Z J +,wealsodefne { V(L) = m = ( m j : (j, ) K ) Z K + : : j } m j = m, for all 1 N. } m j = L j, for all 1 j J. Here, by notaton j (and j) we mean R j > 0. For each m U(m), we explot notaton somewhat and defne m j = : j m j,forallj J.Alsodefne ( ) m j m j! = m j : j : j ( m j!). For m Z N +,wedefne (m) as (16) (m) = (( m U(m) j J m j ) m j : j : j ( Rj C j ) mj ).

13 OPTIMAL SCHEDULING 2219 We shall defne (m) = 0 f any of the components of m s negatve. The storeand-forward allocaton (SFA) assgns rates accordng to the functon φ : Z N + R N +, so that for any m ZN +, φ(m) = (φ (m)) N, wth φ (m) = (m e ) (17), (m) where, recallng that m e s the same as m at all but the th component, ts th component equals m 1. The bandwdth allocaton φ(m) s the statonary throughput of jobs on the routes of a mult-class queung network (descrbed above), condtonal on there beng m jobs on each route. A pror t s not clear f the above descrbed bandwdth allocaton s even admssble, that s, satsfes (14). Ths can be argued as follows. The φ(m) can be related to the statonary throughput of a mult-class network wth a fnte number of jobs, m, on each route. Under ths scenaro (due to fnte number of jobs), each queue must be stable. Therefore, the load on each queue, Rφ(m), must be less than the overall system capacty C. That s, the allocaton s admssble. The precse argument along these lnes s provded n, for example, [20], Corollary 2 and [38], Lemma 4.1. The SFA nduces a product-form nvarant dstrbuton for the number of packets watng n the bandwdth-sharng network and s nsenstve. We summarze ths n the followng result. THEOREM 4.1. Consder a bandwdth-sharng network wth Rλ < C. Under the SFA polcy descrbed above, themarkovprocessy(t) s postve (Harrs) recurrent, and M(t) has a unque statonary probablty dstrbuton π gven by (18) where π(m) = (m) N λ m for all m Z N +, (19) J ( ) C j = C j=1 j : j R j λ s a normalzng factor. Furthermore, the steady-state resdual workload of packets watng n the network can be characterzed as follows. Frst, the steady-state dstrbuton of the resdual workload of a packet s ndependent from π. Second, n steady state, condtoned on the number of packets on each route of the network, the resdual workload of each packet s unformly dstrbuted on [0, 1], and s ndependent from the resdual workloads of other packets. Note that statements smlar to Theorem 4.1 have appeared n other works, for example, [3], [38], Proposton 4.2, and [20]. Theorem 4.1 s a summary of these statements, and for completeness, t s proved n Appendx A.

14 2220 D. SHAH, N. S. WALTON AND Y. ZHONG The followng property of the statonary dstrbuton π descrbed n Theorem 4.1 wll be useful. PROPOSITION 4.2. Consder the setup of Theorem 4.1, and let π be descrbed by (18). Defne a measure π on Z K + as follows: for m Z K +, π( m) = 1 J (( ) m j ( ) Rj λ mj ) (20). m j=1 j : j C : j j Then, for any L Z +, ({ }) ({ }) N J (21) π m : m = L = π m : m j = L. j=1 We relate the dstrbuton π to the statonary dstrbuton of an nsenstve multclass queung network wth a product-form statonary dstrbuton and geometrcally dstrbuted queue szes. PROPOSITION 4.3. Consder the dstrbuton π defned n (20). Then, for any L = (L 1,...,L J ) Z J +, (22) π( m 1 = L 1,..., m J = L J ) = ( m j ) V(L) π ( ( m j ) ) = J j=1 ρ L j j (1 ρ j ), where ρ j = ( : j R j λ j )/C j. Usng Theorem 4.1 and Propostons 4.2 and 4.3, we can compute the expected value and the probablty tal exponent of the steady-state total resdual workload n the system. Recall that the total resdual workload n the system at tme t s gven by N W (t). PROPOSITION 4.4. Consder a bandwdth-sharng network wth Rλ < C, operatng under the SFA polcy. Denote the load nduced by λ to be ρ = ρ(λ)(< 1), and for each j, let ρ j = ( R j λ )/C j. Then W( ) has a unque statonary probablty dstrbuton. Wth respect to ths statonary dstrbuton, the followng propertes hold: () The expected total resdual workload s gven by [ N ] E W = 1 J ρ j (23). 2 1 ρ j=1 j

