Order Optimal Delay for Opportunistic Scheduling in Multi-User Wireless Uplinks and Downlinks

Transcription

1 PROCEEDINGS OF THE 44TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, SEPT Order Optmal Delay for Opportunstc Schedulng n Mult-User Wreless Uplnks and Downlnks Mchael J. Neely Unversty of Southern Calforna mjneely Abstract We consder a one-hop wreless network wth ndependent tme varyng channels and N users, such as a multuser uplnk or downlnk. We frst show that general classes of schedulng algorthms that do not consder queue backlog necessarly ncur average delay that grows at least lnearly wth N. We then construct a dynamc queue-length aware algorthm that stablzes the system and acheves an average delay that s ndependent of N. Ths s the frst analytcal demonstraton that O(1) delay s achevable n such a mult-user wreless settng. The delay bounds are acheved va a technque of queue groupng together wth basc Lyapunov stablty and statstcal multplexng concepts. Index Terms Queueng Analyss, Stablty, Stochastc Control, Lyapunov Functon, Satellte Communcaton I. INTRODUCTION In ths paper, we nvestgate the fundamental delay scalng laws n a mult-user wreless system wth N tme varyng data lnks, such as a mult-user uplnk or downlnk. Packets arrve to the system accordng to ndependent stochastc arrval streams, wth one arrval stream for each lnk, and are stored n separate queues to awat transmsson. Tme s slotted, and the system can support a transmsson over at most one lnk per tmeslot. Channel condtons on each lnk vary ndependently every slot accordng to ON/OFF Bernoull processes, so that a lnk can transmt exactly one packet durng a tmeslot when t s n the ON state, and cannot transmt n the OFF state. Such ON/OFF channel states mght arse from channel fluctuatons or fadng due to user moblty. Every tmeslot, a network controller vews the condtons on each channel and chooses exactly one lnk to transmt. Ths system model s central to the study of channelaware (or opportunstc ) schedulng n wreless systems, and the model along wth many generalzatons have been extensvely consdered n the lterature 1]-]. Landmark work by Tassulas and Ephremdes n 1] characterzes the capacty regon of ths model, consstng of the set of all arrval rate vectors the system can be confgured to stably support. The work n 1] also proposes the Largest Connected Queue (LCQ) schedulng polcy, and uses a Lyapunov drft argument to show that ths polcy stablzes the system (and thus maxmzes throughput) whenever nput rates are nteror to the capacty regon. Furthermore, the work n 1] uses a stochastc couplng argument to show that, n the specal case of a symmetrc system wth dentcal nput rates for each user Ths work s supported n part by one or both of the followng: the Natonal Scence Foundaton grant OCE 05034, the DARPA IT-MANET program. and dentcal channel probabltes for each lnk, the LCQ polcy mnmzes average delay. Ths delay optmalty result s generalzed n 4] 10], where a delay optmal polcy s developed for selectng transmsson rates wthn the polytope capacty regon assocated wth the Gaussan multple access channel, and n 14] where generalzatons to mult-server systems are consdered. However, these delay optmalty results hold only n cases when the system exhbts perfect symmetry n traffc rates and channel statstcs. Indeed, these works use the stochastc couplng technque of 1], whch seems to requre ths symmetry. Further, the actual average delay acheved by these strateges s unknown, even n these symmetrc cases. Work n 7] computes upper bounds on the delay of stablzng largestqueue type strateges. However, these bounds grow lnearly n the number of users N. Specfcally, the delay bound s gven by O(N/(1 ρ)), where ρ s a parameter such that 0 < ρ < 1 and represents the fracton the nput rate vector s away from the capacty regon boundary. Whether or not optmal delay can grow sub-lnearly wth N has remaned an mportant open queston, and s a queston that we resolve n ths paper. Usng the smple ON/OFF channel model, we frst show that, for general classes of schedulng algorthms that use channel state nformaton but do not consder queue backlog, average delay must grow at least lnearly wth N. We then construct a smple dynamc control polcy called Largest Connected Group that uses both queue state and channel state nformaton. We apply ths polcy to the smple symmetrc system where all data rates and channel probabltes are the same, and show the polcy yelds average delay that s ndependent of N. Specfcally, we compute an upper bound on average delay that s O( log(1/(1 ρ)) 1 ρ ). Ths s the frst analytcal demonstraton that such delay s possble. Next, we derve a smlar result for large classes of asymmetrc systems,.e., systems wth heterogeneous traffc rates and channel probabltes. Prevous work n the area of wreless schedulng s found n 5]8]9] for systems wth an nfnte backlog of data, and a clearng problem n a system wth N lnks and a fxed amount of data s treated n 11]. Stable schedulng and queueng s consdered for satellte, wreless, and ad-hoc moble systems n 1]]3]6]7]1]13]. The work n 6] develops delay optmalty results n the lmt as the system loadng ρ approaches 1, but does not provde asymptotc results n the number of users. Indeed, the analyss n 6] uses a flud lmt and a heavy traffc lmt that may suggest each of

