IEEE/AM TRANSATIONS ON NETWORKING, VOL. X, NO. XX, XXXXXXX 20X Achevng Optma Throughput Utty and Low Deay wth SMA-ke Agorthms: A Vrtua Mut-hanne Approach Po-Ka Huang, Student Member, IEEE, and Xaojun Ln, Senor Member, IEEE Abstract SMA agorthms have recenty receved sgnfcant nterests n the terature for desgnng wreess contro agorthms. SMA agorthms ncur ow compexty and can acheve the optma capacty under certan assumptons. However, SMA agorthms suffer the starvaton probem and ncur arge deay that may grow exponentay wth the network sze. In ths paper, our goa s to deveop a new agorthm that can provaby acheve hgh throughput utty and ow deay wth ow compexty. Towards ths end, we propose a new SMA-ke agorthm, caed Vrtua-Mut-hanne (VM-) SMA, that can dramatcay reduce deay. The key dea of VM-SMA to avod the starvaton probem s to use mutpe vrtua channes (whch emuate a mut-channe system) and compute a good set of feasbe schedues smutaneousy (wthout constanty swtchng/re-computng schedues). Under the protoco nterference mode and a sngehop utty-maxmzaton settng, VM-SMA can approach arbtrary cose-to-optma system utty wth both the number of vrtua channes and the computaton compexty ncreasng ogarthmcay wth the network sze. Further, once VM-SMA converges to the steady-state, we can show that under certan assumptons on the utty functons and the topoogy, both the expected packet deay and the ta dstrbuton of the HOL (headof-ne) watng tme at each nk can be bounded ndependenty of the network sze. Our smuaton resuts confrm that VM- SMA agorthms ndeed acheve both hgh throughput utty and ow deay wth ow-compexty operatons. Index Terms arrer sence mutpe access (SMA), utty maxmzaton, dstrbuted schedung, Markov chan, starvaton, vrtua mutpe channes. I. INTRODUTION A centra probem to the desgn of wreess contro agorthms s how to schedue the nk transmssons n the presence of nterference. Among the many desgn goas for wreess schedung, three of them are perhaps the most mportant. Frst, n order to support the ncreasng amount of the traffc paced on wreess networks, the contro agorthm shoud acheve hgh capacty. Second, the packet deay shoud be sma to meet the appcatons servce requrements. Thrd, for arge networks, the contro agorthms shoud have ow computatona compexty and ow communcaton overhead, and preferaby can be mpemented n a dstrbuted manner. Exstng agorthms n the terature acheve dfferent tradeoffs among capacty, deay, and compexty. It s we known that max-weght agorthms [2] can attan the argest capacty Ths work has been partay supported by the Natona Scence Foundaton through grant NS-064345, NS-072484, and NS-083999. Earer verson of ths paper has appeared n IEEE INFOOM 3 []. Po-Ka Huang and Xaojun Ln are wth the Schoo of EE, Purdue Unversty (Ema: huang3@purdue.edu, nx@purdue.edu). regon of the network, based on whch many wreess crossayer contro agorthms have been deveoped to optmze varous performance objectves such as the system throughput, utty/farness, and power effcency [3,4]. However, for many probem settngs, the max-weght agorthm ncurs exponenta compexty as the network sze ncreases. Hence, they are nfeasbe even for medum network sze. Further, t s a centrazed agorthm that requres goba nformaton. There are a number of ow-compexty agorthms that can be vewed as approxmatons to the max-weght agorthm (see [3] and the reference wthn). However, these agorthms can ony guarantee a fracton of the optma system capacty. Recenty, a cass of SMA (arrer Sense Mutpe Access) agorthms were studed n [5,6]. SMA agorthms are attractve because they are fuy dstrbuted and have O() compexty that s ndependent of the network sze. Further, SMA agorthms can be shown to acheve the optma capacty under certan assumptons. However, SMA agorthms have been observed to have arge deay that may grow exponentay wth the network sze [7,8], whch makes the usefuness of the capacty gan questonabe because most appcatons (even as smpe as web-browsng) requre some eve of ow deay. In summary, these exstng agorthms have been unabe to acheve a three goas,.e., hgh throughput, ow deay and ow compexty at the same tme. Ths unsatsfactory state-of-art has ed to the conjecture that perhaps there s a fundamenta tradeoff among these three dmensons. For exampe, n an eegant resut n [9], the authors show that there exst worstcase topooges such that even to acheve a dmnshngy sma fracton of the optma throughput capacty, ether the compexty or the deay must grow exponentay wth the network sze. Based on ths mpossbty resut, t may seem an unattanabe endeavor to dramatcay reduce the deay of SMA agorthms wthout sacrfcng ther hgh throughput or ow compexty. However, whe ths mpossbty resut s theoretcay ntrgung, t does not expan why for easer topooges, one cannot dramatcay reduce the deay of SMA-ke agorthms. In fact, for topooges wth bounded degree, [0,] have shown that neary-optma throughput can ndeed be acheved wth compexty and deay that do not grow wth the network sze. The agorthms n [0,] are st qute compex. Nonetheess, they do suggest the possbty that for a certan cass of network topooges, there may be potenta to address the deay probem of SMA-ke agorthms, whch may then ead to a ow-compexty and dstrbuted agorthm wth both provaby hgh throughput and provaby ow deay.
2 IEEE/AM TRANSATIONS ON NETWORKING, VOL. X, NO. XX, XXXXXXX 20X odd schedue even schedue Fg.. A torus nterference graph wth the odd and even schedues. Unfortunatey, mprovng the deay performance of SMA remans a chaengng probem n spte of a number of recent works [6,8,9,] [7]. The man dffcuty of reducng the deay of SMA may be expaned by the foowng exampe. onsder a n-by-n torus nterference graph as shown n Fg., where each crce represents a unt-capacty nk, and the edge between two crces means that these two nks can not transmt smutaneousy. Suppose that the target rates of a nks are 0.5. In a typca dscrete-tme verson of the standard SMA agorthms [6], t s possbe to adjust the parameters so that the SMA agorthms w stay at ether the even or the odd schedue wth probabty cose to 0.5 to attan the target rates. However, once SMA fnds ether the even or the odd schedue, t w be ocked to ths schedue for a ong tme before t can swtch to the other schedue [8], and the correspondng nactve nks w be starved of servce n ths perod. Ths probem s known as the starvaton probem for SMA. Hence, t s dffcut for SMA to attan ow deay. In ths paper, we propose a new agorthm, caed Vrtua- Mut-hanne (VM-) SMA, that can acheve both provaby hgh throughput utty and provaby ow deay wth ow compexty. The man novety of our proposed agorthm to sgnfcanty reduce deay s to take advantage of mutpe (physca- or vrtua-) channes. Ths man dea can be expaned as foows. Suppose that there are two separate channes for the same topoogy n Fg., and each has haf the bandwdth of the orgna system. If one channe uses the odd schedue, and the other channe uses the even schedue, then each nk can acheve both the target rate of 0.5 as we as ow deay (because each nk s served at a constant rate at a tmesots). Note that the key dea here s to compute a set of good schedues for mutpe channes at once rather than computng one schedue at a tme and constanty swtchng/recomputng schedues. Thus, the starvaton probem s avoded. Athough ths dea seems to be qute natura, there are three man dffcutes to generaze t to arbtrary networks. Frst, n many systems, we may ony have one physca channe. How can we st reduce deay by usng mutpe channes? Second, even f we have more than one channe, how can we desgn a ow-compexty and dstrbuted agorthm to compute the rght mut-channe schedues? Thrd, the number of channes n the above exampe s two, but t may not be suffcent for genera topooges. W the number of channes requred to approach near-optma performance be exceedngy arge such that the compexty ncreases to a prohbtve eve? Our proposed VM-SMA agorthms precsey address these dffcutes. Frst, for systems wth ony one physca channe, we ntroduce the noton of vrtua channes. Specfcay, there are vrtua channes, and each vrtua channe can have a dfferent schedue. By randomy choosng a vrtua channe and usng the correspondng schedue at each tme sot, we can then emuate the behavor of mutpe physca channes. (See Secton III-B for the correspondng dstrbuted mpementaton.) Second, assumng that each nk has a utty functon U (R ) of ts rate R, our proposed VM- SMA agorthms teratvey update the schedues across a vrtua channes to optmze the tota system utty. In each teraton, VM-SMA ony requres oca nformaton exchange and ncurs a ow compexty that ncreases neary wth the number of vrtua-channes. Thrd, we rgorousy quantfy the throughput and deay performance of the VM- SMA agorthms wth respect to. Specfcay, for an arbtrary network topoogy, et L denote the tota number of nks. We show that when ϵ 0. and 2 og L 3ϵ, VM-SMA agorthms can aocate an expected rate vector R = [R ] to each 2 nk such that L = U (R ) ( ϵ) L = U ([R ϵ]+ ), where R = [R ] s the rate vector wth maxmum system utty. Thus, VM-SMA agorthms can acheve cose-tooptma system utty wth the number of channes (and hence the correspondng compexty) ncreasng very sowy (O(og L)) wth the network sze. For deay, we show that, once VM-SMA agorthms converge to the