LARGEST WEIGHTED DELAY FIRST SCHEDULING: LARGE DEVIATIONS AND OPTIMALITY. By Alexander L. Stolyar and Kavita Ramanan Bell Labs

Transcription

1 The Annals of Appled Probablty 200, Vol., No., 48 LARGEST WEIGHTED DELAY FIRST SCHEDULING: LARGE DEVIATIONS AND OPTIMALITY By Alexander L. Stolyar and Kavta Ramanan Bell Labs We consder a sngle server system wth N nput flows. We assume that each flow has statonary ncrements and satsfes a sample path large devaton prncple, and that the system s stable. We ntroduce the largest weghted delay frst (LWDF) queueng dscplne assocated wth any gven weght vector α = α α N. We show that under the LWDF dscplne the sequence of scaled statonary dstrbutons of the delay ŵ of each flow satsfes a large devaton prncple wth the rate functon gven by a fntedmensonal optmzaton problem. We also prove that the LWDF dscplne s optmal n the sense that t maxmzes the quantty [ ] mn α lm = N n n log P ŵ >n wthn a large class of work conservng dscplnes.. Introducton... Qualty of servce requrements and the LWDF dscplne. A very mportant and challengng problem n the desgn of hgh-speed communcaton networks s that of provdng qualty of servce (QoS) guarantees, usually specfed n terms of loss probabltes or delays of packets n the network. The control of delays s often of crucal mportance, especally for real-tme applcatons lke vdeo. For example, the QoS requrements for each traffc flow (customer class) can be specfed n terms of a deadlne T and an allowed volaton probablty δ. More precsely, f there are N classes of users and ŵ s the statonary customer delay of the th class, then one would lke to determne f there exsts a polcy that would meet the gven QoS constrants (.) P ŵ >T δ for = N In ths paper we restrct ourselves to a sngle node and address a related asymptotc queston. Suppose a set of postve constants (weghts) α α 2 α N s fxed. For a nonnegatve random varable X, let us use the notaton β X = lm log P X >n n n Receved October 999; revsed February AMS 2000 subject classfcatons. 60F0, 90B2, 60K25. Key words and phrases. Queueng theory, queueng delay, large devatons, rate functon, optmalty, flud lmt, control, schedulng, qualty of servce (QoS), earlest deadlne frst (EDF), LWDF.

2 2 A. L. STOLYAR AND K. RAMANAN assumng the lmt exsts. We ask the followng queston. Is there a dscplne G such that the asymptotc QoS requrements (.2) β ŵ G α for = N are satsfed, where ŵ G s the statonary delay for class wth dscplne G? [Observe that when T s large and δ s small, the problem (.2) can be related to (.) by settng α = T / log δ for all.] We address the problem (.2) by consderng the followng equvalent optmzaton problem: (.3) max mn α β ŵ G G = N The problems are equvalent n the sense that (.2) has a soluton f and only f the maxmum n (.3) s greater than or equal to. We ntroduce the largest weghted delay frst (LWDF) queueng (schedulng) dscplne, whch s parametrzed by a weght vector α α N. Roughly speakng, ths dscplne always chooses for servce the longest watng customer from the queue for whch the current weghted delay ŵ t /α s maxmal. Under the assumpton that the nput flows have statonary ncrements and satsfy a sample path large devaton prncple, and the system s stable, we prove that LWDF s an optmal soluton to the problem (.3). Ths mples that f there s any dscplne G whch solves problem (.2), then LWDF does so. Note that f all weghts α are equal, LWDF reduces to the smple FIFO dscplne. Also note that LWDF s nvarant wth respect to the stochastc structure of the nput flows, n the sense that the algorthm tself reles only on the weghts α whch characterze the QoS requrements of the flows, and not on the structure of the nput flows. To provde some ntuton behnd our result, note that f for a gven dsc- are well defned, then plne G all β ŵ G (.4) mn α β ( ( ŵ G ) = β = N max = N ŵ G ) α Thus the problem (.3) s reduced to that of maxmzng (over all dscplnes G) the rate of decay of the tal of the statonary dstrbuton of the maxmal weghted delay ˆr G = max = N ŵ G /α. We solve ths problem usng large devatons technques. Frst, we establsh a large devatons prncple (LDP) for the sequence of scaled statonary maxmal weghted delays under the LWDF dscplne. In partcular, we characterze J, defned to be the quantty n (.4) wth G equal to the LWDF dscplne, n terms of a fnte-dmensonal optmzaton problem. Then we establsh optmalty of the LWDF dscplne by showng that the value of (.4) for any other dscplne G s bounded above by J. The same technque used to establsh the LDP for the maxmal weghted delay can also be used to derve the LDP for each sequence of scaled statonary delays ŵ under the LWDF dscplne. Also t s clear that a Prorty dscplne can be vewed as the lmt of the LWDF dscplne wth parameters