2 PROCEEDINGS OF THE 44TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, SEPT. 006 the N queues s usually non-empty. In our analyss, we provde an average delay bound for a fxed loadng factor ρ < 1, and obtan delay that s ndependent of N by schedulng to ensure that each queue s usually empty. Ths provdes an advantage n the case when there are many users and ρ s a fxed fracton away from the capacty regon boundary. However, whle our ) bound n ths paper has a better asymptotc n N than the prevous O(N/(1 ρ)) bound n 7], t has a slghtly worse asymptotc n ρ. Much work n the area of dynamc schedulng s developed for computer networks and swtchng systems, ncludng work n 3]4]5]6] that uses Lyapunov stablty theory. The work n 4] consders max-weght-match (MWM) schedulng n an N N packet swtch wth..d. traffc (such as Bernoull or Posson), and shows that average delay s no more than O(N/(1 ρ)). Varous methods of queue groupngs are used wth Lyapunov functons n 6]7]8] to acheve low complexty schedulng. Whle 6]7]8] does not prmarly focus on delay, t s nterestng to note that f an N N swtch s half loaded (ρ < 1/) wth ndependent Bernoull or Posson nputs, then smlar queue groupngs together wth the Lyapunov delay technque of 4] can be used to show that average delay s O(1/(1 ρ)) under maxmal match schedulng. However, ths result does not seem to extend to cases when ρ > 1/. Work n 9] uses a smple framebased algorthm for an N N swtch to show t s possble to acheve an average delay of O(log(N)/(1 ρ) ), for any value ρ < 1. Our results n the present paper parallel our prevous work n 9] for swtch schedulng. However, the problem formulaton and soluton technque s qute dfferent here, as the frame-based approach n 9] does not appear tractable wth stochastc channel condtons. Here, we pursue a novel queue groupng approach, and show that the average delay of our wreless system can be bounded ndependently of the number of users N, for any value of ρ < 1. In the next secton, we formulate the problem and revew the system capacty regon from 1]. In Secton III we show that a large class of backlog-unaware schedulng algorthms necessarly ncur average delay that grows at least lnearly wth N. In Secton IV we develop our backlog-aware Largest Connected Group algorthm and show t yelds average delay that s ndependent of N. O( log(1/(1 ρ)) 1 ρ II. PROBLEM FORMULATION Consder an N queue system that evolves n dscrete tme wth ntegral tmeslots t 0, 1,,...}. Let Q (t) represent the number of packets n queue at the begnnng of slot t (for 1,..., N}). Let A (t) represent the number of new packet arrvals durng slot t, and let µ (t) represent the transmsson rate (n unts of packets) durng slot t. The dynamc equaton for each queue 1,..., N} s gven by: Q (t + 1) = maxq (t) µ (t), 0] + A (t) (1) Each queue contans data that must be transmtted over a dstnct lnk wth tme varyng channels. Let S (t) ON, OF F } represent the channel state of lnk durng slot t. Assume these channel states are..d. over tmeslots and ndependent across channels, and let q represent the ON probablty for channel : q = P rs (t) = ON] The channel states are assumed to be known to the network controller at the begnnng of each slot. Every slot t, the network controller chooses transmsson decson varables µ(t) = (µ 1 (t),..., µ N (t)) subject to the constrants: µ (t) 0, 1} 1,..., N} µ (t) = 0 f S (t) = OF F N µ (t) 1 () The above constrants specfy that at most one lnk can be chosen for transmsson on any tmeslot, and that exactly one packet can be transmtted over a gven lnk durng a tmeslot n whch S (t) = ON, whle no packets can be transmtted over a channel that s OF F. Ths system model can be used to represent a mult-user wreless or satellte downlnk, where all packets arrve to a sngle node that nternally stores data n separate queues for transmsson to ts proper destnaton. Alternatvely, the system can represent a mult-user wreless uplnk, where each user has ts own data that must be transmtted to a central access pont. In ths uplnk scenaro, the queues are dstrbuted over the dfferent users. It s assumed n ths case that the access pont determnes whch user transmts on every slot by sendng permsson sgnals over a dedcated control channel. Defnton 1: A dscrete tme queue Q(t) wth a general arrval and server rate process s strongly stable f: 1 1 t 1 lm sup E Q(τ)} < t t A network of queues s sad to be strongly stable f each queue s strongly stable. Throughout ths paper, we use the term stablty to refer to strong stablty. The goal s to desgn a schedulng algorthm that stablzes the system whle keepng tme average backlog and average delay as small as possble. A. The Capacty Regon Suppose arrvals A (t) are..d. over tmeslots, and let λ = E A (t)} represent the packet arrval rate of stream (for each 1,..., N}). Let λ = (λ 1,..., λ N ) represent the arrval rate vector. The network capacty regon Λ s the closure of the set of all rate vectors λ for whch a stablzng algorthm exsts. For a system of queues (N = ), the capacty regon s gven by all rate vectors (λ 1, λ ) that satsfy: λ 1 q 1, λ q λ 1 + λ q 1 + (1 q 1 )q These nequaltes are clearly necessary for stablty, as otherwse one or both queues would have an nput rate that exceeds the transmsson rate capabltes of the system. It s 1 We note that f a queue Q(t) s strongly stable and also evolves accordng to an aperodc, rreducble Markov chan, then the lm sup on the left hand sde n the stablty defnton above can be replaced wth a regular lmt that represents the steady state backlog.