steady-state, the expected packet deay of nk equas to /R, and the ta dstrbuton of ts head-of-ne (HOL) watng tme can be asymptotcay bounded (see Lemma 5 for detas). More mportanty, we show that under the assumpton of ogarthmc utty functons, as ong as some suffcent condtons for the network topoogy and utty functons are satsfed, the deay of each nk w not grow wth the network sze. Our smuaton resuts confrm that the proposed VM-SMA agorthms ndeed acheve both hgh throughput and ow deay. Further, t can qucky adapt to network traffc changes. In summary, the proposed VM-SMA agorthm acheves both provabe cose-to-optma throughput utty and provabe ow deay (that does not grow wth the network sze under sutabe condtons) wth ow-compexty operatons. The standard SMA agorthm has been extended to mutchanne networks n [8]. However, the deay ssue was not dscussed. There have been a number of recent studes that try to quantfy and mprove the deay performance of SMA agorthms [8,] [7]. However, none of them can attan the same eve of throughput/deay performance and ow compexty as we reported here. The work n [5] compares the deay performance as one tunes the parameter of a cass of SMA agorthms. Smary, the work n [7] ntroduces a threshod for each nk. A nk w mmedatey renqush transmsson opportunty f the queue ength drops beow the threshod. However, t s uncear whether such modfcatons w fundamentay and provaby ater the exponenta order of the starvaton tme. In [4], SMA agorthms wth deadne constrants are studed under the settng of a compete graph (where each nk nterferes wth every other nk). In another work [8], the authors propose to perodcay reset or unock the schedue to an nta empty schedue. They show that for
HUANG et a.: AHIEVING OPTIMAL THROUGHPUT UTILITY AND LOW DELAY WITH SMA-LIKE ALGORITHMS 3 a torus nterference graph ke Fg., the deay can be made ndependent of the network sze. However, t seems dffcuty to generaze the resuts from [8,4] to more genera network topooges. [2,3] show that f the offered oad s suffcenty sma (as a functon of the maxmum degree of the nterference graph), both the starvaton tme and the deay of SMA can be reduced to O(og L) [2] or even O() [3]. However, for such resuts to hod, the offered oad must be reduced sgnfcanty from the optma capacty. Smary, [6] consders usng mutpe physca channes for SMA, but under the constrant that each nk can occupy at most one channe. Thus, the resutng capacty coud aso be far from optma. Our dea of usng mutpe channes s aso nspred by [9]. However, t s dffcut to modfy the agorthm n [9] to optmze goba system utty. To the best of our knowedge, ony the agorthms n [0,] can acheve smar goas as ours,.e., both provaby hgh throughput and provaby ow deay wth ow compexty. Both [0] and [] partton the network nto non-nterferng peces wth fnte sze. In [0], a maxweght agorthm s used n each partton. However, n order to approach the optma capacty, the sze of each partton must be arge. Hence, the max-weght agorthm n each partton requres coordnaton among nodes that are ncreasngy far apart. In [], SMA s run n each partton. However, f the partton sze s arge, the actua deay n each partton may st be arge. Fnay, our resuts do not contradct wth the mpossbty resut of [9] (dscussed earer) because our suffcent condtons on network topoogy may excude the worst-case topoogy of [9]. Further, both our noton of deay (steady-state deay vs. transent deay) and our system settng (wth vs. wthout congeston contro) are dfferent (see the end of Secton III-D for detaed dscussons). The rest of the paper s organzed as foows. The system mode s presented n Secton II. In Secton III, we present the VM-SMA agorthm aong wth the performance anayss. We dscuss the mpementaton ssues n Secton IV and the smuaton resuts n Secton V. Then we concude. II. SYSTEM MODEL onsder a wreess network wth N nodes and L nks, where each node represents a communcaton devce, and each nk corresponds to a par of transmttng node and recevng node. We assume the so-caed protoco nterference mode,.e., two nks nterfere wth each other f they can not transmt data at the same tme. Let E be the set of nks that nterfere wth nk. We assume that the nk nterference reatonshp s batera,.e., f E 2, then 2 E. Two nks and 2 are caed neghbors f and 2 nterfere wth each other. We consder a tme-sotted system, where each sot has unt ength. Assume that the wreess network has ony one physca channe. The capacty of each nk s assumed to be,.e., each nk can transmt at most one unt-szed packet The basc dea of SMA has been extended to more genera nterference modes, e.g., those based on SINR [20,2], athough the agorthms there are key to suffer smar starvaton probem and arge deay as n [7,8]. Thus, we expect that the key nsghts of our work can aso be extended to reduce deay under these more genera nterference modes. n one tme sot. To represent a schedue, we w use a L- dmenson vector such that the th eement s f nk s ncuded n the schedue and 0 otherwse. Assocated wth each nk s a one-hop fow,.e., packets of the fow w mmedatey eave the network after t traverses the nk. We assume that each fow s nfntey backogged,.e., at the transport ayer t aways has packets to send. Further, each fow at nk has a utty functon U ( ) assocated wth t. If the ong-term average rate of nk s R, then U (R ) represents the satsfactory eve of the correspondng fow [3]. We assume that each utty functon s postve, nondecreasng, strcty concave, and twce dfferentabe [22]. Further, we assume that t s bounded on the doman D = [0, ]. Let U( R) = L = U (R ). Let Ω denote the capacty regon of the wreess network, whch s gven by the set of a rate vectors R = [R ] such that there exsts a contro pocy that can support the ong-term average rate R at each nk. As we have dscussed n Secton I, we are nterested n both hgh throughput utty and ow deay. For hgh throughput utty, we am to sove the foowng optmzaton probem: max R 0 = L U (R ), R = [R ] Ω. () Let R = [R ] be the optma souton of probem (). Note that probem () s a cross-ayer contro probem [3] because t nvoves two contro mechansms. Frst, the transport ayer of the fow at each nk does congeston contro and determnes how packets can be njected at the ong-term average rate R. Second, at the MA ayer, the system determnes how to schedue the nk transmssons to support the ong-term average rate R at a nks. Under nfnte-backog settng, the deay of a packet s defned as the dfference between the tme when the transport ayer njects the packet nto the buffer and the tme when ths packet s served by the nk. Athough ths defnton seems to be a natura defnton of deay, t unfortunatey does not fuy capture the effect of the possbe starvaton probem [8]. For exampe, consder a queue wth a snge buffer. A new packet s added to the buffer mmedatey after the od packet s served. Suppose that for every 000 packets, t takes ony one tme-sot to serve each of the frst 999 packets. However, the 000 th packet suffers starvaton for 000 tme-sots. In ths case, the expected packet deay (average over a packets) s.99. Thus, the effect of the starvaton probem s not obvous. Due to ths reason, we ntroduce another noton of deay. Specfcay, at each tme sot, we study the tme that the head-of-ne (HOL) packet has wated n the system. In the above exampe, the expected HOL watng tme across tmesots woud be around 250. Hence, the negatve mpact of the starvaton probem s more obvous. In ths paper, by ow deay, we mean that both the packet deay and the HOL watng tme shoud be sma. Thus, the goa s to deveop ow-compexty and dstrbuted agorthms that can provaby acheve both hgh throughput utty and ow deay. Remark: Note that n ths paper, we have defned throughput and deay drecty based on a utty-maxmzaton settng. Ths defnton may seem somewhat dfferent from many exstng
4 IEEE/AM TRANSATIONS ON NETWORKING, VOL. X, NO. XX, XXXXXXX 20X works [5,6,8,]. There, throughput optmaty s often defned for a system where packets arrve accordng to some stochastc process, and the deay s defned as the tme from packet arrva to the tme when the packet s served at the MA ayer. However, we beeve that the nfnte-backog mode and the deay defnton n ths paper are mportant to study. Frst, the goa of throughput optmaty s to support the argest possbe capacty regon. In practce, the offered oad may fa outsde the capacty regon, n whch case some form of congeston contro s requred to avod overoadng the system. Further, whenever overoad occurs, t s mportant to mantan some noton of farness across users. The nfnte-backog uttymaxmzaton formuaton s often used to mode congeston contro and farness [3,4], and hence s mportant to study. Second, under such an nfnte-backog mode, the deay at the transport ayer woud have been nfnte. Thus, we focus nstead on the deay after the packet s njected by the transport ayer, whch can be nterpreted as the servce deay at the nk s MA ayer. We note that under the tradtona SMA agorthm, ths deay remans very arge due to the starvaton probem (see Fg. 4(a) n Secton V). Hence, t s mportant to study how to reduce such deay. Fnay, under certan assumptons, t may be possbe to extend the key nsghts of our work to the settng wth packet arrvas. We brefy dscuss the man dea and potenta chaenge n Secton III-F. III. VM-SMA