3 LARGEST WEIGHTED DELAY FIRST SCHEDULING 3 α = ε N ε 0. (The lower flow ndex means hgher prorty.) Ths allows a dervaton of the LDP for the prorty dscplne from that of LWDF. We present the correspondng results n Secton 6.4. In addton, n Secton 7 we state an analogous result for the unfnshed work processes. In ths case an optmal dscplne s the largest weghted (unfnshed) work frst (LWWF), whch s defned n Defnton 7.. These results can be obtaned analogously (and n fact more easly) than the results for the delays. The QoS crteron we consder s expressed n terms of probablstc bounds on delays of packets wthn the network. The performance of delays n a queueng system s often harder to analyze than that of queue lengths due to the fact that systems wth delays often do not admt a fnte Markovan state representaton. However, as mentoned earler, delays are often the crucal measure of performance, especally for real tme traffc lke vdeo, and also n certan wreless settngs []. In [, 5] t was shown that the earlest deadlne frst (EDF) dscplne s optmal n the context of determnstc QoS guarantees (or worst case delay bounds) for a sngle node. It s generally beleved that determnstc QoS requrements lead to an overly conservatve admsson polcy, and a consequent decrease n system throughput [8]. Ths motvates our consderaton of probablstc QoS guarantees, whch provde a very advantageous trade-off of a lttle qualty of servce for a large capacty gan. There s a vast body of lterature on QoS guarantees n communcaton networks (see [3, 7, 3, 7, 9, 20, 22] for just a few examples). In much of the work just cted, a polcy s fxed a pror and the focus s then on performance analyss. Our perspectve s dfferent. We fx the desred performance objectve, and then try to determne the polcy that s best suted to acheve t. Other work that takes ths perspectve ncludes [2, 4, 8]. However these papers focus on optmalty n the heavy traffc regme. In contrast, ths paper provdes a proof of optmalty wth respect to a large devatons crteron n a queueng context. Prevous analyses of optmzaton wth respect to a large devatons crtera have concentrated more on unconstraned dffuson processes [6, 9]. Analyss of queueng processes, and n partcular analyss of ther large devaton propertes, s usually further complcated by the dscontnutes ntroduced by the state space constrants. Our optmalty proof entals a large devatons analyss of the LWDF dscplne. Although the exstence of large devaton prncples for sequences of scaled queue length processes have been establshed for a relatvely large class of queueng networks [5], the problem of determnng the rate functon and characterzng optmal paths has remaned elusve for the vast majorty of cases where there are multple classes, even for a sngle node. Most of the results n the sngle node case have been lmted to two classes [20, 9], where the analyss reles heavly on the relatve smplcty of planar geometry, wth the multdmensonal results beng rather few and restrcted to queue lengths [7]. The outlne of the paper s as follows. The basc notaton used throughout the paper s provded n Secton.2. In Secton 2. we state our basc assumptons on the nput flows and the class of dscplnes consdered, and