3 PROCEEDINGS OF THE 44TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, SEPT not dffcult to show that any rate vector (λ 1, λ ) nteror to ths regon can be stablzed. The capacty regon for a system of N queues s shown n 1] to be the set of all rate vectors λ = (λ 1,..., λ N ) that satsfy the nequalty: λ 1 Π I (1 q ) I for each non-empty subset of ndces I 1,..., N}. Thus, the capacty regon s descrbed by a set of N 1 nequalty constrants. An alternate characterzaton of the capacty regon can be gven n terms of all possble expected transmsson rate vectors that can be acheved by a statonary randomzed schedulng polcy, as shown below. Lemma 1: (Statonary Randomzed Polces 17]1]) A rate vector λ = (λ 1,..., λ N ) s n the capacty regon Λ f and only f there exsts a statonary control strategy that chooses a transmsson rate vector µ(t) = (µ 1 (t),..., µ N (t)) as a (potentally random) functon of the observed channel state vector S(t) = (S 1 (t),..., S N (t)) such that µ(t) satsfes () for all t, and such that the expected transmsson rate yelds: E µ (t)} = λ for all 1,..., N} The expectaton above s wth respect to the statonary dstrbuton for the channel state vector S(t) and the potentally random transmsson decson that depends on S(t). Note that n the specal case of a symmetrc system where q = q for all 1,..., N}, then the largest symmetrc rate vector (λ, λ,..., λ) that s n the capacty regon s gven by the vector wth λ = r N /N for all 1,..., N}, where r N s the probablty that at least one lnk s n the ON state durng a tmeslot: r N = 1 (1 q) N (3) Ths can be seen from Lemma 1 by defnng the statonary polcy that chooses a transmsson lnk ndependently and unformly over all lnks that are ON. Specfcally, ths polcy yelds E µ (t)} = E µ j (t)} for all, j by symmetry, and also yelds E µ 1 (t) +... µ N (t)} = r N, so that E µ (t)} = r N /N for all 1,..., N}. B. The Sngle-Queue Lower Bound A smple lower bound on the average backlog (and hence, by Lttle s Theorem 30], average delay), can be obtaned by usng the multplexng nequalty 31]. Specfcally, the multplexng nequalty states that the total queue backlog N Q (t) n a system of N queues descrbed by (1) s greater than or equal to the backlog n a correspondng sngle queue system wth an nput and servce rate process gven by the sum of the processes n the mult-queue system. That s, gven a sngle queue system Q sngle (t) wth dynamcs: ] N N Q sngle (t + 1) = max Q sngle (t) µ (t), 0 + A (t) then we have: N Q (t) Q sngle (t) for all t Assume the followng tme averages Q and Q sngle exst: 1 t 1 Q = lm t t 1 Q sngle = lm t t t 1 E Q (τ)} E Q sngle (τ)} It follows that the tme average backlog satsfes: N Q Q sngle Defnng λ tot = (λ λ N ) and defnng W and W sngle as the average packet delay n the mult-queue and snglequeue system, respectvely, we have by Lttle s Theorem 30]: λ tot W = N Q, λ tot W sngle = Q sngle Thus, we have the followng sngle-queue delay bound: W W sngle However, note that the constrants () specfy that µ (t) 0, 1}, and can only be 1 f there exsts a channel such that S (t) = ON. The Q sngle (t) queue wll be smallest f we assume µ (t) = 1 whenever possble, and hence the dynamcs of Q sngle (t) reduce to: Q sngle (t + 1) = maxq sngle (t) µ sngle (t), 0] + A sum (t) where A sum (t) = N A (t), and µ sngle (t) s an..d. Bernoull process wth rate µ av, where: µ av = P rµ sngle (t) = 1] = 1 Π N (1 q ) Thus, Q sngle (t) s a smple dscrete tme GI/GI/1 queueng system wth a Bernoull servce process. The average backlog and delay n such a system can be computed exactly: Q sngle = λ tot + E } A sum λ tot µ av (1 ρ) W sngle = λ tot E Asum} λtot µ av (1 ρ) where E A sum} = E Asum (t) }, and ρ = λ tot /µ av (4) In the case when all nputs A (t) are ndependent and Posson wth rates λ, we have: E A sum} = λtot + λ tot and hence the sngle-queue delay bound s gven by: W W sngle = 1 λ tot/ µ av (1 ρ) Ths specfes that the best possble average delay of any schedulng algorthm s O(1/(1 ρ)) when arrvals are ndependent and Posson. On the other hand, n the case when the nputs A (t) are not ndependent, the best possble delay mght be O(N/(1 ρ)). Specfcally, f we have A (t) = A(t) for all 1,..., N}, wth A(t) Posson of rate λ tot /N, then queue 1 receves k (5)

4 PROCEEDINGS OF THE 44TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, SEPT packets on slot t f and only f all other queues receve k packets that slot. It follows that E A sum} = E N A(t) } = Nλ tot + λ tot and hence: W sngle = N + 1 λ tot µ av (1 ρ) (correlated arrval case) (6) The dfference between the O(1/(1 ρ)) and O(N/(1 ρ)) delay bounds n (5) and (6) s due to the statstcal multplexng gans arsng when data streams A (t) are ndependent. Throughout ths paper, we shall assume nputs are ndependent. Our goal s to develop an algorthm that yelds average delay close to the O(1/(1 ρ)) delay assocated wth the sngle-queue bound n (5). III. BACKLOG-UNAWARE SCHEDULING Here we show that f schedulng algorthms are restrcted to a large class of polces that use channel state nformaton but do not use queue backlog nformaton, then average delay necessarly grows at least lnearly wth N. Suppose arrval processes are statonary and ergodc wth rates λ. Let X (t) represent the number of packets that arrve up to tme t, and let X (v)} v t denote the entre arrval hstory up to tme t. We consder statonary schedulng algorthms that choose transmsson rates ndependent of the entre arrval hstory, and hence ndependent