Occupancy distributions of homogeneous queueing systems under opportunistic scheduling

Occupancy distributions of homogeneous queueing systems u nder opp ortunistic sc heduling ∗ Murat Alan yali and Maxim Dashouk Departmen t o f Electrical and Computer Engineering Boston Unive rsit y Abstract W e analyze oppo r tunistic sc hemes for transmissio n scheduling from one of n homo geneous queues whose channel states fluctuate independently . Co nsidered schemes consist of the LCQ p ol- icy , which tra nsmits fro m a long e s t connected queue in the ent ire sys tem, and its low-complexity v ariants that tr ansmit from a longest q ue ue within a randomly chosen subset o f connected q ueues. A Mar k ovian mo del is studied where mean pa c ket transmissio n time is n − 1 and pa c ket ar riv al rate is λ < 1 p er queue. T ransient and equilibrium distributions of queue o ccupancies are obtained in the limit as the sys tem size n tends to infinity . ∗ An earlier versio n of this manuscript app eared at the Information Theory and App li cations W orkshop, UCSD, 2008. 1 In tro duction W e analyze a qu eueing system th a t arises u nder opp ortunistic sc heduling of pac ket transmissions from a collecti on of queues with time-v arying service r at es. The system of interest is motiv ated b y cellular data comm u nicat ions in which a s ingle transceiv er serv es multiple mobile stations through distinct c hann els. T ransmission sc h eduling has b een w ell-studied in this con text u nder the gu id ing pr in ci ple of opp ortunism, whic h b roa dly refers to exploiting c h a nnel v ariations to maximize tran s mission capacit y in the long term. In this w e pap er consider t wo generic opp ortunistic sc heduling p olic ies and obtain asymptotically e xact d escriptions of the r esu lti ng q u eue length d istributions in sym met ric s y s te ms of statisticall y iden tical queues. Explicit analysis of qu eue lengths und e r opp ortunistic sc heduling is generally difficult due to mo del complexities and lac k of closed-form expressions. In related work T assiulas and Ep hremides [10] considered a queueing system u nder an on/off channel m odel in which eac h q u eue is indep endently either connected, and in turn it is eligible for service at a standard rate, or disconnected and it cannot b e serviced. It is sho wn that transmitting fr om a longest conn ec ted queue stabilizes queue lengths if that is at all f easible, and that it m inimize s o ccupancy of sy m metric systems in whic h queues h a v e iden tical load and c hannel statistics. Th is p olicy is coined LCQ . E x p lic it description of queue length distributions under LCQ is not a v ailable, but sev eral b ounds for mean pac k et d el a y are obtained in [3, 6] for LCQ and some of its v arian ts. In more general mo d els th a t admit m ultiple transmiss io n rates and simultaneous transm issio ns, max-weight sc hed uling p olicie s and their v ariations are shown in [9, 7] to asymptotically minimize a r ange of o ccupancy measur es along a certain heavy- traffic limit. In the sp ecial case when one queue can transm it at a time, m a x-w eight trans m it s fr om a qu eue that maximizes the pro duct of instant aneous queue le ngth and transmission rate . T ails of queue length distributions un d er su ch p olicies are studied in [8, 12] via large deviations analysis. Here w e consider a system of n queues u nder a n on/off c h an n el mod el in whic h eac h queue is connected indep enden tly with probabilit y q ∈ (0 , 1]. A con tin u o us-time Marko vian mo del is adopted where pac ke t transmission rate is n and p a c ket arriv al rate is λ < 1 p er queue. It can b e seen that λ is also the load factor of the system; hence the condition λ < 1 is n e cessary to hav e p ositiv e- recurrent queue lengths. W e analyze this system for large v alues of the system size n , under the LCQ scheduling p olicy and un der its lo w-complexit y v ariant, n ame ly LC Q ( d ), that transmits from 2 a longest qu eue within d ≥ 1 r andomly selecte d conn ec ted queues. It is app aren t th a t LCQ( d ) is not particularly suitable for non-sym metric systems, y et our goal here is to obtain a generic ev aluation of its underlying principle, which may b e tailored to sp ecific circumstances. W e establish that as the system size n increases equilibriu m distribution of qu e ue o ccupancies under the LCQ p olicy con v erges to the d ete rministic distribution cen tered at 0. Hence asymp to tically almost all queues are empt y in equilibrium. The n um b er of queues with one p ac ket is Θ(1) and the n um b er of queues w ith more than one pac k et i s o (1) as n → ∞ . In particular maximum queue s ize tends to one. The total num b er of p ack ets in the system is therefore giv en b y th e num b er of non emp t y queues, and this n um b er is sho wn to ha v e the same equilibrium distrib ution as the p ositiv e-recurren t birth-death pro cess with birth rate λ and death rate 1 − (1 − q ) j at state j ∈ Z + . Note that th e latter rate is equal to th e probabilit y of ha v in g at least one conn ected queue within a give n set of j queues, and that the n a ture of the total system o cc upancy may b e an ticipated once maximum queue size is determined to b e on e. The obtained description leads to a symptotic mean pac ket dela y v ia Little’s la w as the rate of pac ket arriv als to the system is readily seen to b e nλ . The analysis tec hnique app lie d to LCQ can b e extended, although with excessiv e tediousness, to