ALGORITHM DESIGN AND ANALYSIS In ths secton, we propose a new ow-compexty Vrtua- Mut-hanne (VM-) SMA agorthm wth provabe hgh throughput utty and provabe ow deay. Snce our agorthm s motvated by the standard SMA agorthm, we w frst brefy descrbe a dscrete tme verson of the standard SMA agorthm [3,6] for sovng probem () and dscuss ts weakness. We w then ntroduce the new VM-SMA agorthm. A. The Standard SMA agorthm Let Q (t) be the number of packets of nk at the begnnng of tme sot t. Let S(t) = [S (t)] be the schedue chosen by the schedung agorthm at tme t. Defne a set S of decson schedues such that every eement n S s a feasbe schedue. Further, each nk must be schedued by at east one schedue n S. Let w (t) be an sutabe ncreasng functon of Q (t) as n [6]. For exampe, w (t) = og(αq (t)), where α s a postve constant. The foowng SMA agorthm s known to sove probem () under certan assumptons. SMA Agorthm: At each tme t, Decson Phase: hoose a decson schedue S D (t) = [S D (t)] randomy n S. Schedung Phase: For each nk, f S D (t) = and S (t) = 0 for every nk E, we have: P ({S (t) = x}) = exp(xw (t)) +exp(w (t)), where x = 0 or. Otherwse, S (t) = S (t ). ongeston ontro: Each nk njects a random number of packets accordng to a Posson dstrbuton wth mean r = arg max r 0 {U (r) βq (t)r}, where β > 0. It can be shown that under a tme-scae separaton assumpton [6], f β s suffcenty sma, r = [r ] w converge to vaues cose to the optma souton of () [3]. In practce, the above tme-scae separaton assumpton must be approxmated by a sma vaue of α. However, when α and β are sma, Q (t) and w (t) tend to be arge. In that case, the agorthm w key to be stuck n one schedue for a ong tme before t swtches to another schedue [8] (as we dscussed n Secton I). In the worst case, the starvaton tme and the deay may grow exponentay wth the network sze [7]. B. Vrtua-Mut-hanne SMA Agorthm We now ntroduce our proposed VM-SMA agorthm that overcomes the dffcutes of starvaton probem and arge deay. As we have mentoned n Secton I, the key to acheve both hgh throughput utty and ow deay s to take advantage of mutpe channes and smutaneousy compute feasbe schedues over a channes. Once a good set of feasbe schedues s found, we can then avod constanty swtchng schedues (and hence the starvaton probem). To make ths dea feasbe n wreess systems wth ony one physca channe, we ntroduce the concept of vrtua channes. Specfcay, for each nk, there are vrtua channes, and we use V = [V,, V ] to denote ts schedue n a vrtua channes, where V k = f nk s schedued n the k th vrtua channe, and V k = 0 otherwse. We use V = [ V ] to denote the goba schedues of a vrtua channes and a nks. When we focus on a specfc vrtua-channe k, there s a feasbe schedue S( V ) k = [V k,, V Lk ] for the network. Note that gven the goba schedue V, the tota number of vrtua channes used by nk s gven by x ( V ) = k= V k. To use these schedues, at each tme sot t, the network randomy chooses a vrtua-channe k(t) unformy from to,..d. across tme-sots. A nks n the network then use the feasbe schedue S( V ) k(t) n ths tme-sot,.e., each nk transmts a packet f V,k(t) =. 2 Note that each nk ony needs to know ts own schedues V. Further, the randomzaton of the vrtua-channe k(t) can be acheved n a dstrbuted manner f a nks are synchronzed, and they have the same randomnumber generator wth a common-seed, whch ony needs to be agreed upon at the begnnng of the system operaton. Wth ths mpementaton, each nk w be schedued for actua transmsson wth probabty equa to r ( V ) = x ( V )/,..d. across tme-sots. Thus, the ong-term average rate of nk s r ( V ), and the average nter-servce tme s /r ( V ). Hence, the deay w key be sma as we w show beow. It remans to deveop a ow-compexty and dstrbuted agorthm for computng the goba schedue V that eads to hgh tota system utty. Specfcay, we seek souton to the foowng optmzaton probem, whch can be vewed as an approxmaton to the orgna optmzaton probem (): L max U (r ( V )), V = S( V ) k s a feasbe schedue, k =, 2,,. 2 We note that the dea of usng a random schedue has some smarty to the statonary randomzed pocy [4] that has been used to anayze the throughput optmaty of other agorthms. However, there are no ow-compexty methods of computng the rght statonary randomzed pocy. (2)
HUANG et a.: AHIEVING OPTIMAL THROUGHPUT UTILITY AND LOW DELAY WITH SMA-LIKE ALGORITHMS 5 Our hope s that when s suffcent arge, the optma souton of (2) w be cose to that of (). We next descrbe our proposed ow-compexty VM- SMA agorthm for sovng (2). Later n Secton III- and III-D, we w study ts throughput and deay as the number of vrtua-channes vares. Because VM-SMA agorthm updates the goba schedue V teratvey over tme, we w use the notaton V (t), V (t), and V k (t) to denote ther correspondng vaues at tme sot t. Smar to the standard SMA agorthm, we defne a set S of decson schedues such that t satsfes the foowng three condtons: () each decson schedue s a feasbe schedue. (2) each nk must be schedued by at east one decson schedue n S. Note that these two condtons are the same as those n the standard SMA agorthm [6]. (3) each nk schedued by a decson schedue can broadcast bts to ts neghbors n a tme sot. We w dscuss n Secton IV how to mpement the decson schedues at each tme sot. Now, we descrbe our agorthm. Vrtua-Mut-hanne SMA Agorthm: At each tme t, Decson Phase: hoose a decson schedue S D (t) = [S D (t)] randomy n S. Update Phase: For each nk, f S D(t) = 0, et V (t) = V (t ). Otherwse, perform Agorthm. Agorthm Update phase for nk f S D (t) = : hoose a permutaton (n, n 2,, n ) of the set {, 2,, } unformy at random. Let f (x) = exp(αu ( x )) and x = k= V k(t ). for = to do f V,n (t ) = for any nk E then V,n (t) = V,n (t ). ese V,n (t) s determned wth the random dstrbuton: = P ({V,n (t) = y}) f (x +y V (t )) f (x V,n (t ))+f (x + V,n where y = 0 or. end f x + = x + V,n (t) V,n (t ). end for Lnk broadcasts V (t) to a of ts neghbors. (t )), (3) Schedung Phase: A common vrtua-channe k(t) s chosen by a nks n the network unformy at random, and each nk transmts a packet f V,k(t) (t) =. ongeston ontro: A nks use the wndow-based fow-contro agorthm wth wndow-sze,.e., a new packet s njected to nk ony f a packet s served. We note a key dfference between the VM-SMA and the standard SMA. Under the VM-SMA (correspondngy standard SMA) agorthm, the random dstrbuton for updatng the schedue of nk s a functon of ts own utty U (x /) (correspondngy ts own queue ength Q (t)). The sgnfcance of ths key dfference s as foows. onsder (3) and V,n (t ) = 0 for an exampe, we have P ({V,n (t) = }) = f (x +)/f (x ) +f (x +)/f (x ) exp( α U ( x )) +exp( α U ( x )) (4) Reca that U ( ) s a strcty concave functon, whch mpes that U ( ) s a strcty decreasng functon. Hence, f a nk has arger x vaue, t s ess key to actvate tsef n a new vrtua channe, and vce versa. Thus, the schedues across vrtua channes are n fact coordnated to reach a state V wth tota system utty cose to the optma vaue. Hence, VM- SMA does not constanty change the schedue and avod the starvaton probem. An addtona beneft s that we can use the wndow-based fow-contro as the congeston contro component, whch further contros the packets n the system and reduces the deay.. Utty Optmaty In ths subsecton, we study the capacty/utty performance when our VM-SMA agorthm reaches steady state. Note that under the VM-SMA agorthm, the goba schedue V (t) behaves as a Markov chan. Hence, we w aso refer to V (t) as the state. Let V be the state space of the Markov chan. We frst ntroduce a proposton that descrbes the statonary dstrbuton of the Markov chan. Proposton : The statonary dstrbuton of the Markov chan V (t) s gven by P ( V ) = Z exp(α L = U (r ( V ))), where Z s a normazaton constant for a V V. Proof: The proof s gven n Appendx A Proposton mpes that the state V wth arger vaue of tota system utty has a hgher chance to be vsted. Further, snce α appears n the exponent n the statonary dstrbuton, as α ncreases, the probabty of reachng the state wth the argest utty, whch s the souton to probem (2), w approach. However, due to the quantzaton effect wth a fnte, the optma utty n (2) may be smaer than the optma utty of the orgna probem (). Hence, the key queston s how many vrtua-channes we need such that the two optma-utty vaues are cose. We note that ths queston s mportant because the compexty of our agorthm aso ncreases wth. In practce, we woud prefer a smaer vaue of. A frst thought for sovng ths queston s to use the aratheodory s theorem [23]. Specfcay, snce the capacty regon of the optmzaton probem n () s a convex hu of a the feasbe schedues, the aratheodory s theorem tes us that the optma souton to () can be wrtten as the convex combnaton of L + feasbe schedues. As a resut, we may guess that we need to be at east Θ(L) so that the optma souton to () can be approxmated by a vad goba schedue V. Somewhat surprsngy, by aowng a sma margn ϵ for error and usng a nove probabstc argument, we can reduce the order of to O(og L). Ths nce resut s presented n the foowng proposton. Reca that R s the optma souton of nk n probem (). Proposton 2: If ϵ 0. and 2 og L 3ϵ 2, then there exsts a state V s n the state space V of the Markov chan such that r ( V s ) R ϵ, for a nk.