4 4 A. L. STOLYAR AND K. RAMANAN then formulate the man result n Theorem 2.2. In Secton 2.2 we ntroduce some basc queueng operators that are used throughout the paper. The LWDF dscplne s defned and analyzed n Secton 3. It s frst defned for dscrete nputs n Secton 3. and then extended n Sectons 3.2 and 3.3 to nclude flud nputs. In Secton 4 we analyze the nfnte-dmensonal varatonal problem that characterzes the rate functon for the sequence of scaled statonary maxmal weghted delays under the LWDF dscplne, although the proof of the LDP s not gven tll Secton 6. Theorem 4.4 shows that the varatonal problem can be reduced to a fnte-dmensonal one usng the fact that the optmal paths have a very smple structure. In Secton 5 we prove a result whch s not used drectly n the dervaton of the man results of the paper, but nevertheless provdes key nsght nto the optmalty property of the LWDF schedulng dscplne. In Secton 6 we present the proof of the man theorem, and derve some mportant corollares. Secton 7 contans a result smlar to our man result, Theorem 2.2, that s vald for unfnshed work processes. We conclude n Secton 8 wth dscussons of some open problems, future work and some practcal mplcatons of our results..2. Basc notaton and defntons. Let [respectvely + ] be the space of RCLL functons (.e., rght contnuous functons wth left lmts) on [respectvely, [0, )]. Unless otherwse specfed, we assume and + are endowed wth the topology of unform convergence on compact sets (u.o.c.). For any s 0 and f N, we defne the norm f s = max sup = N s t s f s For f + N the norm f s s defned smlarly wth s replaced by 0 n the above dsplay. Thus convergence n N or + N s equvalent to convergence n the correspondng norm s for all s>0. For f N we use f t to denote the left lmt lm u t f u of the functon f at the pont t. As measurable spaces, we always assume that and + are endowed wth the σ-algebra generated by the cylnder sets. We now defne some subspaces of and + that wll be used often n the sequel. Let + 0 be the subset of functons h n + such that h 0 =0. Let be the space of functons n whch are nondecreasng, pecewse constant, and have only a fnte number of jumps on any fnte tme nterval, and let + (respectvely, + 0 ) be the subset of nonnegatve functons n + (respectvely, + 0 ) whch are nondecreasng, pecewse constant, and have only a fnte number of jumps on any fnte tme nterval. We use I to denote the subset of nondecreasng functons n, and I + (respectvely, I + 0 ) the subset of nonnegatve nondecreasng functons n + (respectvely, + 0 ). Let be the subset of contnuous functons n I + 0, and let a be the subset of absolutely contnuous functons n. We assume that the subspaces nhert the topology and σ-algebra of the orgnal space. Gven any space N represents the N tmes product space wth the product topology and product σ-algebra defned n the natural way. We defne + N = x N x 0 for = N.

5 LARGEST WEIGHTED DELAY FIRST SCHEDULING 5 In ths paper we use the notaton h n h as n for h h n I + to denote convergence at every pont of contnuty of h. Although we do not use ths fact, we note that (snce all h n and h are nondecreasng) ths convergence s equvalent to convergence n the Skorohod M -topology [23]. Let = F P be a probablty space. We assume that s large enough to support all the ndependent random processes that we use n the paper. We denote by P B the nner measure (wth respect to the probablty P) ofan arbtrary subset B. IfB F, then P B =P B. Gven any subset B of a topologcal space, we use B and B to denote ts closure and nteror, respectvely. The nfmum of a functon over an empty set s nterpreted as. We use to denote the set of ratonal numbers. We let a b denote the mnmum of the scalars a and b. We follow the conventon of usng bold font for stochastc processes and roman font for determnstc processes. We now gve the defnton of a large devaton prncple ([4], page 5). Let be a topologcal space and B a σ-algebra on. Note that B s not necessarly the Borel σ-algebra. Defnton. (LDP). A sequence of random varables X n on takng values n a topologcal space B s sad to satsfy the LDP wth rate functon I f for all B, and lm sup n lm nf n n log P X n nf I x x n log P X n nf x I x where I s a functon wth compact level sets. We defne the scalng operator Ɣ c c > 0, for elements Y N (or N + )as follows: (.5) Ɣ c Y t = c Y ct For a scalar bɣ c b = b/c and for the par b Y Ɣ c b Y = b/c Ɣ c Y. Gven an operator A S S, where S and S belong to the functon spaces defned above, we say that A s scalable f for every f S and c>0, (.6) Ɣ c A f =A Ɣ c f 2. Basc model. In Secton 2. we state our assumptons and the man result, and n Secton 2.2 we ntroduce some basc operators that wll be used throughout the paper.