symmetric max-weig h t p olicies in cases w hen eac h qu e ue can b e serviced indep endent ly at rate nR for some random v ariable R . Ab o v e conclusions ab out LCQ offer su bstan tial insigh t ab out queue o cc upancies in that more general setting. Namely if R exceeds λ with p ositiv e probab ility (note that this c ondition is n ec essary for p ositiv e recurr ence of queue lengths), then stochastic coup lin g with a related LCQ system yields that the maxim um qu eue length in equilibrium tend s to 1 as n → ∞ . In turn, equilibrium distribution of total system o ccupancy should b e exp ected to r esem ble that of a birth-death pro cess with birth rate λ and death r at e E [max { R 1 , R 2 , · · · , R j } ] at state j , wh ere R 1 , R 2 , · · · , R j are indep endent copies of R . W e obtain the equilibr ium distribution { p k } ∞ k =0 of individual qu eue o ccupancy u nder the L CQ( d ) p olicy in the limit as n → ∞ . Sp ecifically p k = v ∗ k − v ∗ k +1 where v ∗ 0 = 1 and v ∗ k = 1 − d q 1 − λv ∗ k − 1 , k = 1 , 2 , · · · . This distribution is shown to ha ve tails that deca y as Θ(( λ/d ) k ) as queue size k → ∞ . Hence, in terms of tail o ccupancy probabilities, the choic e parameter d has the equiv alen t effect of reducing the system load by th e same factor. Y et, for an y fixed d , system o ccupancy und er LCQ( d ) is larger 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 λ n × Mean delay q = 0.1 q = 0.5 q = 0.9 (a) LCQ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 λ Mean delay d = 1 d = 2 d = 4 d = 8 (b) LCQ( d ) Figure 1: Mean pac k et dela ys as n → ∞ . Note the normalization in (a). than th a t of LCQ by a factor of order n . Num er ical v alues of asymptotic mean pac k et dela y un d er LCQ and LCQ( d ) are illustrated in Figure 1. W e also conclude that if d is allo wed to dep end on n so that d → ∞ but d/n → 0 then order of the alluded disparit y redu c es to n/d . Th is su g gests that for m oderate v alues of d and n LCQ( d ) and LCQ ma y b e exp ected to hav e comparable pac k et dela y . The present analysis is based on approximati ng system dynamics v ia asymptotically exact differ- en tial representa tions that amount to functional la w s of large num b ers. Hence b esides the men tioned equilibrium prop erties the pap er also d esc rib es transien t b eha vior of queue o ccupancies. The present analysis of LCQ ( d ) is inspired b y the work o f Vv edensk a ya et al. [11] whic h concerns an analogue of this p olicy that arises in routing and load balancing. It should p erhaps b e noted that the choic e parameter d app ears to ha v e a s ubstan tially m ore pronounced effect in the routing cont ext. O ur conclusions ab out th e LCQ p olicy require a more elab orate tec hnical approac h . Here we apply a tec hniqu e d ue to Kurtz [5] to obtain a su ita ble asymptotic d escriptio n of the sys tem. Relat ed applications of this tec hniqu e can b e found in [1, 4, 13]. The rest of this p aper is organized as follo ws. W e con tinue in Section 2 with formal description of the mo del and the notation adopted in the pap er. The p olicies LC Q( d ) and LCQ are analyzed resp ectiv ely in S ections 3 and 4. T h e pap er concludes with final remarks in Section 5. 4 λ Q1 Channel 1 λ Q2 Channel 2 λ Qn Channel n Scheduler . . . . . . Figure 2: Sk etc h of the considered queu eing system. At most one queu e is serviced a t a time, and eac h queue i can b e serviced only wh en c hannel i is eligible for transmission. 2 Queueing Mo del Consider n queues eac h serving a dedicated stream of pac ket arriv als as illustrated in Figure 2. Arriv als of eac h stream o ccur acco rding to an indep enden t P oisson pro cess with r a te λ < 1 pac ke ts p er un it time and transmission time of eac h pac ket is exp onenti ally d istributed with mean n − 1 , c hosen indep endent ly of the p rior history of th e system. Eac h queue is serviced by a d esig nated c hann el but at most one c hannel can transm it at a ti me. Channel states fluctuate randomly and eac h c hannel is eligible for transmission with prob ab ility q ∈ (0 , 1] indep endent ly of other c hann el s. Queues with eligible c hann els are called c onne cte d . W e assum e that channel states remain constant during pac k et transmission and that they are determined anew, indep endently of eac h other and of the current queue lengths, jus t b efore th e next transmission decision. Let m k ( t ) denote the num b er of queu e s with k or more pac kets at time t , and let u k ( t ) = m k ( t ) n , k = 0 , 1 , 2 , · · · . b e the fraction of su c h queues in the system. In particular 1 = u 0 ( t ) ≥ u 1 ( t ) ≥ · · · ≥ 0 , (1) the sequence { 1 − u k ( t ) } ∞ k =0 is the empirical cumulativ e d istribution fu nctio n of qu eu e o ccupancies, and P k ≥ 1 u k ( t ) is the e mpirical a ve rage queue occupancy in the system at time t . W e denote b y 5 P n and E n resp ectiv ely p robabilitie s and exp ectations asso ciated with system s ize n . In partic ular if q i ( t ) denotes the o cc upancy of the i -th qu e ue at time t then b y the sy m metry of the mo del E n [ u k ( t )] = P n ( q i ( t ) ≥ k ) . Let U denote the collection of sequen ce s u = { u k } ∞ k =0 that satisfy relation (1 ), and en d o w U w it h metric ρ that is defined b y ρ ( u , u ′ ) = sup k > 0 | u k − u ′ k | k , u , u ′ ∈ U. Note that conv ergence in U is equiv alen t to coord inate -wise conv ergence, and that U is compact as eac h co o rdinate lies in a compact in terv al. F or eac h time t let u ( t ) denote the sequence { u k ( t ) } ∞ k =0 . W e represent the tra jectory ( u ( t ) : t ≥ 0) b y the sym b ol u ( · ), an d sa y that u ( · ) conv erges to a giv en tra j ec tory v ( · ) uniformly on compact time- sets (uo c) if for all t > 0 sup 0 ≤ s ≤ t ρ ( u ( s ) , v ( s )) → 0 a.s. as n → ∞ . 