15 OPTIMAL SCHEDULING 2221 () The dstrbuton of the total resdual workload has an exponental tal wth exponent gven by ( 1 N ) lm L L log P (24) W L = θ, where θ s the unque postve soluton of the equaton ρ(e θ 1) = θ. 5. Man result: A polcy and ts performance. In ths secton, we descrbe an onlne schedulng polcy and quantfy ts performance n terms of explct, closed-form bounds on the statonary dstrbuton of the nduced queue szes. Secton 5.1 descrbes the polcy for a generc swtched network and provdes the statement of the man result. Secton 5.2 dscusses ts mplcatons. Specfcally, t dscusses (a) the optmalty of the polcy for a large class of swtched networks wth respect to exponental tal bounds, and (b) the optmalty of the polcy for a class of swtched networks, ncludng nput-queued swtches, wth respect to the average total queue sze. Secton 5.3 proves the man result stated n Secton A polcy for swtched networks. The basc dea behnd the polcy, to be descrbed n detal shortly, s as follows. Gven a swtched network, denoted by SN, wth constrant set S and N queues, let S have rank J and representaton [cf. (6)] S = { x [0, 1] N : Rx C }, R R N J +, C R J +. Now consder a vrtual bandwdth-sharng network, denoted by BN, wth N routes correspondng to each of these N queues. The resource route relaton s determned precsely by the matrx R, andthej resources have capactes gven by C. Both networks, SN and BN, are fed dentcal arrvals. That s, whenever a packet arrves to queue n SN, a packet s added to route n BN at the same tme. The man queston s that of determnng a schedulng polcy for SN; ths wll be derved from BN. Specfcally, BN wll operate under the nsenstve SFA polcy descrbed n Secton 4. By Theorem 4.1 and Propostons 4.2 and 4.3, ths wll nduce a desrable statonary dstrbuton of queue szes n BN. Therefore, f we could use the rate allocaton of BN, that s, the SFA polcy, drectly n SN, t would gve us a desred performance n terms of the statonary dstrbuton of the nduced queue szes. Now the rate allocaton n BN s such that the nstantaneous rate s always nsde S. However, t could change all the tme and need not utlze ponts of S as rates. In contrast, n SN we requre that the rate allocaton can change only once per dscrete tme slot and t must always employ one of the generators of S, that s, a schedule from S. The key to our polcy s an effectve way to emulate the rate allocaton of BN under SFA (or for that matter, any admssble bandwdth allocaton) by utlzng schedules from S n an onlne manner and wth the dscrete-tme constrant. We wll see shortly that ths emulaton polcy reles on S beng monotone; cf. Assumpton 2.1.

16 2222 D. SHAH, N. S. WALTON AND Y. ZHONG To that end, we descrbe ths emulaton polcy. Let us start by ntroducng some useful notaton. Let A( ) = (A ( )) be the vector of exogenous, ndependent Posson processes accordng to whch unt-szed packets arrve to both BN and SN, smultaneously. Recall that A ( ) s a Posson process wth rate λ.let M(t) = (M (t)) denote the vector of numbers of packets watng on the N routes n BN at tme t 0. In BN, the servces are allocated accordng to the SFA polcy descrbed n Secton 4. Let SFA ( ) = ( SFA ( )) R N + denote the cumulatve amount of servce allocated to the N routes n BN under the SFA polcy: SFA (t) denotes the total amount of servce allocated to all packets on route durng the nterval [0,t], fort 0, wth SFA (0) = 0for1 N. By defnton, all components of SFA ( ) are nondecreasng and Lpschtz contnuous. Furthermore, ( SFA (t + s) SFA (t))/s S for any t 0ands>0. Recall that the (rght-)dervatve of SFA ( ) s determned by M( ) through the functon φ( ) as defned n (17). Now we descrbe the schedulng polcy for SN that wll rely on SFA ( ). Let B(τ) = (B (τ)) denote the cumulatve amount of servce allocated n SN by the schedulng polcy up to tme slot τ 0, wth B(0) = 0. The schedulng polcy determnes how B( ) s updated. Let Q(τ) = (Q (τ)) be the queue szes measured at the end of tme slot τ. Let servce be provded accordng to the schedulng polcy nstantly at the begnnng of a tme slot. Thus, the schedulng polcy decdes the schedule db(τ) = B(τ + 1) B(τ) S at the very begnnng of tme slot τ + 1. Ths decson s made as follows. Let D(τ) = SFA (τ) B(τ). We wll see shortly that under our polcy, D(τ) s always nonnegatve. Ths fact wll be useful at varous places, and n partcular, for boundng the dscrepancy between the contnuous-tme polcy SFA and ts dscrete-tme emulaton. Let ρ(d(τ)) be the optmal objectve value n the optmzaton problem PRIMAL(D(τ)) defned n (7). In partcular, there exsts a nonnegatve combnaton of schedules n S such that (25) α σ σ D(τ) and α σ = ρ ( D(τ) ). σ S We clam that n fact, we can fnd nonnegatve numbers α σ, σ S, such that (26) α σ σ = D(τ) and α σ = ρ ( D(τ) ). σ S Ths s formalzed n the followng lemma. LEMMA 5.1. Let D R N + be a nonnegatve vector. Consder the statc plannng problem PRIMAL(D) defned n (7). Let the optmal objectve value to PRIMAL(D) be ρ(d). Then there exsts α σ 0, σ S, such that (26) holds. The proof of the lemma reles on Assumpton 2.1, and s provded n the Appendx. σ S σ S