6 IEEE/AM TRANSATIONS ON NETWORKING, VOL. X, NO. XX, XXXXXXX 20X Proof: The proof s gven n Appendx B Note that snce the rate s aways arger than zero, t then foows from r ( V s ) R ϵ that r ( V s ) max(r ϵ, 0) [R ϵ]+. Wth the hep of Proposton 2, we can prove the frst man theorem n ths paper. Reca that U( r) = L = U (r ). Theorem 3: Under our VM-SMA agorthm, for any ϵ 0., we can choose > 2 og L 3ϵ and α arge enough such that 2 P {U( r( V (t))) L = U ([R ϵ]+ )} ϵ, where R s the optma souton of probem (). Proof: onsder a fxed. Let V max be the souton to (2). Then U( r( V max )) U( r( V )), for a V V. Defne the set A = { V V U( r( V )) = U( r( V max ))}. Denote f( V ) = exp(αu( r( V ))). Let f max = f( V max ). Note that f( V ) = f max for a V A. Snce V s fnte for a fxed, there must exst ϵ > 0 such that f( V ) f max e αϵ for a V / A. Usng Proposton and ths nequaty, we can then show that P ({ V A}) = { V A} f( V ) { V A} f( V )+ { V / A} f( V ) f max A (5). f max A +f max e αϵ ( V A ) Ths mpes that for any ϵ, there exsts α such that P ({ V A}) > ϵ. Further, from Proposton 2, for any ϵ 0., we can choose a fxed 2 og L 3ϵ such that for V A, 2 U( r( V )) U( r( V s )) U ([R ϵ]+ ). The resuts then foows. Theorem 3 states that as ong as α s suffcenty arge, n the steady-state the expected servce rate acheved by VM- SMA and measured at every tme-nstant w acheve a cose-to-optma utty wth hgh probabty. Further, for a gven vaue of ϵ (whch characterzes the optmaty oss), the number of channes ony needs to grow very sowy,.e., ogarthmcay, wth the network sze. Defne r mn as the worst rate for nk among a goba schedues wth the maxmum utty,.e., r mn = mn r ( V ), (6) { V V U( r( V ))=U( r( V max ))} where V max s the souton of probem (2). Theorem 3 aso eads mmedatey to the foowng coroary on the average throughput for each nk and the tota system utty. oroary 4: Under our VM-SMA agorthm, for any ϵ 0., we can choose > 2 og L 3ϵ and α arge enough such 2 that the tme-averaged throughput R of nk has a ower bound R ( ϵ)r mn. Further, U( R) ( ϵ)u([ R ϵ] + ). Proof: To cacuate the throughput of nk, we have to consder the rea schedue, whch s determned both by V (t) and the common vrtua channe k(t) that we choose at each tme sot. Note that P {k(t) = k} = /, k =,,. Further, k(t) s..d. across tme-sots and s ndependent of V (t). Defne the set A = { V V U( r( V )) = U( r( V max ))}. Let P ( V ) be steady state probabty when the state of the Markov chan V (t) s V. From the proof of Theorem 3, we can fnd α such that P { V (t) / A} ϵ. The average throughput of nk can then be derved as foows. R = V V k= P {k(t) = k, V (t) = V }V k = V V k= P { V (t) = V }V k = V V P { V (t) = V }r ( V ) V A P { V (t) = V }r ( V ) V A P { V (t) = V }r mn ( ϵ)r mn Now, combnng Theorem 3 and the fact that U( ) s a concave functon, we can have that U( R) V P ( V )U( r( V )) ( ϵ)u([ R ϵ] + ). Ths concudes the proof. Remark: Under sutabe tme-scae separaton assumptons, the ong-term tme-averaged servce rate under the standard SMA agorthm can aso be shown to acheve the optma utty, whch s smar to oroary 4. However, we emphasze that Theorem 3 s n fact much stronger. Snce VM-SMA chooses a vrtua channe..d. across tme, even over a shorter tme-nterva around tme t, the average servce rate of each nk w be cose to r ( V (t)). Thus, Theorem 3 states that the servce rate averaged even over shorter ntervas w attan cose-to-optma system utty wth hgh probabty. In contrast, due to the starvaton probem, the short-term servce rate of the standard SMA agorthm coud be much more unfar and far away from the optma utty. D. Deay Performance In ths subsecton, we study the deay performance of VM- SMA. A natura defnton for packet deay s the number of tme-sots from the tme when the packet s njected to the buffer of a nk by the congeston contro component at the correspondng source, to the tme when the packet s served by the nk. It turns out that ths packet deay has a smpe characterzaton. Note that snce we use wndow-based contro wth wndow sze, the number of packets n the buffer at each nk s aways. Further, the rate that packets arrve to and depart from nk s gven by R. Hence, by Ltte s aw [24], the expected packet deay (average across packets) at nk must be equa to /R, whch s upper bounded by. ( ϵ)r mn Thus, a hgh throughput at nk mmedatey transates to a ow expected packet deay. However, as we have mentoned n Secton II, the above defnton of packet deay does not fuy capture the effect of starvaton probem. A better metrc s HOL (head-of-ne) watng tme, whch s the tme that the HOL packet at nk has wated n the system. For exampe, an extended perod of starvaton w key cause arge expected HOL watng tme average across tme-sots (pease refer to the exampe n Secton II). The foowng emma shows that under VM- SMA agorthm, the ta dstrbuton of the HOL watng tme w decay qucky. Lemma 5: Under our VM-SMA agorthm, for a fxed nteger d > 0 and any ϵ > 0, there exsts suffcenty arge α so that for each nk, P {HOL watng tme d} ( r mn ) d + ϵ. (7)
HUANG et a.: AHIEVING OPTIMAL THROUGHPUT UTILITY AND LOW DELAY WITH SMA-LIKE ALGORITHMS 7 Proof: Defne the set A = { V V U( r( V )) = U( r( V max ))}. From the proof of Theorem 3, we can fnd α such that P { V (t) / A} ϵ/d. Let E be the event that V (t ) A, t = t d +,, t. Hence, by the unon bound, P (E c ) d ϵ d = ϵ. Let d (t) be the HOL watng tme for nk at tme t. We have that P {d (t) d E} = P {nk s not served for t d + to t E} ( r mn ) d. Use ths nequaty, we can then show that P {d (t) d} = P {d (t) d E}P {E} + P {d (t) d E c }P {E c } ( r mn ) d + ϵ. Ths concudes the proof. Unt now, we have deveoped bounds on the expected packet deay and the ta dstrbuton of the HOL watng tme. These bounds ony depend on r mn and are otherwse ndependent of the network sze. Gven that the deay of SMA agorthms have been observed to grow exponentay wth the network sze [7,8]. Our utmate goa n ths secton s to deveop condtons such that the deay of VM-SMA does not grow wth the network sze. For ths statement to hod, we need to bound r mn away from zero. Unfortunatey, wthout addtona restrctons on the network settng, t s st possbe to construct an exampe such that r mn s cose to 0. Due to the space constrants, we refer the readers to [25] for the exampe. Ths suggests that wthout addtona restrctons, the vaue of r mn may be arbtrary cose to 0. However, ths effect has ess to do wth our VM-SMA agorthm. Rather, t s many determned by the system settng and the rateaocaton correspondng to the maxmum utty. Bascay, when the number of nks n E s arge or the utty functon of nk has a sma dervatve, the optma souton for nk may aready be very sma. Motvated by the exampe, we ntroduce the foowng condtons. Let = E be the number of nks n the neghborhood of nk. Let K be the maxmum number of nks that can be schedued smutaneousy n the neghborhood of nk. In the new condton, we assume that max and max K K for some constants and K. Note that ths condton s very genera and w key hod for a arge cass of network topooges. Further, we assume that the utty functon of a nks s of the form og( + h) og(h). Under these condtons, the next proposton shows that we can have a ower bound for r mn. Proposton 6: Suppose that the utty functons of a nks are of the form og( + h) og(h). Further, assume that 0 < h < ( +2)(K ) and + < K ( +2) h(k ) K,, then under the VM-SMA agorthm, we have r mn K ( +2) h(k ) K +. Proof: The proof s gven n Appendx Wth Proposton 6, t s then easy to show the foowng ower bound for r mn that s ndependent of the network sze. oroary 7: Suppose that the utty functons of a nks are of the form og( + h) og(h). If 0 < h 2( +2)(K ) and 4K( + )( + 2), then under the VM-SMA agorthm, we have that r mn 4K( +2). Proof: Snce h 2( +2)(K ), we know that h < 2( +2)(K ) for a. As a resut, we have that K ( +2) h(k ) K 2K( +2). Further, from 4K( + )( + 2), we know that for a, 4K( + )( + 2) and + 4K( +2). Hence, K ( +2) h(k ) K + Proposton 6, we then have r mn 4K( +2). 4K( +2). By Proposton 6 and oroary 7 show that as ong as the number of nterferng neghbors of each nk s bounded, by makng the utty functons suffcenty steep (.e., a sma h) and by usng a suffcenty arge (but constant) number of channes, can be bounded from beow by a constant ndependent of the network sze. Hence, the average deay of each nk does not grow wth the network sze, and the ta dstrbuton of HOL watng tme s guaranteed to decay qucky. We note that n [9], the authors show that for open-oop systems, even at an offered oad that s a sma fracton of the optma capacty, there exsts network topooges such that ether the compexty or the deay must grow exponentay wth the network sze. We note that our resut dffers from [9] n two mportant aspects. Frst, the deay defnton s dfferent. We are nterested n the steady state deay,.e., after the Markov chan converges to the statonary dstrbuton. The convergence tme (whch we dd not capture n ths paper) may st be exponenta n the network sze. In contrast, the deay n [9] s the worst-case deay across tme and hence aso captures the transent phase. Second, we use a cosed-oop system (wth congeston contro) n contrast to an open-oop system n [9]. Hence, our resuts do not contradct wth those of [9]. r mn E. omputatona ompexty and ommuncaton Overhead In ths subsecton, we dscuss the computatona compexty and communcaton overhead of the VM-SMA agorthm. For each nk, the computatona compexty of the VM- SMA agorthm s O(). Ths s because a of the computatons can be carred out n parae at a nks, and every nk ony needs to go through the vrtua channes. Further, fndng the random permutaton n the update phase s of compexty O() through the use of the Fsher-Yates shuffe [26]. Regardng the communcaton overhead, note that each nk can use a -bt vector to denote ts schedue V across a vrtua channes. At each tme t, a nk schedued by S D (t) may need to broadcast ths -bt vector to a of ts neghbors. Hence, the communcaton overhead s aso O(). From Theorem 3 and oroary 4, we can see that = O(og L), whch grows very sowy wth the network sze. Hence, our agorthm s scaabe to arge networks. 3 In practce, a bts may be put nto a snge contro packet n one tme-sot. For exampe, f the sze of a contro packet s 250 bytes, t can accommodate = 2000 vrtua channes. Even for a arge network wth L = 000 nks, = 2000 vrtua channes are suffcent to reduce ϵ to be 0.05 (see Theorem 3). Hence, the correspondng capacty/utty reducton w be very sma. If a data packet of ength 2000 bytes (ke n 802.) s transmtted n each tme-sot, such a contro packet corresponds to a ow overhead of 2.5%. In practce, ths overhead can be further reduced by reducng the frequency of updates. For exampe, f the operaton n the update phase s performed every 5 tmesots, the communcaton overhead s further reduced to 2.5%. 3 In comparson, the compexty of the standard SMA agorthms s O() and s ndependent of the network sze [5,6]. However, they suffer the arge deay probem as mentoned n Secton I.