3 LCQ( d ) W e f ocus on LCQ ( d ) w hic h randomly and indep endently selects d connected qu eues and transmits from a longest q u eue within this collection. F or conv enience of analysis we assum e that rep etitions are allo w ed in the selection pr ocedure, and that if all selected qu eues are empty or n o connected queu e exists at a scheduling instan t then the scheduler mak es a new selection after id ling for the transm iss io n time of a hypothetical pac ket. T h is latter assumption can b e seen to imp ly that sc h eduling instances form a P oisson pro cess of rate n . F or k = 1 , 2 , 3 , · · · let e k = { e k ( i ) } ∞ i =0 where e k ( i ) = 1 { i = k } . Here and in the rest of the pap er 1 {·} denote s 1 if its argumen t is true and 0 otherwise. Jumps of the p r ocess u ( · ) are of th e form ± n − 1 e k for some k . Namely u ( · ) c hanges b y + n − 1 e k whenev er some queue with exactly k − 1 p a c kets has a n e w arriv al, and by − n − 1 e k whenev er a p a c ket transmission is sc h e duled f r om a queue w ith exactly k pac k ets. The num b er of qu eues with k − 1 pac kets at time t is 6 giv en by n ( u k − 1 ( t ) − u k ( t )); hence the former ev en t o ccurs at instan taneous rate nλ ( u k − 1 ( t ) − u k ( t )). The latter e v ent o ccurs if and o nly if, up on completion of a pac k et transmission, ( i ) there exists a connected qu eue and ( ii ) the sc h eduler insp ects at least one connected qu eu e with k pac k ets but none with more than k pac kets. T o d ete rmine the instanta neous rate of this eve n t let τ b e a sc heduling instan t and let α n = 1 − (1 − q ) n . Namely α n is the probabilit y that there exists a conn e cted queue at time τ . Since c hannel states are assigned indep enden tly of qu eu e lengths a t time τ , a connected queue at this time has s trict ly less than k pac k ets with probabilit y 1 − u k ( τ ). Therefore, given that a connected qu eue exists, the maxim um queue length insp ected by the scheduler is equal to k with (conditional) probabilit y (1 − u k +1 ( τ )) d − (1 − u k ( τ )) d . Since sc heduling instan ts o ccur at constant rate n , in sta n taneous rate of transmissions f rom a queue of size k is nα n ((1 − u k +1 ( t )) d − (1 − u k ( t )) d ). In particular u ( · ) is a time-homogeneous Mark ov pro cess wh ose generator can b e sketc hed as u ←    u + n − 1 e k at rate nλ ( u k − 1 − u k ) u − n − 1 e k at rate nα n ((1 − u k +1 ) d − (1 − u k ) d ) , k = 1 , 2 · · · . (2) It offers some con venience in the subsequent discussion to represent the pro cess u ( · ) via the “random time c hange” construction of [2, Chapter 6]. Namely u ( t ) = u (0) + ∞ X k =1 n − 1 e k A k − 1  nλ Z t 0 u k − 1 ( s ) − u k ( s ) ds  − ∞ X k =1 n − 1 e k D k  nα n Z t 0 (1 − u k +1 ( s )) d − (1 − u k ( s )) d ds  (3) where A k − 1 ( · ) , D k ( · ), k = 1 , 2 , · · · , are mutually indep endent P oisson pro cesses eac h with un it rate. In informal terms, the pr ocesses A k ( · ) and D k ( · ) clo c k respective ly arriv als to and departures from some queue with length k , and the construction (3) is b ased on suitably exp editing these pro cesses to matc h th e instan taneous transition rates giv en in (2). Martinga le deco mp osition of the Poisson pro cesses us ed in (3) yields u ( t ) = u (0) + ∞ X k =1 e k Z t 0  λ ( u k − 1 ( s ) − u k ( s )) ds − α n ((1 − u k +1 ( s )) d − (1 − u k ( s )) d )  ds + ε ( t ) , (4) where ε ( t ) = { ε k ( t ) } ∞ k =0 is suc h that eac h co ordinate pro cess ε k ( · ) is a r ea l-v alued martingale adapted to the filtration generated b y u ( · ). 7 Theorem 3.1 E very subse quenc e of { n } has a fu rt her su b se quenc e along which u ( · ) c onver ges in distribution to a differ entiable pr o c ess v ( · ) such that v 0 ( t ) ≡ 1 and d dt v k ( t ) = λ ( v k − 1 ( t ) − v k ( t )) − (1 − v k +1 ( t )) d + (1 − v k ( t )) d k = 1 , 2 · · · . (5) Pro of. The sequence of p rocesses u ( · ) : n = 1 , 2 , · · · is tigh t in the Sk orokho d sp a ce D U [0 , ∞ ) of r ight conti n uous functions with left limits in U [2, Ch a pter 3.5]. Therefore ev ery subsequence has a fu rther sub s e quence that con verges in d istribution. By Sk orokho d’s Emb edding T h eo rem [2, Theorem 3.1.8] the pr o cesses can b e reconstructed in an a ppropriate probability sp a ce if necessary so that the conv ergence occurs almost surely . Since jum ps of u ( · ) h a v e mag nitudes that scale with n − 1 , the limit p rocess is contin uous a nd con v ergence of u ( · ) can b e taken uo c [2 , Theorem 3.10.1]. T o describ e a limit p rocess v ( · ) n ote that u 0 ( t ) ≡ 1, and so ε 0 ( t ) ≡ 0. F or k = 1 , 2 , · · · the martingale ε k ( · ) is squ are in tegrable. This pro cess has O ( n ) jumps p er unit time an d eac h jump is of size n − 1 , hence its qu a dratic v ariation v anish es as n → ∞ . In turn, Do ob’s L 2 inequalit y [2, Pr oposition 2.2.16 ] implies that ε k ( · ) → 0 uo c as n → ∞ . Since α n → 1 and con vergence of u ( · ) is uo c, th e k th in tegral in equalit y (4) con verges to Z t 0  λ ( v k − 1 ( s ) − v k ( s )) ds − ((1 − v k +1 ( s )) d − (1 − v k ( s )) d )  ds. Therefore v ( · ) satisfies equalit y (4) with α n = 1 and ε ( t ) ≡ 0 . Differenti al representa tion of that equalit y is (5).  