of current queue backlog. Specfcally, we consder the class of schedulng polces that yeld transmsson rates wth the followng property for all 1,..., N}: E µ (t) X (v)} v 0 } = E µ (0)} =µ (7) Ths s a large class of polces, ncludng all of the statonary randomzed schedulng polces used n Lemma 1. Perodc polces (such as round robn schedulng) can also be ncluded n ths class f the phase of the ntal perod s unformly randomzed. Theorem 1: (Backlog Unaware Schedulng) Consder any schedulng algorthm that satsfes (7) and stablzes the system wth fnte average backlogs Q and average delay W. Then: (a) For all t, we have: E Q (t)} E U (t)} where U (t) represents the unfnshed work (or fractonal packets) at tme t n a contnuous tme queueng system wth the same arrvals A (t) but wth a constant transmsson rate µ (and hence determnstc servce tmes 1/µ ). (b) Suppose there are symmetrc channel probabltes q = q and symmetrc rates λ = λ tot /N for all 1,..., N}. Assume λ tot r N (where r N = 1 (1 q) N s the maxmum system output rate). If the arrval streams are contnuous tme Posson processes, then average delay necessarly satsfes: W N r N (1 ρ) where ρ =λ tot /r N. (c) For asymmetrc systems, let r max represent the maxmum possble sum output rate: r max = 1 Π N (1 q ) Let γ 1 and γ be postve constants less than 1. If there are at least γ 1 N arrval processes wth transmsson rates at least γ λ tot /N, then average delay s at least γ 1γ N/(r max ), and hence grows at least lnearly wth N. Proof: See Appendx. IV. THE QUEUE GROUPING ALGORITHM Here we develop a dynamc algorthm that nvolves queue groupng, and we show the algorthm has average delay that s ndependent of N. We frst revew the delay result from 7] that provdes a (loose) upper bound on the average delay of the LCQ polcy from 1]. Recall that the LCQ polcy chooses to transmt over the ON lnk wth the largest queue backlog (breakng tes randomly and unformly), and s shown n 1] to stablze the system whenever nput rates are nsde the capacty regon Λ, and to mnmze average delay n the specal case of a symmetrc system. Assume channel states are ndependent wth probabltes q for 1,..., N}. Let λ = (λ 1,..., λ N ) be the rate vector, and suppose that there exsts a value ɛ > 0 such that λ+ɛ Λ (where ɛ=(ɛ, ɛ,..., ɛ)). Thus, we assume λ s strctly nteror to the capacty regon, and that a postve value ɛ can be added to each component to yeld another vector that s wthn the capacty regon. Lemma : (Delay of LCQ 7]) Suppose arrval vectors A(t) are..d. over tmeslots, and that λ + ɛ Λ. Then: (a) The LCQ polcy stablzes the system and yelds average delay that s upper bounded as follows: W λ tot + N E } A N λ (8) λ tot ɛ where E } A = E A (t) }, and λ tot = N λ. (b) If arrval streams A (t) are ether Bernoull or Posson wth symmetrc rates λ = λ tot /N for 1,..., N}, f q = q for all 1,..., N}, and f λ tot = ρr N for some value ρ such that 0 < ρ < 1 (where r N s defned n (3)), then average delay satsfes: W N λ tot/ r N (1 ρ) Note that part (b) follows mmedately from part (a) by usng ɛ = r N /N λ tot /N. The upper bound on average delay s O(N/(1 ρ)). The lemma holds for arrval vectors wth components that are arbtrarly correlated, and hence n ths sense the asymptotc wth N s tght (recall the sngle-queue bound (6) n the case of correlated arrvals). A. Intuton for Queue Groupng Here we assume arrval streams A (t) are..d. over tmeslots, and also ndependent of each other. To provde ntuton on the advantages of queue groupng, defne q mn = mn 1,...,N} q, and compare the system of N parallel queues (wth channel probabltes q q mn for all 1,..., N}) The dervaton n 7] consders a more general system wth varable transmsson rates that can be any real number, and obtans a slghtly dfferent bound n ths case, but stll wth the O(N/(1 ρ)) structure. The exact expresson (8) follows as a specal case of Theorem n the case N = K.

5 PROCEEDINGS OF THE 44TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, SEPT to a sngle queue system wth a Bernoull server wth rate q mn and wth an arrval process gven by A sum (t), the sum of the ndvdual A (t) arrval processes. It can be shown that f the N queue system schedules accordng to any work conservng schedulng polcy (.e., a polcy that always serves a nonempty ON queue f one s avalable), the resultng backlog s stochastcally less than the backlog n the sngle queue system (we do not requre ths result n our analyss, and hence omt the proof for brevty). It follows that f λ tot < q mn, then the average delay n the mult-queue system s no more than the average delay n the sngle queue system. In partcular, f the nput processes A (t) are Posson or Bernoull, then we have: W 1 λ tot/ q mn λ tot Therefore, delay n ths case does not grow lnearly wth N. Further, ths result holds whenever the nput rate vector s wthn a factor ρ of the capacty regon boundary, for any value ρ such that 0 < ρ < γ, where γ=q mn /r max. To see ths, note that ths result holds for any rates such that λ < q mn, and let Λ denote the closure of ths regon. It follows that: γλ γ λ λ 0, } λ r max = λ λ 0, } λ γr max = λ λ 0, } λ q mn = Λ where the frst ncluson follows because λ r max s a necessary condton for λ Λ (t s not necessarly suffcent). Thus, Λ contans the set γλ. However, ths sngle-queue comparson does not apply when γ ρ < 1. To acheve a larger fracton of the capacty regon, we can assemble each of the N queues of the system nto K dstnct groups. Intutvely speakng, each