8 IEEE/AM TRANSATIONS ON NETWORKING, VOL. X, NO. XX, XXXXXXX 20X Note that at each tme-sot, we can st randomy choose a vrtua-channe schedue for transmsson. Thus, reducng the frequency of updates w ony affect the convergence of the agorthm, but t w not affect the capacty and deay once the Markov chan converges to ts statonary dstrbuton. F. VM-SMA under Exogenous Packet Arrvas Throughout ths paper, we have focused on an nfntebackog settng where the packet njecton to the system can be controed. In ths subsecton, we brefy dscuss how the key dea of VM-SMA may potentay be generazed to the settng where packets arrve accordng to an exogenous stochastc process. The basc dea s as foows. Assume that we know the average packet-arrva rate λ of each nk, and we know that the vector λ = [λ ] s strcty nsde the capacty regon Ω,.e., there exsts ϵ such that λ + ϵ Ω. If we can et the VM-SMA agorthm compute a rate-aocaton vector R = [ R] such that r mn > λ + γ, where γ < ϵ, then the end-to-end packet deay may aso be sma. To see ths, assume wthout oss of generaty that the packet arrvas at each nk are..d. across tme. As we can see from the proof of Lemma 5, whenever the event A occurs (whose probabty s cose to ), the servce at each nk stochastcay domnates an..d. servce wth probabty r mn. Hence, as ong as r mn > λ + γ, we expect that the end-to-end packet deay at nk w be O(/γ). The chaenges, however, are to desgn a utty functon to compute such vector R and to show that even though there s a sma probabty that the event A does not occur, the expected deay w st be O(/γ). We refer the readers to [25] for an exampe that resoves the frst chaenge, and we eave the second chaenge for future work. IV. IMPLEMENTATION In ths secton, we dscuss two mpementaton ssues. Frst, t s we known that for SMA agorthms, there s a tradeoff between optmaty and convergence speed dependng on the vaue of α. In practce, we want to choose a sutabe α that s not too arge to shorten the convergence tme. However, when α s not very arge, we aso observe a common source of performance degradaton, whch can be expaned as foows. Suppose that the Markov chan has found the optma state V max, and a nk s actve n vrtua channe k. When α s not very arge, there s a substanta probabty that nk w turn tsef off n vrtua channe k. If a other nks n ts neghborhood,.e., E, have nterferng nks that are actve n vrtua channe k, no nks n E can be turned on n vrtua channe k. Hence, nk may turn tsef on agan n vrtua channe k ater, and for nk, the transmsson opportuntes durng ths perod are ost unnecessary. Such performance degradaton can be easy avoded wth the foowng mproved agorthm. We ca the schedue computed by VM-SMA as the soft schedue. In addton, we now ntroduce hard schedue as foows. Whenever a nk s turned on n vrtua channe k by the soft schedue (.e. V k = ), we aso turn on nk n vrtua channe k n the hard schedue. However, even f V k = 0, we w ony turn off nk n the vrtua channe k n the hard schedue when a neghborng nk of nk decdes to use vrtua channe k (.e., t turns tsef on n vrtua channe k by the soft schedue). As a resut, the hard schedue w gve up transmsson opportuntes unt the ast mnute, and we can avod the throughput oss due to the reason descrbed above. Note that our earer throughput/deay resuts st hod. Now, we formay descrbe ths smpe fx. We defne an addtona hard vrtua schedue H k for a vrtua-channe schedue V k, and we w ca V k the soft vrtua-channe schedue or smpy the vrtua schedue. The update of soft vrtua schedue V k w reman the same, and the update of hard vrtua schedue H k s as foows. Update Hard Vrtua Schedue: At each tme t, If V k (t) =, we et H k (t) =. If V k (t) = 0, we et H k (t) = 0 when nk receves a broadcast from any nk wth V k(t) =, and E. Otherwse, we et H k (t) = H k (t ). Now, we modfy the schedung phase as foows. Schedung: At tme t, a common vrtua channe k s chosen by a nks n the network unformy at random, and each nk uses H k (t) as the rea schedue at tme t. Note that t s easy to see that each hard vrtua-channe schedue w eventuay become a maxma schedue. (Reca that a maxma schedue must be feasbe, and no more nks can be added to the schedue wthout nterferng wth the nks that has been schedued.) The reason s that for the k th hard vrtua schedue of a the nks,.e., (H k, H 2k,, H Lk ), once t becomes a maxma schedue, t w reman as a maxma schedue n the rest of the tme because each nk ony renqushes the transmsson opportunty when one of ts neghbors s actvated by the agorthm. Further, the k th hard vrtua schedue of a the nks w have a trend to become a maxma schedue startng from tme 0. Ths confrms our cam that we can avod wastng transmsson opportunty when the soft vrtua schedue V (t) jumps n the Markov chan. It s easy to see that the throughput performance of ths mproved schedung agorthm usng hard schedue s aways better than the agorthm descrbed n Secton III-B. Hence, the anaytca resuts n Secton III- and III-D st hod. Second, we brefy dscuss how to choose the decson schedue at each tme sot. Reca n Secton III-B that at each tme sot, we choose a decson schedue S D wth a fxed probabty dstrbuton from a set S of decson schedues. Further, the set S of decson schedues shoud meet condtons (), (2), and (3). Note that for the standard SMA agorthms [6], the authors provde a random backoff agorthm for computng decson schedues that satsfy the frst two condtons. There are a number of ways to satsfy the remanng thrd condton. One possbty s to assume that there s a separate contro channe (e.g. usng DMA). On the other hand, f such a separate contro channe s not avaabe, the other possbty s to make the decson schedue more sparse. Specfcay, two actve nks n the decson schedue are not neghbors, and the two actve nks do not share any common neghbors. As a resut, f a nk s not schedued by the decson schedue, t w not receve more than one broadcast transmsson. Such a decson schedue can be computed n each tmesot va a random-backoff-based agorthm smar to that of
HUANG et a.: AHIEVING OPTIMAL THROUGHPUT UTILITY AND LOW DELAY WITH SMA-LIKE ALGORITHMS 9 F sub-sot 3 mn-sot bts one tme sot Fg. 2. The composton of a tme sot. data transmsson [6] wth addtona sgnang messages to resove confcts at common neghbors. The detas are as foows. We frst descrbe the composton of a tme sot as shown n Fgure 2. The tme sot s composed of three phases. The frst phase ncudes F sub-sot, and each sub-sot has three mn-sots. A decson schedue w be seected dstrbutvey n the frst phase. The second phase s the tme requred for each nk schedued by the decson schedue to broadcast the bts of nformaton for the updated vrtua-channe schedues. The thrd phase s the actua data transmsson. The decson schedue can be computed n a dstrbuted fashon as foows. At the begnnng of each tme sot, we assgn each nk a mark bt equa to 0. Each nk w choose a vaue from to F + to attempt to be ncuded n the decson schedue. If a nk chooses F +, t w not attempt at a. Suppose that a nk chooses, F. If nk hears any successfu transmsson attempts before sub-sot, t w set ts mark bt to, and t w not attempt to be ncuded n the decson schedue n ths tme-sot. On the other hand, f nk does not hear any successfu transmsson attempts before sub-sot, the transmtter of nk transmts a RTS n the frst mn-sot. Ths RTS sgna ndcates to ts neghbors that the nk attempts to be ncuded n the decson schedue. The recever of nk w transmt a TS n the second mn-sot to confrm the recepton of RTS sgna. For any other nk k that does not choose the th sub-sot, t sends a sgna n the thrd mn-sot under one of the two condtons: () f there s a confct because ts mark bt s, and t senses any message transmssons n the frst mn-sot, or (2) t senses a confct due to more than one RTS transmssons n the frst mn-sot. At the end of the th sub-sot, f the transmtter of nk does not receve a TS from the recever for the RTS transmttng n the frst mn-sot, whch mpes that nk codes wth other transmssons n the frst mn-sot, the attempt fas. Further, f nk hears any sgnas n the thrd mn-sot, the attempt fas. Otherwse, nk s ncuded n the decson schedue. Ths agorthm ensures that the nks seected n the decson schedue do not have common neghbors, and hence ther neghborng nks can receve the -bt broadcast wthout nterference. Further, each nk has a postve probabty to be seected n the decson schedue because there s a postve probabty that a the neghborng nks w not attempt to be ncuded n the decson schedue. V. SIMULATION We evauate the VM-SMA agorthm n two topooges wth our ++ smuator. Specfcay, we smuate the mproved schedung agorthm and a random backoff-based decson-schedue agorthm dscussed n Secton IV. For both topooges, there s a one-hop fow assocated wth each nk, and ts utty functon s gven by og(0 5 + ) og(0 5 ). Further, the smuaton tme s 5,000 sots. We frst study a 8-by-8 torus nterference graph as descrbed n Secton I and compare our agorthm wth the standard SMA agorthm. 