17 OPTIMAL SCHEDULING 2223 There could be many possble nonnegatve combnatons of D(τ) satsfyng (26). If there exst nonnegatve numbers α σ, σ S, satsfyng (26) wth α σ 1 for some σ S, then choose σ as the schedule: set db(τ) = σ.ifnosuchdecomposton exsts for D(τ),thensetdB(τ) = σ,where σ s a soluton (tes broken arbtrarly) of (27) maxmze σ over σ S, σ D(τ). Here frst observe that for all tme τ, db(τ) D(τ), sod(τ) 0. Hence, 0 s a feasble soluton for the above problem, as 0 S. The above s a complete descrpton of the schedulng polcy. Observe that t s an onlne polcy, as the vrtual network BN can be smulated n an onlne manner, and, gven ths, the schedulng decson n SN reles only on the hstory of BN and SN. The followng result quantfes the performance of the polcy. THEOREM 5.2. Gven a strctly admssble arrval rate vector λ, wth ρ = ρ(λ) <1, under the polcy descrbed above, the swtched network SN s postve recurrent and has a unque statonary dstrbuton. Let ρ j = ( R j λ )/C j, j = 1, 2,...,J be the same as n Proposton 4.4. Wth respect to ths statonary dstrbuton, the followng propertes hold: (1) The expected total queue sze s bounded as [ N ] E Q 1 ( J ) ρ j (28) + K(N + 2), 2 1 ρ j=1 j where K = max σ S ( σ ). (2) The dstrbuton of the total queue sze has an exponental tal wth exponent gven by ( 1 N ) lm L L log P (29) Q L = θ, where θ s the unque postve soluton of the equaton ρ(e θ 1) = θ Optmalty of the polcy. Ths secton establshes the optmalty of our polcy for nput-queued swtches, both wth respect to expected total queue-sze scalng and tal exponent. General condtons under whch our polcy s optmal wth respect to tal exponent are also provded. Scalng of queue szes. We start by formalzng what we mean by the optmalty of expected queue szes and of ther tal exponents. We consder polces under whch there s a well-defned lmtng statonary dstrbuton of the queue szes for all λ such that ρ(λ)<1. Note that the class of polces s not empty; ndeed,

18 2224 D. SHAH, N. S. WALTON AND Y. ZHONG the maxmum weght polcy and our polcy are members of ths class. Wth some abuse of notaton, let π denote the statonary dstrbuton of the queue-sze vector under the polcy of nterest. We are nterested n two quanttes: (1) Expected total queue sze.let Q be the expected total queue sze under the statonary dstrbuton π, defnedby Q = E π [ Q ]. Note that by ergodcty, the tme average of the total queue sze and the expected total queue sze under π are the same quantty. (2) Tal exponent. Letβ L (Q), β U (Q) [, 0] be the lower and upper lmts of the tal exponent of the total queue sze under π (possbly or 0), respectvely, defned by (30) (31) ( 1 β L (Q) = lm nf l l log P π ( 1 β U (Q) = lm sup l l log P π ) Q l ) Q l. If β L (Q) = β U (Q), then we denote ths common value by β(q). We are nterested n polces that can acheve mnmal Q and β(q). For tractablty, we focus on scalngs of these quanttes wth respect to S (equvalently, N) and ρ(λ), as1/(1 ρ(λ)) and N ncrease. For dfferent λ and λ, t s possble that ρ(λ) = ρ(λ ), but the scalng of Q, for example, could be wldly dfferent. For ths reason, we consder the worst possble dependence on 1/(1 ρ) and N among all λ wth ρ(λ) = ρ. Note that we are consderng scalngs wth respect to two quanttes, ρ and N, and we are nterested n two lmtng regmes, ρ 1andN. The optmalty of queue-sze scalng stated here s wth respect to the order of lmts ρ 1and then N. As noted n [29], takng the lmts n dfferent orders could potentally result n dfferent lmtng behavors of the object of nterest, for example, Q. For further dscusson, see Secton 6. It should be noted, however, that whenever the tal exponent s optmal, ths optmalty holds for any ρ and N. Optmalty of the tal exponent. Here we establsh suffcent condtons under whch our polcy s optmal wth respect to tal exponent. Frst, we present a unversal lower bound on the tal exponent, for a general sngle-hop swtched network under any polcy. We then provde a condton under whch ths lower bound matches the tal exponent under our polcy. Ths condton s satsfed by both nput-queued swtches and the ndependent-set model of wreless networks. Consder any polcy under whch there exsts a well-defned lmtng statonary dstrbuton of the queue szes for all λ such that ρ(λ) <1. Let π 0 denote and