sngle group can be compared to a correspondng sngle queue system wth a Bernoull transmsson rate of q mn. The advantage s that now we only requre the sum of transmsson rates wthn each group to be less than q mn (so that larger nput rate vectors can generally be supported). Each group s then treated as a sngle queue, and the LCQ algorthm s appled to that system of K queues, yeldng an O(K) delay result va Lemma. In the next secton we make ths ntuton precse. B. The Largest Connected Group (LCG) Algorthm Below we specfy the queue groupng algorthm for a general set of groups. We then dscuss ntellgent ways to form the groups for both symmetrc and asymmetrc systems. Let G 1,..., G K } represent any general groupng of the queue ndces 1,..., N} nto dsjont sets, where K s the number of groups. Specfcally, we assume each group G k s a non-empty subset of the set 1,..., N}, and the unon of all K groups s equal to the set of all queue ndces 1,..., N}. For each group ndex k 1,..., K}, defne: A sum,k (t) = A (t) G k Q sum,k (t) = Q (t) G k λ sum,k (t) = G k λ Further defne the ndcator functon 1 k (t) to take the value 1 f group G k has at least one ndex that corresponds to a non-empty queue wth an ON channel state, so that Q (t) > 0 and S (t) = ON. The Largest Connected Group (LCG) Algorthm: Every tmeslot t, the network controller observes the queue backlogs and current channel states, and selects the group ndex k 1,..., K} that maxmzes Q sum,k (t)1 k (t), breakng tes arbtrarly. It then chooses to transmt over any lnk G k that corresponds to a non-empty queue wth a channel that s ON,.e., any non-empty connected queue of the selected group. If there are no such queues for slot t, reman dle. For all k 1,..., K}, defne: q mn,k = mn G k q Now defne Λ K as the K dmensonal capacty regon of a system wth K queues wth Bernoull ON probabltes q mn,k for k 1,..., K}. That s, Λ K s the set of all non-negatve rate vectors ω = (ω 1,..., ω K ) such that ω k 1 Π k I (1 q mn,k ) k I for all subsets I 1,..., K}. Theorem : (LCG Performance for General Groups) Suppose channels are ndependent wth ON probabltes q for 1,..., N}, and arrval vectors A(t) are..d. wth rate vector λ. If there exsts a value ɛ > 0 such that: (λ sum,1 + ɛ, λ sum, + ɛ,..., λ sum,k + ɛ) Λ K then the system s stable, and: Q where: λ tot + K k=1 E A sum,k Q = lm sup t ɛ 1 t 1 t } ] K k=1 λ sum,k N E Q (τ)} If arrval processes A (t) are ndependent and ether Bernoull or Posson, then: λ tot ] K k=1 λ sum,k Q (9) ɛ Proof: The frst part of the theorem s proven n the next secton usng a Lyapunov drft argument. Inequalty (9) then follows mmedately by notng that f A sum,k (t) =

6 PROCEEDINGS OF THE 44TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, SEPT G k A (t) s a sum of ndependent Bernoull or Posson processes wth rates λ, then: E A sum,k} λsum,k + λ sum,k Note that the LCG algorthm breaks tes arbtrarly. However, ntuton from the LCQ algorthm n 1] suggests that servng larger queues tends to yeld better delay performance. Thus, an ntutvely good te breakng rule s to serve the queue wth the largest backlog among all tes under LCG. If there are further tes under ths rule, then break the tes randomly and unformly over all groups. Ths te breakng rule also ensures the vector queueng process Q(t) evolves accordng to a dscrete tme Markov chan, n whch case Foster s crteron 3] can be used to ensure the chan has a vald steady state wth steady state queue occupances Q. If nputs are ndependent and Bernoull or Posson, then the expresson (9) can be smplfed to Q λ tot /ɛ, and hence by Lttle s Theorem the average delay satsfes W 1/ɛ. To smplfy notaton, for the remander of ths paper we assume that such steady state lmts exst whenever the system s stable. C. Choosng Groups for Symmetrc Systems Consder a symmetrc system such that q = q for all 1,..., N}, and defne a loadng parameter ρ such that 0 < ρ < 1. Defne the group sze K as: log(/(1 ρ)) K = (10) log(1/(1 q)) where x denotes the smallest nteger greater than or equal to x. Note that K s chosen ndependently of the number of queues N. For smplcty, assume that N s a multple of K, so that we form dstnct groups G 1,..., G K, each wth N/K elements. Suppose that all nput rates are dentcal, so that λ = λ tot /N for all 1,..., N}. Assume that λ tot = ρr N (where r N s gven n (3)), so that the rate vector s a factor of ρ away from the capacty regon boundary. Theorem 3: (Symmetrc Performance) Consder a unformly loaded symmetrc system as descrbed above, wth a group sze K gven by (10). If N s a multple of K, and f nputs are ndependent and ether Bernoull or Posson, then the LCG algorthm stablzes the system and yelds: Q Kλ tot λ tot r N (1 ρ) Therefore, average delay satsfes: W K λ tot r N (1 ρ) log(/(1 ρ)) r N (1 ρ) log(1/(1 q)) + λ tot r N (1 ρ) The above result demonstrates that average delay satsfes: ( ) log(1/(1 ρ)) W O 1 ρ Ths s the frst analytcal demonstraton that average delay does not grow wth N. Recall that the sngle-queue lower bound of (5) mples that no algorthm can acheve an average delay less than O(1/(1 ρ)). Hence, the LCG algorthm performs optmally n N, and dffers from the optmal performance n ρ by a logarthmc factor log(1/(1 ρ)). Proof: (Theorem 3) Note that Λ K n ths case s the capacty regon assocated wth a symmetrc system of K queues wth ndependent Bernoull channels, each wth ON probablty q. Defne r K as the largest sum rate from ths K queue system. It follows