6 6 A. L. STOLYAR AND K. RAMANAN 2.. Assumptons and man results. We consder a multclass sngle server queueng system wth N nput flows that satsfy the followng assumptons. Assumpton 2. (Input flows). () Each flow f s a random process on wth statonary ncrements that takes values n. () The flows f = N, are mutually ndependent. () For each, and every T<, the sequence of processes f n f n 0, n = 2 restrcted to 0 T satsfes a LDP wth rate functon JT gven by (2.) { T ( JT f = L f s ) ds f f a, 0 otherwse, where f n = Ɣ n f s the scaled verson of f as defned n (.5), L s a convex functon takng values n 0 such that L λ =0 for some λ 0, L x > 0 for x λ, and lm x L x /x =. (v) The server s not overloaded. In other words, N λ < The process f t represents the cumulatve amount of work of class (measured n terms of the requred servce tme) that has entered the system by tme t. A jump n f at tme t corresponds to a customer arrval, wth the servce tme of that customer equal to the sze of the jump f t f t. Consder the class of queueng (or schedulng) dscplnes G such that:. G s work conservng. 2. Schedulng decsons at any tme t are ndependent of both the future of the process and the hstory of the process before the begnnng of the current busy perod. Let G be an arbtrary dscplne n. Gven an nput flow sample path f N, let ˆτ G t be the arrval tme of the longest watng class customer present n the system at tme t, ncludng the customer(s) beng served at tme t. [We wll show later that ths functon s well defned for almost all sample paths of the nput flow process f = f f N.] By conventon we assume that ˆτ G s rght contnuous, and set ˆτ G t =t f there s no class customer n the system at tme t. We refer to ˆτ G = ˆτ G t <t< as the class backlog process. Suppose we are gven the set of postve weghts, 0 <α α 2 α N Then we defne the class delay ŵ G and weghted delay ˆr G processes and the maxmal weghted delay ˆr G process n terms of the class backlog process as

7 LARGEST WEIGHTED DELAY FIRST SCHEDULING 7 follows. For t 0 and = N, (2.2) ŵ G t = t ˆτG t (2.3) ˆr G t = ŵg t /α (2.4) ˆr G t = max ˆr G t We are nterested n dentfyng a dscplne that s optmal n the sense that t maxmzes the exponental decay rate of the statonary dstrbuton of the maxmal weghted delay ˆr G. Our man result s that the largest weghted delay frst (LWDF) dscplne ntroduced n Secton 3 s optmal n a large devatons sense as formulated below n Theorem 2.2. Theorem 2.2. There exsts J < such that the followng hold. () For the LWDF schedulng dscplne defned n Secton 3, the upper bound ( ) (2.5) lm sup n n log P n ˆr 0 > J holds, where ˆr 0 s the statonary maxmal weghted delay assocated wth the LWDF dscplne. () For any G, we have the lower bound ( ) (2.6) lm nf n n log P n ˆrG 0 > J where ˆr G 0 s the statonary maxmal weghted delay assocated wth the dscplne G. () Moreover, J solves the followng fnte-dmensonal optmzaton problem: (2.7) J = mn j x x j γ j α γ L x = subject to j N x > 0 j x > = and wth α N+ = by conventon. j = <γ= x α j j+ = α x α j

8 8 A. L. STOLYAR AND K. RAMANAN Remark. A rgorous defnton of the statonary maxmal weghted delay ˆr G for an arbtrary dscplne G, and the proof of the theorem s presented n Secton 6. As we wll see n Secton 6, ˆr G need not n general be a measurable functon on the probablty space. To allow for ths generalty, we use the noton of nner measure n the lower bound [Theorem 2.2() above] Operators assocated wth a queueng dscplne. Consder a fxed dscplne G. Suppose the server s empty at tme 0, whch means that the left lmt of the unfnshed work n the system s equal to 0. Let a path f + N descrbe the nput process startng at tme 0. [If f 0 > 0, t means that a type customer arrved at tme 0 requrng a servce tme f 0.] By assumpton (2) n the defnton of the class, for each ˆτ G 0 =0 and the evoluton of ˆτ G from tme 0 onwards depends only on the path f, and not on the system behavor before t = 0. Gven dscplne G, we defne ÂG to be the operator  G N + I N + 0 that maps f nto the correspondng ˆτ G = ˆτ G t t 0 = N. Its clear that for any f + N ˆτG = ÂG f s such that for = N, ˆτ G 0 =0 and ˆτG t t for t 0 Once agan by assumpton (2) n the defnton of the class, the dscplne G s completely characterzed by ts behavor wthn a busy perod. However the ponts n tme when new busy perods of the system start do not depend on the queueng dscplne, snce all dscplnes n are work conservng. Thus the operator ÂG assocated wth a dscplne G completely characterzes the evoluton of all ˆτ G t for all t. Moreover, f the dscplne G s nonpreemptve, then the operator ÂG completely characterzes the dscplne tself. We also ntroduce a related operator, R G N whch maps an nput flow path f + N nto the correspondng maxmum weghted delay ˆr G = ˆr G t t 0. Clearly, each operator R G s unquely determned by the correspondng operator ÂG. 3. The largest weghted delay frst dscplne. In ths secton we defne the largest weghted delay frst (LWDF) dscplne. In Secton 3. we defne the LWDF dscplne for dscrete nput paths takng values n N + 0.In Secton 3.2 we ntroduce the noton of a vrtual backlog process, whch s then used n Secton 3.3 to extend the defnton of LWDF to flud nput flow paths f N.