4 Note that because of symmetry, the optma rate R for each nk under ths settng w be 0.5. For the standard SMA agorthm, the weght s chosen as w (t) = og(0.5q (t)), and the parameter β for congeston contro s 0.. For our VM-SMA agorthm, we et = 30 and α = 29. The correspondng ϵ = 0.3. 5 We denote the node n the top eft corner of the torus as node 0 and the node rght next to t as node. Note that node 0 s actve n the odd schedue, and node s actve n the even schedue (See Fg. ). The average throughput, average deay across packets, and average HOL watng tme across the tme-sots for node 0 are presented n the foowng tabe. throughput deay HOL watng tme SMA 0.427 59 372.8 VM-SMA 0.479 2.09 2.0 The resut shows that our agorthm can ndeed acheve throughput cose to the optma rate. Further, the deay performance s exacty equa to the nverse of the average throughput. In contrast, both the packet deay and the HOL watng tme of the standard SMA agorthm are 80 and 70 tmes arger, respectvey. 6 In Fgure 3(a), we pot the ta dstrbuton of the HOL watng tme under both agorthms. The HOL watng tme of our agorthm decays qucky, whch confrms Lemma 5 and expans why the average HOL watng tme s sma. In contract, the HOL watng tme of SMA decays sowy (see Fg. 3(b)) due to the starvaton probem. To gve a sense of the convergence tme of our agorthm, n Fg. 3(c), we pot the tme evouton of the nstantaneous system utty L = U (r ( V (t))) under dfferent α. The smuaton resuts show that the utty approaches very cose to the optma utty after 00 tme sot. Further, the utty s arger than the ower bound gven by ( ϵ)u([ R ϵ] + ) (oroary 4) even for a sma vaue of. Note that when α s arger, the utty vaue after convergence s aso coser to the optma vaue of (). However, arger α aso ncurs onger convergence tme. Before we proceed to a arger topoogy, we comment on the choce of α for dfferent vaues of. Through our smuaton, we observe that a rue of thumb s to choose α to be proportona to. The reason can be expaned by equaton (4). For a fxed vaue of r = x, as ncreases, n order to mantan the same probabty of addng another vrtua channe, we shoud ncrease α proportona to. Our smuaton studes ndcate that as ong as the rato α/ 4 Note that we use ths smpe and symmetrc topoogy frst because t aows us to easy compare wth the optma souton. 5 Note that Theorem 3 requres ϵ = 0. and a correspondng = 277. In ths smuaton settng, we ntentonay choose a sma to show that n reaty we can aso use sma to acheve good performance. 6 Note that the expected packet deay s an average over packets, and the expected HOL watng tme s an average across tme. As n the cassca Inspector s Paradox [27], these two ways of takng expectaton w ead to dfferent vaues when the nter-servce tme s not memoryess, whch s the case for the standard SMA agorthm.
0 IEEE/AM TRANSATIONS ON NETWORKING, VOL. X, NO. XX, XXXXXXX 20X Fg. 3. The smuaton resuts for the 8-by-8 torus graph. (a) (b) (c) (a) (b) (c) Fg. 4. The smuaton resuts for a random network wth 00 nodes and 00 nks. s fxed, the tradeoff between convergence and optmaty s roughy the same for dfferent vaues of. As ths rato ncreases, the optmaty s mproved at the cost of a onger convergence tme. Thus, we w use ths rue of thumb n the foowng smuaton resuts. Next, we smuate our agorthm under a arger random topoogy wth 00 nodes and 00 nks. We set the maxmum node degree to be 4, and under ths constrant, each nk s generated by randomy choosng two nodes. 7 We assume that each nk w nterfere wth the nks that are two-hop away (.e., the two-hop nterference mode). In addton to the standard SMA agorthm, we w aso compare our agorthm wth two other agorthms: the constant-tme (T) dstrbuted agorthm [28] and the we-known maxmum weghted matchng agorthm (MWM) [3]. However, We cauton that snce these agorthms ncur very dfferent computatona compexty and communcaton overhead, t s dffcut to conduct a competey far comparson. Because our proposed VM- SMA agorthm, the standard SMA agorthm, and the T agorthm requre ony one round of oca contro-message exchange (ncudng channe sensng) n each teraton, we w compare ther performance drecty. On the other hand, the MWM agorthm s a centrazed agorthm that requres the queue ength nformaton of a nks. Further, ts compexty may be exponenta under the two-hop nterference mode. In order to make the comparson sghty more far, we smuate a verson of the MWM agorthm that exchanges queue ength 7 We set a maxmum on the node degree smpy to guarantee a ower bound of r mn as suggested n Proposton 6 and oroary 7. nformaton and computes the schedue once every 00 tmesots. We ca ths agorthm LMWM (Low-frequency MWM). Note that even at the reduced frequency, the LMWM agorthm s st very costy to mpement due to ts hgh compexty and the requrement of coectng goba queue ength nformaton. In Fg. 4(a), we report the throughput and deay tradeoff of each agorthm. Specfcay, the y-axs s the average packet deay, and the x-axs s the average error percentage of the rate vectors computed by these agorthms under varous parameter settng. The error percentage for each nk s defned by max[optma rate aocated rate, 0]/optma rate. Thus, the most desrabe agorthm w correspond to a pont cose to the orgn, where t attans both hgh capacty and ow deay. As we can see n Fg. 4(a), for the VM-SMA agorthm, when we vary (whch s abeed next to each pont) and set α = 0.48, the VM-SMA consstenty acheves ow error percentage and ow deay. On the other hand, the performance curves for SMA and LMWM (each pont s abeed wth the vaue of β, the step sze used n the congeston contro) are sgnfcanty worse. For T agorthm, the curve (each pont s abeed wth the vaue of the backoff wndow sze) exhbts arge error percentage because T can ony acheve a fracton of the capacty regon. The performance of MWM agorthm s aso ncuded n Fg. 4(a) for a reference. However, MWM ncurs much hgher compexty. Hence, the VM-SMA agorthm s the ony one that can attan hgh throughput utty and ow packet deay wth ow compexty. Smar to the torus nterference graph, we aso report the ta dstrbuton of the HOL watng tme. Specfcay, we et
HUANG et a.: AHIEVING OPTIMAL THROUGHPUT UTILITY AND LOW DELAY WITH SMA-LIKE ALGORITHMS = 00 and choose a node where the optma aocated rate s 0.277. As shown n Fg. 4(b), the ta dstrbuton under our agorthm agan decays qucky. In contrast, the ta dstrbuton of the HOL watng tme under SMA agorthm does not decay even after 00. Fnay, we smuate our agorthm under the possbty of traffc changes and observe how our agorthm adapts wth the change. Specfcay, we et = 00 and α = 48. Further, we turn off the traffc on haf of the nks after 4000 tme-sots and turn on the traffc after 8000 tme-sots. Then, we record the evouton of the tota utty compared wth the optma utty computed offne. The resut s shown n Fg. 4(c). We can see that our agorthm adapts we to the traffc changes, and the nstantaneous utty qucky approaches to the optma utty. VI. ONLUSION In ths paper, we propose a vrtua-mut-channe (VM-) SMA agorthm that can provaby acheve hgh throughput utty and ow deay wth ow compexty for wreess networks under the protoco nterference mode and a snge-hop utty-maxmzaton settng. The key dea of VM-SMA s to resove the starvaton probem of standard SMA agorthms by emuatng a mut-channe system wth vrtua channes and computng mutpe feasbe schedues smutaneousy. VM-SMA nherts the dstrbuted nature of SMA, and the compexty of each nk grows neary wth. We use a nove probabstc argument to show that VM-SMA can approach arbtrary