that the symmetrc rate vector ω = (r K /K,..., r K /K) s contaned n Λ K. Further note that λ sum,k = λ tot /K for all k 1,..., K}. To ensure that the condtons of Theorem hold, we desre to fnd a value ɛ > 0 such that: (λ sum,1 + ɛ,..., λ sum,k + ɛ) Λ K It suffces to show that λ sum,k + ɛ r K /K, whch s equvalent to showng: To ths end, note by (10) that and hence: It follows that: λ tot + ɛk r K (11) K log(1/(1 q)) log(/(1 ρ)) (1 q) K (1 ρ)/ r K = 1 (1 q) K (1 + ρ)/ (1 + ρ)r N / Therefore: r K λ tot = r K ρr N (1 + ρ)r N / ρr N = r N (1 ρ)/ It follows that choosng ɛ =r N (1 ρ)/(k) ensures that (11) s satsfed. The result follows by applyng nequalty (9) from Theorem. D. Asymmetrc Systems Consder a general asymmetrc system wth N queues and ndependent channels wth ON probabltes q } for 1,..., N}. Defne q mn = mn 1,...,N}] q. Defne a loadng parameter ρ such that 0 < ρ < 1, and choose the group sze K as follows: Further defne: K = log(/(1 ρ)) log(1/(1 q mn )) r a = 1 (1 q mn ) K r max = 1 Π N (1 q ) (1) We assume that N K. Note that r a s the maxmum output rate n a system of K queues wth ndependent Bernoull channels wth probablty q mn, and r max s the maxmum output rate of the asymmetrc system of N queues. Consder heterogeneous nput rates (λ 1,..., λ N ), and wthout loss of generalty assume λ 1 λ... λ N. Defne λ tot = N λ, and assume that λ tot = ρr max, so that the

7 PROCEEDINGS OF THE 44TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, SEPT rate vector s at most a dstance ρ away from the capacty regon boundary. Place the queues nto groups as follows: Defne G 1 as the set of all ndces 1,..., M 1 }, where M 1 s the smallest nteger such that M 1 λ λ tot /K. Then defne G as the set of all ntegers M 1 + 1,..., M }, where M s the smallest nteger such that M =M λ 1+1 λ tot /K. Proceedng ths way, we form groups G 1,..., G K by successvely packng the nputs nto groups untl the last nput added makes the sum rate for that group exceed λ tot /K. It follows that all groups k 1,..., K} satsfy: λ sum,k λ tot /K + λ (13) where λ = max 1,...,N} λ. To proceed, we make the followng addtonal assumpton concernng the sze of ths largest nput rate λ: λ (1 ρ)r max /(3K) (14) Note that the average sze of each nput s gven by λ tot /N = ρr max /N. Because N can be much larger than K, ths addtonal assumpton (14) states that the largest nput s upper bounded by a number much larger than the average. Theorem 4: (Asymmetrc Performance) Consder an asymmetrc system as descrbed above, and assume the group sze K satsfes (1). Assume that N K, and that the largest nput rate λ satsfes (14). If nputs are ndependent and ether Bernoull or Posson, then the LCG algorthm stablzes the system and yelds: 3K λ tot ] K k=1 λ sum,k Q r max (1 ρ) and hence average delay satsfes: 3K 1 ] K λ tot k=1 λ sum,k W r max (1 ρ) Because K satsfes (1), we agan see that average delay s O( log(1/(1 ρ)) 1 ρ ), and hence s ndependent of N. Proof: (Theorem 4) Smlar to the proof of the symmetrc case, the nequalty (1) can be used to show: r a = 1 (1 q mn ) K (1 + ρ)/ and hence (usng (13) and (14)): r a K λ sum,k r max(1 ρ) 6K for all k 1,..., K} (15) However, note that the capacty regon assocated wth K queues, each wth ndependent Bernoull channels wth probabltes q mn, s a subset of Λ K (ths s because the set Λ K has queues wth probabltes q mn,k q mn for all k 1,..., K}). Therefore, the vector ω = (r a /K,..., r a /K) s contaned n the set Λ K. It follows from (15) that we can defne ɛ as follows: ɛ = r max (1 ρ)/(6k) The result follows by pluggng ths value of ɛ nto (9) of Theorem. V. LYAPUNOV ANALYSIS Here we use Lyapunov drft theory to prove Theorem of the prevous secton. We begn wth a smple modfcaton of an mportant Lyapunov drft result from 17]4]. A. Lyapunov Drft Let Q(t) represent a vector process of dscrete tme queues that evolves accordng to some probablty law. Let L(Q) be a non-negatve functon of the queue vector, called a Lyapunov functon. Defne the condtonal Lyapunov drft (Q(t)) as follows: 3 (Q(t)) =E L(Q(t + 1)) L(Q(t)) Q(t)} (16) Lemma 3: (Lyapunov Drft 17]4]) Suppose there s a non-negatve functon L(Q), a non-negatve process B(t), and a value ɛ > 0 such that for all tme t and all possble Q(t), we have: (Q(t)) E B(t) ɛh(t) Q(t)} where h(t) represents a non-negatve process that mght depend on the queue state. Then: 1 t 1 1 t 1 E B(τ)} lm sup E h(τ)} lm sup t t t t ɛ Choosng quadratc Lyapunov functons often leads to drft expressons where h(t) s lnear n Q(t), so that the above result can be used to bound frst moments of queue congeston. B. Proof of Theorem Defne the Lyapunov functon: ( L(Q) = 1 K k=1 G k Q Thus, we have that L(Q(t)) s the sum of squares of the total backlog assocated wth each group k 1,..., K}: L(Q(t)) = 1 ) K (Q sum,k (t)) k=1 To compute (Q(t)), defne for each k 1,..., K} µ sum,k (t) = G k µ (t) Because the sum transmsson rate s no more than 1, µ sum,k (t) represents the transmsson rate offered to group k durng slot t. Defne µ sum,k (t) to be the actual number of packets transmtted by group k durng ths slot (so that µ sum,k (t) 0, 1} and can only be 1 f the group has a nonempty connected queue durng slot t). For each group k, we have: Q sum,k (t + 1) = Q sum,k (t) µ sum,k (t) + A sum,k (t) 3 Strctly speakng, the condtonal drft should use notaton (Q(t), t) as a general drft may also depend on t, but we use the smpler notaton (Q(t)) to formally represent the rght hand sde of (16).