9 LARGEST WEIGHTED DELAY FIRST SCHEDULING The LWDF dscplne for dscrete nputs. In Secton 2. we ntroduced the backlog, delay, weghted delay and maxmal weghted delay assocated wth any queueng dscplne G and nput flows n + N. Recall that a dscplne s sad to be nonpreemptve f the servce of a customer cannot be nterrupted before ts servce has been completed. We now defne the largest weghted delay frst (LWDF) dscplne for nput flows n + N. Defnton 3. [The largest weghted delay frst (LWDF) dscplne]. The LWDF dscplne s a nonpreemptve, work conservng dscplne that always chooses for servce the longest watng (.e., head-of-the-lne) customer of the flow whch has the maxmal weghted delay ˆr t = ˆr t. In case of a te, by conventon the LWDF dscplne chooses the class wth the hghest ndex. Note that LWDF s frst-n-frst-out wthn each class. We drop the subscrpt G for the LWDF dscplne and denote the operators ÂG and R G assocated wth the LWDF dscplne smply by  and R, respectvely Vrtual delays for LWDF. In the last secton we ntroduced the operator  that characterzes the LWDF dscplne. We now ntroduce the related operator A + N I + N 0 whch maps f + N (.e., nput flows startng at tme 0) nto the vrtual backlog process τ I+ N 0 defned below. The man reason for ntroducng the vrtual backlog process s that t enables one to extend the defnton of the LWDF dscplne to contnuous nput sample paths f N, as shown n the next secton. To defne the operator A, as n the defnton of the operator  we frst assume that the system s empty at tme 0 and consder a fxed f + N whch descrbes the nput flows startng at tme 0. Consder ˆτ = Âf, and the correspondng processes ŵ ˆr for = N, and Rf =ˆr = max ˆr. We defne the vrtual backlog τ t t 0 for the flow as follows. At any tme t when the system s empty, τ t = ˆτ t =t. Wthn a busy perod of the system, let t 0 <t < <t k be a (fnte) sequence of tme nstants when a new customer arrval or a servce completon occurs. Then wthn ths busy perod the functons τ are nondecreasng pecewse constant (namely, constant n each nterval [t m t m+ ]) and are defned by the followng recurson. For every = N, let (3.) τ t 0 = ˆτ t 0 =t 0 (3.2) τ t m+ = max { τ t m t m+ α ˆr t m+ } We wll refer to the recurson (3.2) as the dscrete system jump rule (for the LWDF dscplne). It smply descrbes how far ahead each τ s moved after a servce of a customer s completed.