Let U o denote th e set of system states in w hic h a verag e queue occup ancy is fin it e. That is, U o = { u ∈ U : ∞ X k =1 u k < ∞} . Let v ∗ = { v ∗ k } ∞ k =0 ∈ U b e defined b y setting v ∗ 0 = 1 and v ∗ k = 1 − d q 1 − λv ∗ k − 1 , k = 1 , 2 , · · · . (6) Since 1 − d p 1 − λv ∗ k − 1 ≤ λv ∗ k − 1 it follo ws th a t v ∗ k ≤ λ k ; in particular v ∗ ∈ U o . It can b e readily v erifi ed b y substitution that v ∗ is an equilibr ium p oint for the differentia l system (5). The follo win g lemma establishes that v ∗ is the unique stable equilibrium for tra jectories that start in U o . 8 Lemma 3 .1 L et v ( · ) solve the differ ential system (5) with initial state v (0) ∈ U o . Then lim t →∞ v k ( t ) = v ∗ k , k = 1 , 2 , · · · . (7) W e pro vide a pro of based on the follo wing auxiliary result: Lemma 3 .2 L et v + ( · ) and v − ( · ) solve the differ e ntia l system (5) with r esp e c tive i ni tial c onditions v + (0) , v − (0) ∈ U such that v + k (0) ≥ v − k (0) for al l k . Then v + k ( t ) ≥ v − k ( t ) for al l k and al l t > 0 . Pro of. S upp ose that the lemma is incorrect and let t > 0 b e the fir s t instan t suc h that v + k ( t ) = v − k ( t ) and d dt v + k ( t ) < d dt v − k ( t ) for some k . Let i b e the largest index k that satisfies th is condition at time t . Then b y (5) d dt v + i ( t ) − d dt v − i ( t ) = λ ( v + i − 1 ( t ) − v − i − 1 ( t )) + (1 − v − i +1 ( t )) d − (1 − v + i +1 ( t )) d . The righ t hand side of this equalit y is nonn egativ e due to the c h oi ce of t (since otherwise either the condition v + i − 1 ( t ) ≥ v − i − 1 ( t ) and or the co ndition v + i +1 ( t ) ≥ v − i +1 ( t ) m ust b e v iolated b efore time t ). This con tr ad icts with the definition of t ; therefore no suc h t exists and the lemma holds.  Pro of of Lemma 3.1 Le t v + ( · ) and v − ( · ) b e solutions to (5) with resp ectiv e initial states v + (0) and v − (0) that are defined b y setting v + k (0) = max { v k (0) , v ∗ k } and v − k (0) = min { v k (0) , v ∗ k } for k = 0 , 1 , 2 , · · · . By Lemma 3.2 v − k ( t ) ≤ v k ( t ) , v ∗ k ≤ v + k ( t ) , for all k , t. (8) Equalit y (5) and definition (6) of v ∗ giv e d dt ∞ X i = k v ± i ( t ) = λv ± k − 1 ( t ) + (1 − v ± k ( t )) d − 1 = λ ( v ± k − 1 ( t ) − v ∗ k − 1 ) + (1 − v ± k ( t )) d − (1 − v ∗ k ) d , or, in in tegral form, ∞ X i = k v ± i ( t ) − ∞ X i = k v ± i (0) = Z t 0 λ ( v ± k − 1 ( s ) − v ∗ k − 1 ) ds + Z t 0 ((1 − v ± k ( s )) d − (1 − v ∗ k ) d ) ds. (9) 9 Note that since v + 1 ( t ) ≥ v ∗ 1 = 1 − d √ 1 − λ it follo w s that d dt ∞ X i =1 v + k ( t ) = λ + (1 − v + 1 ( t )) d − 1 ≤ 0 . Hence P ∞ i = k v + i ( t ), and therefore P ∞ i = k v − i ( t ), is b ounded by P ∞ k =1 v + k (0) un iformly for all t . In tu rn equalit y (9) yields     Z t 0 λ ( v ± k − 1 ( s ) − v ∗ k − 1 ) ds + Z t 0 ((1 − v ± k ( s )) d − (1 − v ∗ k ) d ) ds     ≤ ∞ X i = k v + i (0) . The b ound on the right hand side is fi nite since v + (0) ∈ U o due to the hyp othesis v (0) ∈ U o . Note that, owing to the in equ a lit y (8), neither one of the t w o inte grands ab o v e c han ges sign. Hence if the first in tegral con verges as t → ∞ then so do es the second one, implyin g fur ther that lim t →∞ v ± k ( t ) = v ∗ k . (10) Since v ± 0 ( t ) ≡ v ∗ 0 = 1, th is is clearly the case f o r k = 1. Ind uctio n on k confirms that equalit y ( 10) holds for all k . The desired conclusion (7) no w follo ws from the prop ert y (8).  Theorem 3 .1, whic h establishes conv ergence o v er finite time in terv als, is c omplemen ted next b y sho w ing that equilibrium d istribution of u ( · ) con v erges as n → ∞ to the deterministic measure concen trated at v ∗ . Theorem 3.2 The pr o c ess u ( · ) i s er go dic. L et u ∗ = { u ∗ k } ∞ k =0 denote the e q u ilibrium r andom variable. F or ε > 0 lim n →∞ P n ( ρ ( u ∗ , v ∗ ) > ε ) = 0 . (11) In p articular lim n →∞ E n [ u ∗ k ] = v ∗ k for k = 0 , 1 , 2 , · · · . Pro of. Let U n = { u ∈ U : nu k ∈ Z + for k ≥ 0 } . Note that ( u ( t ) : t ≥ 0) is irr educible in U n and U n is compact; therefore the p r ocess is ergo dic and has a unique equilibrium distribution concen trated on U n . In that e quilibrium the rate of arriv als to queues with occupancy k o r higher should b e equal to the r a te of departures from suc h queues. That is, E n [ λu ∗ k − 1 ] = 1 − E n [(1 − u ∗ k ) d ] ≥ E n [ u ∗ k ] , f or k ≥ 1 , where the inequalit y follo w s s ince (1 − u ∗ k ) d ≤ (1 − u ∗ k ). Therefore E n [ u ∗ k ] ≤ λ k and in turn E n [ P ∞ k =1 u ∗ k ] ≤ λ/ (1 − λ ). Let U o,λ , { u ∈ U : P ∞ k =1 u k ≤ λ/ (1 − λ ) } so that P n ( u ∗ ∈ U o,λ ) = 1. 10 Supp ose th a t (11) is false so that for some in finite subsequence { n ′ } of { n } and some δ > 0 P n ′ ( ρ ( u ∗ , v ∗ ) > ε ) > δ . (12) Due to Lemma 3.1 and compactness of U o,λ one can c h o ose t ( ε ) su c h that if v (0) ∈ U o,λ then ρ ( v ( t ) , v ∗ ) < ε/ 2 for t ≥ t ( ε ). Let u (0) h a v e the same d istribution as u ∗ and let v (0) = u (0). By Theorem 3.1 there is a f urther subsequence { n ′′ } of { n ′ } such that P n ′′ ( ρ ( u ( t ( ε )) , v ( t ( ε ))) > ε/ 2) < δ wh enev er n ′′ is large enough. Sin ce P n ′′ ( u ∗ ∈ U o,λ ) = 1, th e choice of t ( ε ) implies that P n ′′ ( ρ ( u ( t ( ε )) , v ∗ ) > ε ) < δ f or those v alues of n ′′ . Ho we v er u ( t ( ε )) and u ∗ ha ve identica l distributions as the latter is in equilibrium; leading to a con tradiction with (12). Hence n o s e quence { n ′ } and constan t δ > 0 satisfy (12); so (11 ) holds. By definition of ρ equ a lit y (11) imp lie s that eac h en try u ∗ k of u ∗ con verge s in probabilit y to the constan t v ∗ k ; since 0 ≤ u ∗ k ≤ 1, so d oes E n [ u ∗ k ].  