8 PROCEEDINGS OF THE 44TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, SEPT Squarng both sdes of the above equalty and usng the fact that µ sum,k (t) = µ sum,k (t) (because the value s ether 0 or 1) yelds: Q sum,k (t+1) = Q sum,k(t) + B k (t) + Q sum,k (t)a sum,k (t) Q sum,k (t) µ sum,k (t) where B k (t) = µsum,k (t) + A sum,k (t) ] A sum,k (t) µ sum,k (t) Takng condtonal expectatons and summng over all k yelds: (Q(t)) = E B(t) Q(t)} + K k=1 Q sum,k(t)λ sum,k K k=1 Q sum,k(t)e µ sum,k (t) Q(t)} (17) where B(t) = K k=1 B k(t), and where we have used the fact that arrvals are..d. over slots and hence have expected values that are ndependent of the current queue state. Gven Q(t) and the channel states, the LCG algorthm s desgned to choose transmsson rates that maxmze the expresson k Q sum,k(t) µ sum,k (t) over all possble transmsson decsons durng slot t that are subject to the constrants: µ sum,k (t) 0, 1} for all k 1,..., K} (18) K k=1 µ sum,k(t) 1 (19) µ sum,k (t) 1 k (t) for all k 1,..., K} (0) Hence, t also maxmzes the condtonal expectaton of ths expresson gven Q(t). It follows that the LCG algorthm mnmzes the fnal term n the drft expresson (17) over all feasble transmsson rate decsons that satsfy the constrants (18)-(0) durng slot t. Therefore, we have: (Q(t)) E B(t) Q(t)} + K k=1 Q sum,k(t)λ sum,k K k=1 Q sum,k(t)e µ k (t) Q(t)} (1) where (µ 1(t),..., µ K (t)) represents any transmsson rate decson vector that satsfes (18)-(0). Now recall that, accordng to the condtons of Theorem, we have: (λ sum,1 + ɛ,..., λ sum,k + ɛ) Λ K where Λ K s the capacty regon of a vrtual system wth K ndependent queues wth channel probabltes q mn,k for k 1,..., K}. Let S v (t) represent the channel states of ths vrtual system (havng ndependent entres wth P rsk v(t) = ON] = q mn,k for all k 1,..., K}). By Lemma 1, we know there exsts a statonary randomzed control polcy that makes transmsson decsons (µ v 1(t),..., µ v K (t)) as a (potentally random) functon of S v (t), such that: E µ v k(t)} = λ sum,k + ɛ for all k 1,..., K} () Now, for each group G k (k 1,..., K}), we defne an ndex (k) G k as follows: If Q sum,k (t) = 0, then choose any queue G k and label ths choce (k). If Q sum,k (t) > 0, choose any queue G k such that Q (t) > 0, and defne ths queue as (k). For each k 1,..., K}, let H k be an ndependent Bernoull varable wth P rh k = 1] = q mn,k /q (k). Note that ths s a vald probablty because q mn,k q (k). Now defne vrtual channel states S v (t) = (S1 v (t),..., SK v (t)) as follows: Sk(t) v ON f S = (k)(t) = ON and H k = 1 0 otherwse It follows that the vrtual channels S v (t) are ndependent Bernoull channels wth P rsk v(t) = ON] = q mn,k for all k 1,..., K} (regardless of Q(t)), whch s exactly the rght dstrbuton to correspond wth the vrtual system for the capacty regon Λ K. Furthermore, Sk v (t) = ON} mples that S (k)(t) = ON}. Now defne a vrtual transmsson rate vector µ v (t) = (µ v 1(t),..., µ v K (t)) accordng to the statonary randomzed control polcy that chooses µ v (t) based only on S v (t), and yelds (). It follows that the vrtual transmsson rates µ v k (t) yeld () regardless of Q(t). Further, ths vrtual rate vector s feasble for the vrtual system, and so t has at most one non-zero entry, and for each entry k 1,..., K} t satsfes µ v k (t) = 0 f Sv k (t) = OF F. Now choose actual transmsson rates µ k (t) = µv k (t) f Q sum,k (t) > 0, and µ k (t) = 0 f Q sum,k(t) = 0. It follows that the (µ 1(t),..., µ K (t)) vector satsfes the constrants (18)-(0). Indeed, t nherts the constrants (18)-(19) from the (µ v 1(t),..., µ v K (t)) vector. Constrant (0) s satsfed because f µ k (t) = 1, then Q sum,k(t) > 0 and Sk v (t) = ON (so that S (k)(t) = ON), mplyng that there s at least one non-empty connected queue n group G k. Furthermore, for any k 1,..., K} such that Q sum,k (t) > 0, we have: E µ k(t) Q(t)} = E µ v k(t) Q(t)} = E µ v k(t)} (3) = λ sum,k + ɛ (4) where (3) follows because the dstrbuton of the vrtual transmsson vector µ v (t) does not depend on the queue state Q(t), and (4) follows from (). For any k 1,..., K} such that Q sum,k (t) = 0, we clearly have E µ k (t) Q(t)} = 0. Therefore, pluggng these expressons for E µ k (t) Q(t)} nto the fnal term on the rght hand sde of (1) yelds: (Q(t)) E B(t) Q(t)} + K k=1 Q sum,k(t)λ sum,k K k=1 Q sum,k(t)(λ sum,k + ɛ) and thus: (Q(t)) E B(t) Q(t)} ɛ K k=1 Q sum,k(t) (5) The nequalty (5) s n the exact form for applcaton of the Lyapunov drft lemma (Lemma 3) wth h(t) = k Q sum,k(t), and hence: 1 t 1 K 1 t 1 E B(τ)} lm sup E Q sum,k (τ)} lm sup t t t t ɛ k=1 } Because E A sum,k (t)} = λ sum,k, E A sum,k (t) = } E A sum,k, and E µ sum,k (t)} 1 for all t, the process B(t) satsfes E B(t)} B for all t (where B s a fnte