cose to the optma tota system utty wth (and hence the compexty) ncreasng ogarthmcay wth the network sze. Further, we derve suffcent condtons for the network topoogy and utty functons under whch the expected packet deay and the dstrbuton of ts head-ofne (HOL) watng tme can be bounded ndependenty of the network sze. Our smuaton resuts show that VM-SMA agorthm ndeed acheves both hgh throughput utty and ow deay wth ow-compexty operatons. In the future work, we w study how to extend ths nove dea to the wreess networks wth arrvas and mut-hop traffc. APPENDIX A PROOF OF PROPOSITION It s easy to verfy that the Markov han s rreducbe and aperodc. The reason s that every feasbe state V can communcate wth state 0, and the perod of state zero s. By [27, Theorem 4.3.3], f we can fnd a statonary dstrbuton of the Markov chan, ths statonary dstrbuton w be the unque dstrbuton. Now, we w show that P ( V ) = Z exp( L = U ( x ( V ) )) s the correct dstrbuton. We w demonstrate the correctness by verfyng the oca baance equatons. onsder two states V and V 2. If V has a transton to V 2, t mpes that there exsts a decson schedue S D such that for every nk that s schedued by S D, t s feasbe to change the state from V to V 2. Further, f nk s not schedued by S D, then V = V 2. Otherwse, there s no transton from V to V 2. Now, based on these condtons, we aso know that there s a transton from V 2 to V. The reason s that for every nk that s schedued by the same decson schedue S D, t s feasbe to change the state from V 2 to V. Further, f nk s not schedued by S D, then V 2 = V. Note that there may be mutpe decson schedues that can be used to change the state from V to V 2. We w ca these decson schedues feasbe decson schedues. Further, for any nk that s schedued by the decson schedue, any permutaton can be used to change the state from V to V 2. Now, et P ( S D ) be the probabty that decson schedue S D s chosen. Aso, et n = (n,, n ) be a vector represent a permutaton of the set {, 2,, }, and et n = (n,, n ). Note that n s the permutaton n n reverse order. To verfy the oca baance equatons, we ony need to show that P ( V ) {feasbe S D } P ( S D ) { S D } ( P,n P,n )! n = P ( V 2 ) P ( S D ) }( P P,n,n ).! {feasbe S D } { S D n (8) Note that P,n s the probabty of changng V to V 2, whch s the th teraton of updatng the vrtua-channe schedues from V to V 2 wth permutaton n. Smary, P s the probabty of changng the state from V 2 to, whch s the ( + ) th teraton of updatng the V vrtua-channe schedues from V 2 to V wth permutaton n. Let f (x) = exp(αu ( x )). Aso, et x2 = x ( V 2 ), and x = x ( V ). Reca that for any that s not schedued by S D, we have that V 2 = V. It mpes that x2 = x f s not schedued by S D. We can pug n P ( V ) = Z exp(α L = U ( x )), =, 2, and rewrte equaton (8) to the foowng equaton: = {feasbe S D } P ( S D ) Z P ( S D ) Z { S D } ( P,n P,n )f (x! ) n P,n P,n )f (x 2! ). }( {feasbe S D } { S D n (9) If we can show that gven any permutaton n = (n,, n ), we have that f (x )P,n P,n = P,n P,nf (x 2 ), (0) then equaton (9) w be true, and we coud fnsh the proof. Now, we show that equaton (0) s true. By equaton (3), P,n and P can be descrbed by fractons, and we w show that the denomnator (resp. numerator) on the eft hand sde of equaton (0) s equa to the denomnator (resp. numerator) on the rght hand sde of equaton (0). We frst focus on at the numerator. Note that the number of vrtua channes actvated by nk,.e., x, determnes the probabty of updatng the schedue of the vrtua channe. Further, note that the product operaton on the eft hand sde of equaton (0) mpes that we proceed the operaton wth the order dependng on the permutaton vector (n,, n ). Smary, for the rght hand sde of equaton (0), we proceed the product operaton wth the order dependng on the permutaton vector (n,, n ). Hence, we
2 IEEE/AM TRANSATIONS ON NETWORKING, VOL. X, NO. XX, XXXXXXX 20X know that for the eft hand sde of equaton (0), the vaue of x rght before updatng the schedue of vrtua channe n k s k x + = (V 2 V ). () Smary, for the rght hand sde of equaton (0), the vaue of x rght before updatng the schedue of vrtua channe n k s k+ x 2 + (V = Aso, we have the equaty between x and x2 x 2 = x + = (V 2 V 2 ). (2) V ). (3) From equatons (3), (), and (2), we know that the numerator on the eft hand sde of equaton (0) s equa to f (x ) f (x + k= k = (V 2 V )). (4) Smary, we know that the numerator on the rght hand sde of equaton (0) s equa to f (x 2 ) f (x 2 + k= k (V = V 2 )). (5) It then foows from equaton (3) that equaton (4) equas to equaton (5). Hence, the numerator on the eft hand sde of equaton (0) equas to the numerator on the rght hand sde of equaton (0). Now, we focus on the denomnator of equaton (0). We w show that the denomnator of P,n k equa to that of P for a k. Specfcay, from equatons (3),,n k (), and (2), we know that the denomnator of P,n k s f (x + k = (V 2 V + f (x + k = (V 2 and the denomnator of P,n k )) V ) + {V,n =0} {V =}),,n k k s f (x 2 + k+ = (V V 2 + f (x 2 + k+ = (V Let y = x + k = (V 2 (6) as )) V 2 ) + {V 2,n =0} {V 2 =}).,n k k (6) (7) V ). We can rewrte equaton f (y) + f (y + {V,n =0} {V =}). (8),n k k By equaton (3), we can aso rewrte equaton (7) as f (y + V 2 V,n k + f (y + V 2,n k,n k) + {V 2,n =0} {V 2 =}).,n k k (9) It s then easy to show that under a four possbe condtons,.e, (V 2,n k V,n k, V,n k) = (0, 0), (0, ), (, 0), or (, ), equaton (8) s equa to equaton (9). Hence, the denomnator of P,n k s equa to that of P,n k. Now, we have proved that equaton (0) s true. Ths mpes that equaton (8) s true, and we have verfed the oca baance equatons. Ths concudes the proof. APPENDIX B PROOF OF PROPOSITION 2 We frst prove one emma that s requred for provng Proposton 2. We omt the proofs due to the space constrants. Detas can be found n the technca report [25]. Lemma 8: If 0 < ϵ 0. and ϵ < r <, ( r + ϵ) og( r + ϵ ) + (r ϵ) og( r ϵ ) > 3 r r 2 ϵ2 Now, we can prove Proposton 2. Proof of Proposton 2: Reca that R = [R ] s the optma rate aocaton of probem (). By aratheodory s Theorem [23], R can be represented as a convex combnaton of L + feasbe schedues S, S 2,, S L+,.e., R = L+ = α S, where α 0 and L+ = α =. Now, consder the foowng random system. There are tras. For each tra k =,,, choose a S from S,, S L+ based on the probabty dstrbuton P ({ S = S }) = α, for a, ndependenty across k. Let r s be the number of tmes nk s schedued by ths random set of schedues dvded by. Then, n order to show that there exsts V s such that r ( V s ) R ϵ, for a nk, t s suffcent to show that the foowng event occurs wth a postve probabty: R rs ϵ for a nks. Note that by unon bound, we have P ( {R rs ϵ}) L = P (R rs > ϵ). Hence, t s suffcent to show that P (R r s > ϵ) <, for a nks. (20) L onsder a fxed nk. Intutvey, snce the probabty that nk s actve n each tra s equa to R,..d. across tras, (20) shoud hod when s suffcenty arge. Precsey, note that (20) hods trvay f R ϵ. Further, f R =, then nk w be actvated by a schedues S, and P (R rs > ϵ) = 0. Hence, we ony need to consder the possbty that ϵ < R <. (2) Let h be a random varabe that represents the number of tmes that nk s actvated by the seres of random schedues, and r s = h. Note that h s a Bnoma random varabe, and ts moment generatng functon s E[e ht ] = (R e t + R ). Usng the hernoff bound, we then have, for a t > 0, P (R rs > ϵ) = P ( h > ϵ R ) e t(r ϵ) E[e ht ] = e f(t), (22) where f(t) = t(r ϵ) + og(r e t + R ), t > 0. To get the best bound, we sove the mnmum of f(t) = t(r ϵ) + og(r e t + R ), t > 0. We frst observe that f (t) = R e t ( R ) > 0, t > 0, (R e t + R )2