W e conclud e the discussion of LCQ ( d ) with a r ela tionship b et we en d and the tail p robabiliti es of equilibrium queue o ccupancy: Theorem 3.3 v ∗ k = Θ(( λ/d ) k ) as k → ∞ . Pro of. The assertion is immediate for d = 1 so w e consider the case d > 1. Equalit y (6), together with T a ylor expansion of d √ 1 − x around x = 0 yields v ∗ k = λ d v ∗ k − 1 + 1 d ∞ X i =2 ( λv ∗ k − 1 ) i i ! i − 1 Y j =1 ( j − 1 d ) , k = 1 , 2 , · · · . (13) The second term on the righ t hand side is nonnegativ e; therefore v ∗ k ≥ ( λ/d ) v ∗ k and lim in f k →∞ v ∗ k ( λ/d ) k ≥ 1 . (14) W e define c k , v ∗ k / ( λ/d ) k and co mplete the proof b y sho w in g that { c k } ∞ k =0 is u niformly b ounded . Let β k b e d efi ned as β k , 1 + ∞ X i =1 ( λv ∗ k − 1 ) i 2 ( i + 2)! i +1 Y j =2 ( j − 1 d ) < 1 + ∞ X i =1 ( λv ∗ k − 1 ) i (15) so that equalit y (13) can b e rearranged as v ∗ k = λ d v ∗ k − 1 + ( λv ∗ k − 1 ) 2 2 d (1 − 1 d ) β k . (16) 11 Since v ∗ k ≤ λ k there exists a finite k o suc h that v ∗ k − 1 < 1 /d for k ≥ k o . The b ound in (15) implies that β k < 1 / (1 − λ/d ) < d/ ( d − 1) for suc h k ; in tur n b y (16) c k < c k − 1 + c 2 k − 1 d 2 ( λ/d ) k , k > k o . It can b e verified by induction on k > k o that c k < c k o (1 + λ + λ 2 + · · · + λ k − k o ) : (17) Namely , if (17) holds for k then it holds also for k + 1 if c k o (1 + λ + λ 2 + · · · + λ k − k o ) 2 λ k o / (2 d k ) < 1. This latter condition can b e v erified based on the b ound c k o ( λ/d ) k o = v k o < 1 /d , which follo ws fr om the definition of k o . Inequalit y (17) implies the un ifo rm b ound c k < c k o / (1 − λ ); therefore lim su p k →∞ v ∗ k ( λ/d ) k < c k o 1 − λ . (18) The theorem follo ws due to (14) and (18).  4 LCQ Giv en m ( τ ) at a sc heduling instan t τ , the maxim um o ccupancy o v er all connected queues at time τ is equal to k = 1 , 2 , · · · with probabilit y (1 − q ) m k +1 ( τ ) − (1 − q ) m k ( τ ) ; hence under th e LC Q p olicy u ( · ) is a time-homogenous Mark ov pro cess with ju mp rates u ←    u + n − 1 e k at rate nλ ( u k − 1 − u k ) u − n − 1 e k at rate n ((1 − q ) nu k +1 − (1 − q ) nu k ) . (19) This pro cess can b e constructed as in Section 3, so that u k ( t ) = u k (0) + Z t 0  λ ( u k − 1 ( s ) − u k ( s )) − (1 − q ) m k +1 ( s ) + (1 − q ) m k ( s )  ds + ε k ( t ) (20) where ε k ( · ) is a martingal e that v anish es as n → ∞ . The sequen c e of processes u ( · ) : n = 1 , 2 , · · · con verge s in distribu tio n along sub sequences of { n } , but identifying a limit is relativ ely m o re inv olv ed than for the LCQ( d ) p olicy since the pro cess m ( · ) fl uctuate s p ersisten tly for all v alues of n and the in tegrand in (20) do es n o t conv erge. Rather than this in tegrand, here we study the b eha vior of the in tegral in (20) via an a v eraging tec hn ique due to Kur tz [5]. In reading this section the reader ma y find it helpful to consult related applications of this tec h nique in [1, 4, 13]. 12 Let Ω d en o te the set of sequences ω = { ω k } ∞ k =0 suc h that ω k ∈ Z + ∪ {∞} , ω 0 = ∞ , and ω k ≥ ω k +1 . Define the mapping h : Ω 7→ [0 , 1] ∞ b y setting h ( ω ) = { (1 + ω k ) − 1 } ∞ k =0 , ω ∈ Ω , with the un d erstanding that 1 + ∞ = ∞ and 1 / ∞ = 0. Let Ω b e endow ed with metric ρ o defined by ρ o ( ω , ω ′ ) = ρ ( h ( ω ) , h ( ω ′ )) , ω , ω ′ ∈ Ω . In particular Ω is compact with resp ect to the induced top olog y . W e den o te by L the collection of measures µ on the pro duct space [0 , ∞ ) × Ω suc h that µ ([0 , t ) × Ω) = t for e ac h t > 0. Let L b e endo wed with the top ology corresp onding to weak conv ergence of measures restricted to [0 , t ) × Ω for eac h t . Since Ω is compact, so is L due to Prohoro v’s Theorem. Let ξ b e a random mem b er of L defined by ξ ([0 , t ) × A ) = Z t 0 1 { m ( s ) ∈ A } ds, t > 0 , A ∈ B (Ω) . Here B (Ω) d e notes Borel sets of Ω. Note that equalit y (20) can b e expressed in terms of ξ as u k ( t ) = u k (0) + Z t 0 λ ( u k − 1 ( s ) − u k ( s )) ds − φ k +1 ( t ) + φ k ( t ) + ε k ( t ) where φ k ( t ) , Z t 0 (1 − q ) m k ( s ) ds = Z [0 ,t ) × Ω (1 − q ) ω k ξ ( ds × d ω ) . Compactness of L implies th at eac h su bsequence of { n } has a further subsequence along which ξ con verge s in d istribution. This pr o p ert y is also p ossessed by u ( · ), and therefore by the pair ( u ( · ) , ξ ). The f ollo wing d efinition is useful in c haracterizing p ossible limits of ( u ( · ) , ξ ): F or fixed u ∈ U let ω u ( · ) denote the Mark ov pro cess w it h states in Ω and with the follo wing transition rates: ω u ←    ω u + e k at rate λ ( u k − 1 − u k ) ω u − e k at rate (1 − q ) ω u k +1 − (1 − q ) ω u k . (21) See Figure 3 for a partial illustration of this pr ocess. The pro cess ω u ( · ) b ears a certain resem b lance to m ( · ) = n u ( · ) , which can b e obser ved by insp ecting th e generators (19) and (21), though it should b e noted that in (21) u = { u k } ∞ k =0 is a constant and has n o binding to instanta neous v alues of ω u ( · ). W e also p oin t out th a t ω u ( · ) ev olve s on a compactified state s p ac e and it is redu cible d ue to the states that inv olv e ∞ ; hence it has m u lt iple equilibrium distrib utions in general. 