9 PROCEEDINGS OF THE 44TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, SEPT constant). It follows that the queueng network s strongly stable. Further, t can be shown that E Q (t)/t} 0, and that: It follows that: lm sup t 1 t lm sup t t t 1 t E µ sum,k (τ)} = λ sum,k λ E B(τ)} = tot + k E A sum,k whch completes the proof of Theorem. VI. CONCLUSIONS } ] k λ sum,k We have nvestgated the fundamental delay propertes for opportunstc schedulng n a mult-user wreless system wth tme varyng channels. It was shown that a large class of schedulng algorthms that do not consder queue backlog necessarly ncur average delay that grows at least lnearly wth the number of users N. We then proved t s possble to acheve an average delay that s ndependent of N by consderng queue backlog and usng a smple queue groupng technque. The technque enables the computaton of analytcal delay bounds for large scale systems n terms of smaller systems, and may offer nsght nto other problems of networkng analyss and desgn. APPENDIX PROOF OF THEOREM 1 The proof closely follows our prevous work n 9]. Proof: (Theorem 1 part (a)) Consder a partcular queue, and assume that Q (0) = 0. Consder the system vewed n contnuous tme, where µ (t) s vewed as a contnuous tme process that s constant on unt ntervals, so that µ (t) = µ ( t ) for all real tmes t. Let X (t) represent the total number of packets that have arrved from stream up to tme t. Let Q (t) represent the fractonal packets n ths system wth the same arrvals but operatng wthout the tmeslot structure. It s not dffcult to show that: Q (t) Q (t) for all real tme t (6) } and hence E Q (t)} E Q (t) for all t. Further, the value of Q (t) s gven by: Q (t) = sup X (t) X (τ) τ 0 t t τ ] µ (v)dv Takng expectatons of both sdes wth respect to the stochastc arrval process X (descrbng X (u) for all u such that 0 u t) yelds: } E Q (t) = E X E µ X sup τ 0 t X (t) X (t τ) E X sup X (t) X (t τ) τ 0 = E X sup τ 0 t t τ t τ X (t) X (t τ) ]} µ (v)dv ]} E µ (v) X } dv t t τ ]} µ dv where the frst nequalty follows by Jensen s nequalty together wth the fact that the sup( ) operator s convex. The fnal equalty follows because (from property (7)), the expected transmsson rate does not depend on the arrval hstory and s equal to µ for all tme. However, note that the fnal expresson on the rght hand sde s equal to E X U (t)}, where U (t) s the unfnshed work n a contnuous tme queueng system wth the same nputs but wth a constant server rate µ for all tme. Therefore, we obtan the lower bound: } E Q (t)} E Q (t) E U (t)} for all t completng the proof of part (a) of Theorem 1. Proof: (Theorem 1 part (b)) Suppose the system s symmetrc so that q = q and λ = λ tot /N for all 1,..., N}, and that nputs are Posson. By part (a), we know that E Q (t)} E U (t)}, where U (t) s the unfnshed work n an M/D/1 queue wth constant servce tme 1/µ. Takng t to nfnty yelds the steady state value, and hence: Q U. The steady state unfnshed work n an M/D/1 queue wth arrval rate λ λ and constant servce tme 1/µ s equal to (µ λ, whch ) can be computed by addng λ /(µ ), the average porton of a packet remanng n the server, to the expresson for the average number of packets n the buffer of an M/D/1 queue 30]. Because λ = λ tot /N, we have: N N λ tot /N Q (µ λ tot /N) Note that N µ r N (as the sum transmsson rate cannot exceed 1 (1 q) N ). Therefore, the rght hand sde n the above nequalty s greater than or equal to the soluton to: Mnmze: Subject to: N λ tot/n (µ λ tot/n) N µ r N The above optmzaton seeks to mnmze a convex symmetrc functon of (µ 1,..., µ N ) over the smplex constrant, and s mnmzed at the symmetrc pont µ = r N /N for all 1,..., N}. Therefore: N Q = λ tot (r N /N λ tot /N) Nλ tot r N (1 ρ) where ρ=λ tot /r N. Dvdng both sdes by λ tot and usng Lttle s Theorem proves the result. Proof: (Theorem 1 part (c)) Agan from part (a), we have that E Q (t)} E U (t)}, where U (t) s the unfnshed work n a queue wth a packet arrval process of rate λ and a constant server queue of rate µ. By Lttle s Theorem, the steady state expected number of packets n the server s equal to λ /µ, and hence the expected unfnshed work n the server s equal to λ /(µ ). Ths s certanly a lower bound on the

10 PROCEEDINGS OF THE 44TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, SEPT expected total unfnshed work n the system, and hence: N Q = N λ µ N λ nf (µ ) P µ rmax] (7) µ ( N ) 1 λ (8) r max where (7) follows because we agan have µ r max, and (8) holds because the soluton to the convex optmzaton problem n the prevous lne s gven by µ = λ /(r max j λj ), whch can be proven wth a smple Lagrange multpler argument. Because there are at least γ 1 N λ values that are greater than or equal to γ λ tot /N, the rght hand sde of (8) s greater than or equal to γ1γ Nλ tot /(r max ). Dvdng by λ tot bounds the average delay and proves the result. REFERENCES 1] L. Tassulas and A. Ephremdes. Dynamc server allocaton to parallel queues wth randomly varyng connectvty. IEEE Trans. on Inform. Theory, vol. 39, pp , March ] N. Kahale and P. E. Wrght. Dynamc global packet routng n wreless networks. Proc. IEEE INFOCOM, ] M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, and P. Whtng. 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