HUANG et a.: AHIEVING OPTIMAL THROUGHPUT UTILITY AND LOW DELAY WITH SMA-LIKE ALGORITHMS 3 whch foows from equaton (2). Therefore, f(t) s strcty convex, and there s one goba mnmum. Hence, we fnd the mnmum by sovng f (t) = (R t ϵ) + R e = 0 R t e + R (23) e t = ( R )(R ϵ) R ( R +ϵ). Substtutng (23) nto (22), we can then show that P (R [ rs > ϵ) ( R ϵ R ) (R ϵ) ( ) R R +ϵ] R +ϵ. Hence, a suffcent condton of (20) to hod s > og L ( R +ϵ) og( R +ϵ R )+(R ϵ) og( R ϵ R ). (24) (25) It then foows from 2 og L 3ϵ 2, Lemma 8, equaton (25), and equaton (24) that P (R rs > ϵ) < L f ϵ < R <. Hence, equaton (20) s true, and we fnsh the proof. APPENDIX PROOF OF PROPOSITION 6 We frst prove one emma that s requred for provng Proposton 6. In the foowng emma, we assume that 0 < h < ( +2)(K ). We omt the proofs due to the space constrants. Detas can be found n the technca report [25]. Lemma 9: onsder nk and n nks, 2,, n, where nk nterferes wth a n nks, but these n nks do not nterfere wth each other. Suppose that the rate of nk s r, and the rate of nk s r. Further, suppose that n K, and the utty functon of a nks s og( + h) og(h). ) Suppose that < ( +2)(+K ) (K )h K + and +2. If r = 0 and r +2, for a, then U (r + ) + n = U (r ) > U (r ) + n = U (r ). 2) Suppose that < K ( +2) h(k ) K. If 0 < r < K ( +2) h(k ) K + and r +2 +, then U (r + ) + n = U (r ) > U (r ) + n = U (r ). Proof of Proposton 6: We prove ths proposton by contradcton. Suppose that there s a nk wth r mn K ( +2) h(k ) K + <. Let I = E {}. Ths mpes that there s a V such that V V, U( r( V )) = U( r( V max )),.e., U( r( V )) U( r( V )), for a V V, and r ( V ) <. In the seque, we refer to the K ( +2) h(k ) K + rate r ( V ) as the optma rate of each nk. We w construct another souton V V such that U( r( V )) > U( r( V )), whch then eads to a contradcton. From the defnton of r( V ), r( V ) can be descrbed by the convex combnaton of schedues, where each schedue has L dmensons. Denote these schedues as S,, S. Let a be the coeffcent of S. Note that a = / for a. Let S = { S, }. Now, we can truncate the schedue S to the set I,.e., we ony keep the eements of S that correspond to nks n I. Let S be the truncated verson of S. We then know that the optma rate for a nks n I can aso be descrbed by the convex combnaton of schedues,.e., S,, wth dmenson +. Let S = { S, }. Next, we w show that the optma rate for a nks n I can actuay be descrbed by the convex combnaton of k nonempty schedues from S wth postve coeffcent, where k s at most + 2. Frst, S s not empty for each. Otherwse, we can ncude nk n S, and we have another goba schedue wth arger utty. Second, we can foow the proof of the aratheodory s theorem to descrbe the optma rate n I as the convex combnaton of +2 schedues n S wth postve coeffcents. Wthout oss of generaty, et these schedues be S,, S k, k + 2. Let a be the weght of S, k. We know that k = a =, a > 0. Note that for each S, there exsts at east one correspondng untruncated schedue S wth a postve coeffcent,.e., a = / > 0, and ts truncaton n I equa to S. onsder the foowng two cases. ase : r ( V ) = 0. In ths case, we know that a these k schedues do not schedue nk. Further, snce k = a =, we can concude that one of the k schedues must have weght no smaer than +2. Wthout oss of generaty, et ths schedue be S, and we have that a +2. Ths mpes that the nks schedued by S have optma rate no smaer than +2. Reca that S s the correspondng untruncated schedue wth coeffcent a = /. Now, construct a schedue S such that ts schedue for nk s the same as S f / I, and t ony schedues nk n I. From the condton + K ( +2) h(k ) < < ( +2)(+K ) K, we can show that (K )h K + and +2. onsder another souton V such that r( V ) = =2 S + S. Suppose that there are n nks schedued n S. Denote these n nks as, =, 2,, n. Note that n K. Then U( r( V )) U( r( V )) = U (r ( V ) + ) U (r ( V )) ( n = U (r ( V )) U (r ( V ) )). For each nk schedued n S, we know that r ( V ) +2. By Lemma 9 (), we then know that U( r( V )) U( r( V )) > 0. Ths mpes that the utty of r( V ) s arger than the utty of r( V ), whch contradcts wth the assumpton that U( r( V )) = U( r( V max )). ase 2: r ( V ) > 0. We know that every feasbe schedue that schedues at east one nk n E can not schedue nk because every nk n E nterferes wth nk. Further, every schedue that schedues nk can not schedue any nks n E. Hence, r ( V ) s equa to the sum of the coeffcent of the k schedues that ony schedue nk, and there exsts at ease one schedue (from the k schedues) that schedues nk. We then know that there are at most + schedues (from the k schedues) that do not schedue nk. (Note that ths step does not hod n ase.) Snce r ( V ) < K ( +2) h(k ), we know that the sum of the weght of the rest k schedues, 0 < k +, s arger than K ( +2) K ( +2) + h(k ) K + +. Hence, we can concude that one of the remanng k schedues must have weght arger than K ( +2) K ( +2)( +) + h(k ) K ( +) +, whch can be shown to be no smaer than +2 +. Wthout oss of generaty, et ths schedue be S, and we have that a > +2 +. Ths mpes K +
4 IEEE/AM TRANSATIONS ON NETWORKING, VOL. X, NO. XX, XXXXXXX 20X have rate arger than +. that the nks schedued by S ℓ +2 s the correspondng untruncated schedue wth Reca that S such that coeffcent a = /. Now, construct a schedue S f ℓ ts schedue for nk ℓ s the same as S / Iℓ, and t ony schedues nk ℓ n Iℓ. Suppose that there are n nks schedued. Denote these n nks as ℓ, =, 2,, n. Note that n S h(kℓ ) )< n Kℓ. Reca that 0 < rℓ (V ℓ+, Kℓ ( ℓ +2) Kℓ ) and from a > ℓ+2 +, we have that ℓ+2 + rℓ (V for any nk ℓ schedued n S. such that r(v ) = another souton V onsder )+ U ( r(v )) U ( r(v )) = Uℓ (rℓ (V =2 S + S. Then n = Uℓ (rℓ (V )) Uℓ (rℓ (V ) )). By ) Uℓ (rℓ (V )) ( )) U ( r(v )) > 0. Lemma 9 (2), we then know that U ( r(v ) s arger than the Ths mpes that the utty of r(v ), whch contradcts wth the assumpton that utty of r(v )) = U ( r(v max )). U ( r(v ) < Snce for both cases, the assumpton rℓ (V Kℓ ( ℓ +2) h(kℓ ) ℓ+ eads to a contradcton, we concude that Kℓ rℓmn Kℓ ( ℓ +2) h(kkℓℓ ) ℓ+. Ths concudes the proof. R EFERENES [] P.-K. Huang and X. Ln, Improvng the Deay Performance of SMA Agorthms: A Vrtua Mut-hanne Approach, n IEEE INFOOM, 203, pp. 2598 2606. [2] L. Tassuas and A. Ephremdes, Stabty Propertes of onstraned Queueng Systems and Schedung Poces for Maxmum Throughput n Muthop Rado Networks, IEEE Trans. Autom. ontro, vo. 37, no. 2, pp. 936 948, 992. [3] X. Ln, N. B. Shroff, and R. Srkant, A Tutora on ross-layer Optmzaton n Wreess Networks, IEEE J. Se. Areas ommun., vo. 24, no. 8, pp. 452 463, 2006. [4] L. Georgads, M. J. Neey, and L. Tassuas, Resource Aocaton and ross-layer ontro n Wreess Networks, Found. Trends Netw, vo., no., pp. 44, 2006. [5] L. Jang and J. Warand, A Dstrbuted SMA Agorthm for Throughput and Utty Maxmzaton n Wreess Networks, IEEE/AM Trans. Netw., vo. 8, no. 3, pp. 960 972, 200. [6] J. N, B. Tan, and R. Srkant, Q-SMA: queue-ength based SMA/A agorthms for achevng maxmum throughput and ow deay n wreess networks, IEEE/AM Trans. 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Knuth, Art of omputer Programmng Voume 2: Semnumerca Agorthms, 3rd ed. Addson-Wesey Pubshng ompany, 997. [27] S. M. Ross, Stochastc Processes, 2nd ed. New York: John Wey & Son, 996. [28] A. Gupta, X. Ln, and R. Srkant, Low-ompexty Dstrbuted Schedung Agorthms for Wreess Networks, IEEE/AM Trans. Netw., vo. 7, no. 6, pp. 846 859, 2009. Po-Ka Huang receved hs B.S. and M.S. n Eectrca and omputer Engneerng from Natona Tsng Hua Unversty, Tawan, n 2004 and 2006, respectvey. He receved hs Ph.D. degree n Eectrca and omputer Engneerng from Purdue Unversty, Indana, n 203. He s currenty workng as system engneer at Inte corporaton. Hs research nterests are n the area of wreess networks ncudng wreess cross-ayer contro, resource aocaton, and deay performance anayss. Xaojun Ln (S 02 / M 05 / SM 2) receved hs B.S. from Zhongshan Unversty, Guangzhou, hna, n 994, and hs M.S. and Ph.D. degrees from Purdue Unversty, West Lafayette, Indana, n 2000 and 2005, respectvey. He s currenty an Assstant Professor of Eectrca and omputer Engneerng at Purdue Unversty. Dr. Ln s research nterests are n the anayss, contro and optmzaton of wreess and wrene communcaton networks. He receved the IEEE INFOOM 2008 best paper award and 2005 best paper of the year award from Journa of ommuncatons and Networks. Hs paper was aso one of two runner-up papers for the best-paper award at IEEE INFOOM 2005. He receved the NSF AREER award n 2007. He was the Workshop co-char for IEEE GLOBEOM 2007, the Pane co-char for WION 2008, the TP co-char for AM MobHoc 2009, and the Mnonference co-char for IEEE INFOOM 202. He s currenty servng as an Area Edtor for (Esever) omputer Networks journa, and has served as a Guest Edtor for (Esever) Ad Hoc Networks journa.