13 0 1 2 j λ ( u k-1 -u k ) λ ( u k-1 -u k ) λ ( u k-1 -u k ) 1 ( 1 ) ( 1 ) k q q ω + − − − u 1 2 ( 1 ) ( 1 ) k q q ω + − − − u 1 ( 1 ) ( 1 ) k j q q ω + − − − u Figure 3: T rans it ion rates of ω u k ( · ), th a t is, the k th co ord inate of ω u ( · ). The pro cess has a lso an isolated state ∞ w hic h is not shown. Th e coord inate pr ocess ω u k ( · ) is generally n o t Mark o vian d ue to its dep endence on ω u k +1 ( · ). Theorem 4.1 L et ( v ( · ) , χ ) b e the limit of ( u ( · ) , ξ ) a long a c onver gent subse qu enc e of { n } . a) The limit me asur e χ satisfies χ ([0 , t ) × A ) = Z t 0 π v ( s ) ( A ) ds, t > 0 , A ∈ B (Ω) , wher e, for e ach s > 0 , π v ( s ) is an e q u ilibrium distribution for the pr o c ess ω v ( s ) ( · ) such that π v ( s ) ( ω k = ∞ ) = 1 if v k ( s ) > 0 . b) The limit tr aje ctory v ( · ) sa tisfies d dt v k ( t ) = λ ( v k − 1 ( t ) − v k ( t )) − E π v ( t ) [(1 − q ) ω k +1 − (1 − q ) ω k ] , (22) wher e k = 1 , 2 , · · · and E π v ( t ) denotes exp e ctation with r esp e ct to distribution π v ( t ) . Pro of. Let all pro cesses b e constru c ted on a common prob ab ility space so that con verge nce of ( u ( · ) , ξ ) is almost sure. Conv ergence of u ( · ) is then uo c. W e start by consulting [5, Lemma 1.4] to v erify that the limit measure χ p ossesses a d ensit y so that χ ([0 , t ) × A ) = Z t 0 γ s ( A ) ds, t > 0 , A ∈ B (Ω) , (23) where, for eac h s , γ s is a probabilit y distribu ti on on Ω . W e p roceed by iden tifying these distributions. Let F denote the collecti on of b ounded contin uous functions f : Ω 7→ R such th a t f ( ω ) dep ends on a finite n umb er of en tries in the sequence ω = { ω k } ∞ k =0 ∈ Ω. Giv en f ∈ F define the function 14 G f : Ω × U 7→ R b y setting G f ( ω , u ) , ∞ X k =1 ( f ( ω + e k ) − f ( ω )) λ ( u k − 1 − u k ) + ( f ( ω − e k ) − f ( ω ))( (1 − q ) ω k +1 − (1 − q ) ω k ) for eac h ω ∈ Ω and u = { u k } ∞ k =0 ∈ U . G f is con tinuous due to the conti n uit y of f , and cont in uit y of G f is uniform since the pro duct sp ac e Ω × U is compact. The pro cess f ( m ( · )) satisfies at eac h instant t f ( m ( t )) − f ( m (0 )) = Z t 0 ∞ X k =1 ( f ( m ( s ) + e k ) − f ( m ( s ))) dA k − 1  Z s 0 nλ ( u k − 1 ( τ ) − u k ( τ )) dτ  + Z t 0 ∞ X k =1 ( f ( m ( s ) − e k ) − f ( m ( s ))) d D k  Z s 0 n  (1 − q ) m k +1 ( τ ) − (1 − q ) m k ( τ )  dτ  = n Z t 0 ∞ X k =1 ( f ( m ( s ) + e k ) − f ( m ( s ))) λ ( u k − 1 ( s ) − u k ( s )) ds + n Z t 0 ∞ X k =1 ( f ( m ( s ) − e k ) − f ( m ( s )))  (1 − q ) m k +1 ( s ) − (1 − q ) m k ( s )  ds + µ f ( t ) = n Z t 0 G f ( m ( s ) , u ( s )) ds + µ f ( t ) , where µ f ( · ) is a square-int egrable martingale. Rearranging the last equalit y and expressing the in tegral there in terms of the random measure ξ yields Z [0 ,t ) × Ω G f ( ω , u ( s )) ξ ( ds × d ω ) = Z t 0 G f ( m ( s ) , u ( s )) ds = ( f ( m ( t )) − f ( m (0 )) /n + µ f ( t ) /n. Since f is b ounded, the first term on the right hand side v anishes as n → ∞ . The martingale µ f ( · ) has b ounded jum ps; in turn by Do ob’s L 2 inequalit y µ f ( t ) /n also v anishes. Th erefo re Z [0 ,t ) × Ω G f ( ω , u ( s )) ξ ( ds × d ω ) → 0 . (24) Since u ( · ) con verges uo c to v ( · ) b y hypothesis, uniform con tin u ity of G f implies      Z [0 ,t ) × Ω G f ( ω , u ( s )) ξ ( ds × d ω ) − Z [0 ,t ) × Ω G f ( ω , v ( s )) ξ ( ds × d ω )      → 0 . (25) Finally b y the Con tinuous Mapping Theorem Z [0 ,t ) × Ω G f ( ω , v ( s )) ξ ( ds × d ω ) → Z [0 ,t ) × Ω G f ( ω , v ( s )) χ ( ds × d ω ) . (26) Observ ations (24)–(26) lead to Z [0 ,t ) × Ω G f ( ω , v ( s )) χ ( ds × d ω ) = Z t 0 X ω ∈ Ω G f ( ω , v ( s )) γ s ( ω ) ds = 0 , 15 where the left equalit y is due to (23). This equalit y holds for all t > 0; therefore X ω ∈ Ω G f ( ω , v ( s )) γ s ( ω ) = 0 for almost all s > 0. Since f ∈ F is arb it rary (note that F is dense in contin uous b ounded fu nctio ns on Ω) [2, Prop osition 4.9.2] imp lie s that γ s is an equilibrium distrib u tio n f or the p rocess ω v ( s ) ( · ). Let ε > 0 and [ t 0 , t 1 ] b e an inte rv al suc h that v k ( t ) ≥ ε for t ∈ [ t 0 , t 1 ]. Since v k ( t ) is the limit of u k ( t ) = n − 1 m k ( t ), for an y giv en intege r B ξ ([ t 0 , t 1 ] × { 0 , 1 , 2 , · · · , B } ) = Z t 1 t 0 1 { m k ( s ) ≤ B } ds → 0 as n → ∞ . Hence χ ([ t 0 , t 1 ] × Z + ) = 0 due to the arbitrariness of B . Sin ce ε can b e c hosen a rbitrarily sm all it follo ws that γ t ( Z + ) = 0 for al most all t suc h that v k ( t ) > 0. This c ompletes the pro of o f part a). P art b) follo ws from equalit y (20) sin c e Z t 0 ( u k − 1 ( s ) − u k ( s )) ds → Z t 0 ( v k − 1 ( s ) − v k ( s )) ds due to uo c conv ergence of u ( · ), and φ k ( t ) → Z [0 ,t ) × Ω (1 − q ) ω k χ ( ds × d ω ) = Z t 0 E γ s [(1 − q ) w k ] ds due to the Con tinuous Mapping Theorem.  Theorem 4.1 explains the exten t of the disp arit y b et we en time scales of t w o pro cesses, namely m ( · ) and its normalized v ersion u ( · ): The pro cess m ( · ) displa ys far larger v ariation than its nor- malized ve rsion, so that, in the limit of large n , m ( · ) settles to equilibrium b efore u ( · ) changes its v alue. In particular integral of a b inary-v alued measurable function of m ( · ) is well -appro ximated b y in tegrating an appr opriate equilibrium probabilit y . Pro vid ed that u ( t ) remains close to v ( t ), the pro cess ω v ( t ) ( · ) mimics a slo wed-do wn ve rsion of m ( · ) obs e rv ed around time t ; hence the alluded equilibrium distribu tio n p ertains to ω v ( t ) ( · ). Sp ecification of ω v ( t ) ( · ) requ ires in cl usion of ∞ since en tries of m ( · ) can b e as large as n . Compactifying the augmen ted