19 OPTIMAL SCHEDULING 2225 the statonary dstrbuton of queue szes under ths polcy. The followng lemma establshes a unversal lower bound on the tal exponent. LEMMA 5.3. Consder a swtched network as descrbed n Theorem 5.2, wth schedulng set S and admssble regon {x [0, 1] N : Rx C}. Let π 0 and λ be as descrbed. For each j, let ρ j = N R j λ /C j be defned as n Theorem 5.2. Then under π 0, ( 1 ) (32) lm nf L L log P π 0 Q L mn θ j, j=1,2,...,j where, for each j {1, 2,...,J}, θj s the unque postve soluton of the equaton N ( λ e R j θ 1 ) = θ. PROOF. Consder a fxed j {1, 2,...,J}. Wthout loss of generalty, we assume that C j = 1, by properly normalzng the nequalty (Rx) j C j. In ths case, R j 1forall, snce for each {1, 2,...,N}, e S S, and satsfes the constrant (Re ) j = R j C j = 1. Now consder the followng sngle-server queung system. The arrval process s gven by the sum N R j A ( ), so that arrvals across tme slots are ndependent, and that n each tme slot, the amount of work that arrves s N R j a,wherea s an ndependent Posson random varable wth mean λ, for each. Note that the arrvng amount n a sngle tme slot does not have to be ntegral. Note also that N R j λ = ρ j < 1, snce ρ(λ) = max j ρ j < 1. In each tme slot, a unt amount of servce s allocated to the total workload n the system. Then, for ths system, the workload process W( ) satsfes W(τ + 1) = [ W(τ) 1 ] + N + R j a (τ), where a (τ) s the number of arrvals to queue n the orgnal system n tme slot τ. We make two observatons for ths system. Frst, W( ) s stochastcally domnated by N R j Q ( ), whereq ( ) s the sze of queue n the orgnal system, under any onlne schedulng polcy. Ths s because for all schedules σ S, σ satsfes Rσ C, and hence N R j σ C j = 1foreveryσ S. Second, snce R j 1forall, N R j Q ( ) s stochastcally domnated by N Q ( ). Thus we have lm nf L We now show that ( 1 ) L log P 1 π 0 Q L lm nf L L log P( W( ) L ). lm nf L 1 L log P( W( ) L ) θ j,

20 2226 D. SHAH, N. S. WALTON AND Y. ZHONG where θ j s the unque postve soluton of the equaton N ( λ e R j θ 1 ) = θ. Consder the log-moment generatng functon (log-mgf) of N R j a,thearrvng amount n one tme slot. Snce a s a Posson random varable wth mean λ for each, ts moment generatng functon s gven by Hence the log-mgf s ( N ( f(θ)= exp λ e R j θ 1 )). By Theorem 1.4 of [13], N ( log f(θ)= λ e R j θ 1 ). 1 lm L L log P( W( ) L ) = θj, where θ j = sup{θ >0:logf(θ)<θ}. Snce log f(θ) θ s strctly convex, θ j satsfes j {1, 2,...,J} s arbtrary, so lm nf L 1 L log P π 0 N ( λ e R j θj 1 ) = θj. ( ) Q L mn θ j. j=1,2,...,j For general swtched networks, the lower bound above need not match the tal exponent acheved under our polcy [cf. (29)]. However, for a wde class of swtched networks, these two quanttes are equal. The followng corollary of Lemma 5.3 s mmedate. COROLLARY 5.4. Consder a swtched network as descrbed n Lemma 5.3, wth schedulng set S and admssble regon {x [0, 1] N : Rx 1}. If for all j and, R j {0, 1}, then our polcy acheves optmal tal exponent, for any strctly admssble arrval-rate vector λ. PROOF. Letλ R N + be strctly admssble, that s, Rλ < 1.Let ρ j = R j λ for each j, andletρ = ρ(λ) be the system load nduced by λ. Consder the θj n