state-space Ω of m ( · ) via choice of th e metric ρ o leads to the repre- sen tation (22) of a limit tr a jecto ry v ( · ), but it also entails am biguit y in that repr ese n tation. Namely , Theorem 4.1 do es not sp ecify whic h equilibriu m d istribution for ω v ( t ) ( · ) sh ould b e adopted in (22). While a f ull accoun t of equilibriu m distributions of ω v ( t ) ( · ) app ears difficult, an imp ortan t f e ature of the righ t distribution can b e id en tified: 16 Lemma 4 .1 L et v ( · ) and π v ( · ) b e as sp e cifie d b y The or em 4.1. Given k = 1 , 2 , · · · π v ( t ) ( ω k ∈ Z + and ω k +1 = 0) = 1 for almost a l l t such that v k ( t ) = 0 . Lemma 4.1 will b e instrumental in obtaining a s harp er description for v ( · ), y et an informal explanation ma y s till b e us efu l in pu tti ng it in p ersp ectiv e w ith the queueing system of interest. Note that if v k ( t ) = 0 and v k − 1 ( t ) > 0 then v ( t ) reflects a d istribution with supp ort { 0 , 1 , · · · , k − 1 } . This prop ert y do es not immediately translate in to a b ound on the maximum queue length in the system, sin ce v ( t ) is the limit of u ( t ) = n − 1 m ( t ) and s o the n um b er of queues w it h at least i ≥ k pac kets, m i ( t ), is o ( n ) as n → ∞ . By wa y of in terpreting ω v ( t ) ( · ) as a p ro xy to m ( · ) around time t , Lemma 4.1 indicates that the maxim u m queue size is at m ost one larger than what is deduced fr o m v ( t ) and that the num b er of maximal queues is O (1) as n → ∞ . Pro of of Lemma 4.1 Let [ t 0 , t 1 ] b e an interv al such that v k ( t ) = 0 for t ∈ [ t 0 , t 1 ]. W e pr o v e the lemma b y sho win g that as n → ∞ along the con vergen t subsequence of in terest ξ ([ t 0 , t 1 ] × { ω : w k +1 = 0 } ) = Z t 1 t 0 1 { m k +1 ( t ) ≥ 1 } dt → 0 , (27) ξ ([ t 0 , t 1 ] × { ω : w k ∈ Z + } ) = Z t 1 t 0 1 { m k ( t ) ∈ Z + } dt → t 1 − t 0 . (28) F or eac h in teger l and time t let s l ( t ) , P ∞ i = l m i ( t ). Th is quantit y increases when some queue with size at least l − 1 receiv es a pac k et, and it decreases when transm issio n is sc heduled fr om some queue with s iz e at least l . Giv en u ( t ), these ev en ts occur at resp ectiv e instantaneo us rates nλu l − 1 ( t ) and n  1 − (1 − q ) m l ( t )  . Therefore E n [ s l ( t 1 ) − s l ( t 0 )] = nE n  Z t 1 t 0 λu l − 1 ( t ) −  1 − (1 − q ) m l ( t )  dt  . (29) Consider this equalit y for l = k + 1. By c hoice of the in terv al [ t 0 , t 1 ] n − 1 E n [ s k +1 ( t )] → ∞ X i = k +1 v i ( t ) = 0 and u k ( t ) → v k ( t ) = 0 for all t ∈ [ t 0 , t 1 ]. Consequently E n  Z t 1 t 0  1 − (1 − q ) m k +1 ( t )  dt  → 0 . 17 This leads to (27) since 1 − (1 − q ) m k +1 ( t ) ≥ q 1 { m k +1 ( t ) ≥ 1 } . T o complete the pro of, note that n − 1 E n [ s k ( t )] → 0 for all t ∈ [ t 0 , t 1 ]; therefore (29) ev aluated at l = k implies that for an y op en subset B ⊂ [ t 0 , t 1 ] lim su p n →∞ E n  Z B  1 − (1 − q ) m k ( t )  dt  = lim su p n →∞ E n  Z B λu k − 1 ( t ) dt  < Z B dt. The last inequalit y is strict since λ < 1. Arbitrariness of B implies (28).  Giv en p ositiv e integ er K let U K = { u ∈ U : u k = 0 f o r k ≥ K } . Theorem 4.2 L et v ( · ) and π v ( · ) b e as sp e cifie d by The or e m 4.1 with initial state v (0) ∈ U K for some K . Then for t > 0 a) v ( t ) ∈ U K and π v ( t )  ω K ( t ) ∈ Z + and ω K ( t )+1 = 0  = 1 wher e K ( t ) = min { k : v j ( t ) = 0 for j ≥ k } . b) d dt v k ( t ) =    λv k − 1 ( t ) − 1 < 0 if k = K ( t ) − 1 0 if k ≥ K ( t ) . In p articular v k ( t ) = 0 for k > 0 and t > K/ (1 − λ ) . Pro of. L et t b e an instan t suc h that K ( t ) < ∞ . Lemma 4.1 implies that π v ( t )  ω K ( t ) ∈ Z + and ω K ( t )+1 = 0  = 1 . (30) In particular the coord inate pro cess ω v ( t ) K ( t ) ( · ) p ossesses an equilibrium in Z + . The pr o cess should ha ve equal rates of up-jump s and down-jumps in that equilibriu m, namely E π v ( t ) [(1 − q ) ω K ( t ) ] = 1 − λv K ( t ) − 1 ( t ) . (31) Since v K ( t ) − 1 ( t ) > 0 by definition of K ( t ), Theorem 4.1.a implies that E π v ( t ) [(1 − q ) ω K ( t ) − 1 ] = 0 . (32) 18 Substituting (31) and (32) in equalit y (22) ev aluated at k = K ( t ) − 1 yields d dt v K ( t ) − 1 ( t ) = λv K ( t ) − 2 ( t ) − 1 < 0 . (33) Note also that E π v ( t ) [(1 − q ) ω K ( t )+1 ] = 1 due to (30); hence equalit y (22) for k = K ( t ) giv es d dt v K ( t ) ( t ) = − λv K ( t ) ( t ) = 0 . (34) Since K (0) = K < ∞ by hyp ot hesis, it follo ws via (33) and (34) that K ( t ) is fin ite and nonincreasing in t . Part (a) of the theorem now follo w s b y (30). P art (b) is due to (33) and (34).  Corollary 4.1 If u (0) ∈ U K for some K then lim n →∞ P n ( m 1 ( t ) ∈ Z + , m 2 ( t ) = 0) = 1 (35) for t ≥ K / (1 − λ ) . The system o c cup ancy P ∞ k =1 m k ( t ) c onver ges in distribution to th e e quilibrium value of a birth-de ath pr o c ess with c onstant bi rth r ate λ and de ath r ate 1 − (1 − q ) j at state j . Pro of. Let { n i } b e a subsequ en c e along which ( u ( · ) , ξ ) con v erges and let ( v ( · ) , χ ) denote th e limit. Since u (0) ∈ U K it follo ws that v (0) ∈ U K . Cho ose t 1 > t 0 > K / ( 1 − λ ) so that b y T heorem 4.2.b v ( t ) = { 1 , 0 , 0 , 0 , · · · } for t ∈ [ t 0 , t 1 ]. Let A = { ω ∈ Ω : ω 1 ∈ Z + , w 2 = 0 } . Th en Z t 1 t 0 P n i ( m ( t ) ∈ A ) dt = E n i  Z t 1 t 0 1 { m ( t ) ∈ A } dt  → Z t 1 t 0 π v ( t ) ( A ) dt = t 1 − t 0 , where the last equalit y is d u e to T h eo rem 4.2.a. The ab o ve limit do es not d epend on the particular subsequence { n i } ; th erefore (35) follo ws. The final claim of the corollary is v erifi ed by obs er v in g that for t > K/ (1 − λ ) the coord inate pro ce ss ω v ( t ) 2 ≡ 0 in equilibrium ; and in turn ω v ( t ) 1 is a positive recurrent birth-death pro cess on Z + with birth rate λ and death rate 1 − (1 − q ) j at state j .  It should b e noted that the hyp ot hesis u (0) ∈ U K is necessary f o r the conclusions of Corollary 4.1: If the initial size of a single qu eue is allo we d to grow without b ound with increasing n then, for large v alues of n , that qu eue r ec eiv es service wheneve r it is connected. In effe ct th is r educes the service rate av ailable to the rest of the system by a factor of (1 − q ). In such degenerate cases th e present analysis applies to the subs yste m that is comp osed of queues with b ounded initial o ccupancies, after appropriate adjustment of the service rate. 19 5 Final remarks: LCQ( d n ) Conclusions of Sect ions 3 and 4 rev eal that the system o ccupancies u nder LCQ( d ) an d LCQ d iffer b y a factor o f order n as n → ∞ . More insigh t on this disparit y , esp ec ially for m oderate v alues of d relativ e to n , can b e gained by considering an asymp totic regime in whic h d is allo w ed to dep end on n . Here w e sk etc h asymptotic analysis of LCQ( d n ) in the case lim n →∞ d n = ∞ and lim n →∞ d n n = 0 . The present discussion closely follo w s that of Section 4, hence pro ofs are omitted. Under LCQ( d n ) the representat ion (4) can b e exp r essed as u k ( t ) = u k (0) + Z t 0  λ ( u k − 1 ( s ) − u k ( s )) −  (1 − b k +1 ( s ) d n ) d n − (1 − b k ( s ) d n ) d n  ds + ε k ( t ) where b k ( t ) , d n u k ( t ). Let b ( t ) = { b k ( t ) } ∞ k =0 and let Ω o b e obtained b y augmen ting Ω with n on in - creasing sequences that tak e v alues in R + ∪ {∞} . Define the ran d om measure ξ o b y ξ o ([0 , t ) × A ) = Z t 0 1 { b ( s ) ∈ A } ds, t > 0 , A ∈ B (Ω o ) . Consideration of the pair ( u ( · ) , ξ o ) via an analogue of T heorem 4.1 iden tifies p ossible limits v ( · ) of u ( · ) as solutions to d dt v k ( t ) = λ ( v k − 1 ( t ) − v k ( t )) − E π v ( t ) [ e − ω k +1 − e − ω k ] , k = 1 , 2 , · · · where π v ( t ) is a distribution on Ω o suc h that π v ( t ) ( ω k = ∞ ) = 1 if v k ( t ) > 0 and E π v ( t )  1 { ω k 6 = ∞}  λ ( v k − 1 ( t ) − v k ( t )) + e − ω k − e − ω k +1  = 0 . The line of reasoning emplo y ed in establishing Lemma 4.1 and Theorem 4.2 readily app lie s to v ( · ) and π v ( · ) here, yielding that π v ( t ) ( ω k ∈ R + and ω k +1 = 0) = 1 if v k ( t ) = 0 , and that v 1 ( t ) = 0 for t > K (0) / (1 − λ ). I n turn for such t , b 1 ( t ) = O (1) and b 2 ( t ) = o (1) as n → ∞ . The maximum queue size in equ ilibr ium ther efore tends to one, bu t th e n um b er of queues at t hat o cc upancy is sub stan tially larger than the s ame num b er und er the LCQ p olicy . In p articular for large enough v alues of t the total system o c cupancy P k ≥ 1 m k ( t ) = ( n/d n ) P k ≥ 1 b k ( t ) is O ( n/d n ). 20 References [1] M. Alan y ali, “Asymptotically exact analysis of a loss net work w ith c h annel conti n uit y ,” The Anna ls of A pp lie d P r ob ability , vol. 13, n o. 4, pp. 1474–1 493, 2003. [2] S . N. Eth ie r and T . G. Ku rtz, M a rkov pr o c esses: Char acterization and c onver genc e, Wiley , 1986. [3] A. Gan ti, E. Mod iano, and J. N. Tsitsiklis, “Optimal transmiss ion scheduling in symmetric comm un ica tion mo dels with intermitten t connectivit y ,” IEEE T r ansactions on Information The ory, v ol. 53, no. 3, p p. 998–1008 , 2007. [4] P . J. Hun t and T. G. Kur tz , “Large loss net works,” Sto c h astic Pr o c esses and Their Applic ations , v ol. 53, pp. 363–3 78, 1994. [5] T. G. Ku rtz, “Av er aging for martingale problems and sto c hastic approximat ion,” L e ctur e Notes on Contr ol and Information Scienc es, vo l. 117, pp . 186–209, S pringer-V erlag, 1992. [6] M. J. Ne ely , “Or d er optimal dela y for o pp ortunistic s c heduling in multi-user wireless uplinks and do w nlinks,” 44th Al lerton Confer enc e on Communic ation, Contr ol, and Computing , 2006. [7] S . Shakko ttai, R. Srik ant, and A. L. Stoly ar, “Path wise op timalit y of the exp onentia l sc h eduling rule for wireless channels,” A dvanc es in Applie d Pr ob ability, v ol. 36, no. 4, pp. 1021-1045, 2004. [8] S . Shakko ttai, “Effectiv e capacit y and QoS for wireless scheduling,” IE EE T r ansactions on Autom atic Contr ol, vol. 53, no. 3, pp. 749-761, 2008 . [9] A. L. Stolya r, “Maxwe igh t s c heduling in a generalized switc h: State space collapse and workload minimization in hea vy traffic,” The Annal s of Applie d Pr ob ability, vo l. 14, no. 1, pp. 1-53, 2004. [10] L. T assiulas and A. Ephremides, “Dynamic serve r allo catio n to parallel queues with randomly v arying connectivit y ,” in IEEE T r ansactions on Information The ory , v ol. 39, n o. 2, 1993. [11] N. D. Vv edensk a ya, R. L. Dobrushin , and F. I. Karp elevic h, “Queueing system with selection of the shortest of t wo queues: an asymptotic approac h,” in Pr oblems of Informat ion T r ansmission , v ol. 32, no. 1, pp. 15–27, 1996 . [12] L. Ying, R. S rik ant, A. Eryilmaz and G. Dullerud, “A large deviations analysis of scheduling in wireless net works,” IEEE T r ansactions on Information The ory, vo l. 52, no. 11, pp. 5088- 5098, 2006. 21 [13] S. Zac h a ry an d I. Ziedin s, “A refi n emen t of the Hunt-Kurtz theory of large loss netw orks with an application to virtual partitioning,” The Annal s of Applie d P r ob ability, vo l. 12, pp. 1–22, 2002. 22