21 OPTIMAL SCHEDULING 2227 Lemma 5.3. WhenR j {0, 1} for all j, and, θj s the unque postve soluton of the equaton ρ j ( e θ 1 ) = θ for each j. Usng the relaton ρ = max j ρ j, we see that mn j θj postve soluton of the equaton ρ ( e θ 1 ) = θ. s the unque Comparng ths wth equaton (29) of Theorem 5.2, we see that our polcy acheves the optmal tal exponent. Consder an n n nput-queued swtch, defned n Secton 2.4, and wth the admssble regon descrbed by (13). By Corollary 5.4, t s clear that the tal exponent n nput-queued swtches s optmal under our polcy. Moreover, nput-queued swtches are not the only network model that satsfes the condton stated n Corollary 5.4. For example, consder the ndependent-set model of a wreless network. When the underlyng nterference graph s bpartte, t s easy to see that the admssble regon s characterzed by nequaltes of the form x + x j 1 over all edges (, j) of the graph, and x 1 for solated nodes. More generally, when the underlyng graph s perfect, nequalty constrants characterzng the admssbleregontaketheform x 1, where the summaton s over all vertces of a clque. Ths latter fact follows from a proof of the weak perfect graph theorem, see, for example, Theorem n [22]. Thus the ncdence matrx R has all entres n {0, 1}, and the tal exponent under our polcy s optmal for ths model. Optmalty n nput-queued swtches. Here we argue the optmalty of our polcy for nput-queued swtches. As dscussed above, the scalng of tal exponent s optmal under our polcy for nput-queued swtches. We would argue the optmal scalng of the average total queue sze under our polcy for nput-queued swtches. To that end, as argued n Shah, Tstskls and Zhong [29], when all nput and output ports approach crtcal load, the average total queue sze under any polcy for nput-queued swtch must scale at least as fast as N/(1 ρ), foranyn-port swtch wth N = n 2 queues. For completeness, we nclude the proof for ths lower bound here. As n Secton 2.4, we use double ndexng. LEMMA 5.5. Consder an n-port nput-queued swtch, wth an arrval rate vector λ. Suppose that the loads on all nput and output ports are ρ, that s, nk=1 λ k,l = m λ l,m = ρ, for all l {1, 2,...,n}, where ρ (0, 1). Consder any polcy under whch the queue-sze process has a well-defned lmtng statonary dstrbuton, and let ths dstrbuton be denoted by π 0. Then under π 0, we must have [ n ] nρ E π 0 Q k,l 2(1 ρ). k,l=1

22 2228 D. SHAH, N. S. WALTON AND Y. ZHONG PROOF. We consder the sums of queue szes at each output port, that s, the quanttes n k=1 Q k,l for each l {1, 2,...,n}. Snce at most one packet can depart at each tme slot, n k=1 Q k,l stochastcally domnates the queue sze n an M/D/1 system, wth arrval rate ρ and determnstc servce rate 1. Therefore, for each l {1, 2,...,n}, ρ [ n ] ρ E π 0 Q k,l 2(1 ρ). k=1 Here, 2(1 ρ) s the expected queue sze n steady state n an M/D/1 system. Summng over l gves us the desred bound. The optmalty n terms of the average total queue sze s a drect consequence of Theorem 5.2 and Lemma 5.5. COROLLARY 5.6. Consder the same setup as n Lemma 5.5. Then n the heavy-traffc lmt ρ 1, our polcy s 2-optmal n terms of the average total queue sze. More precsely, consder the expected total queue sze n the dffuson scale n steady state, that s, (1 ρ) Q. Then under our polcy, and lm sup(1 ρ) Q n ρ 1 under any other polcy. lm nf ρ 1 (1 ρ) Q n 2 PROOF. Lemma 5.5 mples that lm nf ρ 1 (1 ρ) Q n 2 under any polcy. For the upper bound, note that by Theorem 5.2, under our polcy, Q J + (N + 2)K. 2(1 ρ) For nput-queued swtches, J 2n, as remarked n Secton 5.2, N = n 2 and K = n. Therefore, we have that under our polcy, the expected total queue sze satsfes (33) Q n 1 ρ + ( n ) n. Now consder the steady-state heavy-traffc scalng (1 ρ)q.wehavethat (34) (1 ρ) Q n + (1 ρ) ( n ) n.