10 0 A. L. STOLYAR AND K. RAMANAN We defned the vrtual delay, vrtual weghted delay and the vrtual maxmal weghted delay n the natural way as follows. For t 0 let (3.3) (3.4) (3.5) w t r t r t = t τ t = w t /α = max r t We let R denote the operator whch maps f N + nto r + 0. The backlog ˆτ and the vrtual backlog τ are clearly closely related. The followng lemma summarzes the relaton between the two. Lemma 3.2. The vrtual delay process τ satsfes the followng propertes: (3.6) (a) For all t 0, τ t ˆτ t for = N whch mples that w t ŵ t and r t ˆr t. In addton, we have (3.7) and (3.8) r t r 2 t r N t τ t τ 2 t τ N t (b) If the servce of a customer of type starts at tme t, then (3.9) τ t = ˆτ t and consequently w t =ŵ t and r t = ˆr t. Moreover, (3.0) (3.) (3.2) (3.3) r t =r t = ˆr t = ˆr t r t r 2 t r N t τ t τ 2 t τ N t ˆk t =arg max j ˆr j t k t =arg max r j t j wth the set k t havng the form (3.4) k t = 2 Proof. Property (3.6) of statement (a) follows drectly from (3.2). As far as the proof of statement (b) s concerned, we only need to prove t wthn a system s busy perod, whch s easly done by nducton of the event tme nstants t m m= 0 usng agan the recurrence (3.2). Then we get (3.7) and (3.8) from (3.) and (3.2).

11 LARGEST WEIGHTED DELAY FIRST SCHEDULING 3.3. Extenson of operator A to flud nputs. We now extend the doman of the operator A (characterzng the LWDF dscplne) that was defned on + N n the last secton to nclude the space of contnuous (or flud ) nput flows f N. As shown n Theorem 3.3, the operator s naturally extended to the space N by contnuty n the product topology nduced by the at the contnuty ponts ( ) convergence. Theorem 3.3. Gven f N, there exsts a unque τ = τ t t 0 = N I N + 0 such that the followng statements hold: (a) For any sequence f n f n + N n= 2 such that (3.5) f n f we have (3.6) = Af n τ (b) Each functon τ s nondecreasng, rght contnuous, wth (3.7) (c) Defne τ 0 =0 and τ t t for t 0 ˆτ t = sup ( ξ f ξ =f τ j t ) t and let w r r be defned n terms of τ, and ŵ ˆr and ˆr n terms of ˆτ as for the dscrete system by the set of equatons (2.2) (2.4) and (3.3) (3.5). Then τ satsfes the followng addtonal condtons: () For = N and all t 0, (3.8) τ t ˆτ t () The followng conservaton law holds for any t 0, (3.9) [ ( )] N N f τ t = t + nf f ξ ξ 0 0 ξ t = = () If t arg mn 0 ξ t N = f ξ ξ and N = f t t 0.e., the flud system at tme t s empty then for every = N, (3.20) τ t =t (v) If for some t r t <r j t and τ j t = ˆτ j t, then there exsts ε>0 such that for any θ t t + ε, (3.2) τ θ =τ t (v) The followng flud system jump rule holds for any t 0, and any N, (3.22) r t =r t ˆr t τ n

12 2 A. L. STOLYAR AND K. RAMANAN (v) For any t 0, (3.23) r t r 2 t r N t (3.24) τ t τ 2 t τ N t and for any t 0, there exsts j such that (3.25) r t =r j t = ˆr j t = ˆr t Theorem 3.3 s a drect consequence of the followng two propostons. Proposton. Consder f N and any sequence f n f n + N such that f n f Then there exsts a subsequence f nl f n and τ I+ N 0 satsfyng propertes (3.7) (3.25) such that (3.26) τ nl τ where τ n l = Af nl. satsfyng proper- Proposton 2. Gven f N, an element τ I+ N 0 tes (3.7) (3.25) s unque. Proof of Proposton. Gven f N and a sequence f n f n + N such that f n f, let τ n = Af n. Snce each functon τ n s nondecreasng, and for any fxed t satsfes (3.27) τ n t t there exsts a subsequence n l n such that τ n l (3.28) q τ q q q 0 where recall that s the set of ratonal numbers. Clearly τ s a nondecreasng functon snce each τ n l s. We extend the defnton of τ to all reals by rght-contnuty, that s for t 0weset (3.29) τ t = nf τ q q>t q It follows mmedately that τ s nondecreasng, and τ n l t τ t at each pont of contnuty t of τ, and thus we obtan (3.30) τ n l τ It remans to show that τ so defned satsfes (3.7) to (3.25). Property (3.8) follows trvally from the defnton of ˆτ. Propertes (3.23) and (3.24) of τ follow

13 LARGEST WEIGHTED DELAY FIRST SCHEDULING 3 from the constructon of τ and the correspondng propertes (3.7) and (3.8) of the prelmtng functons τ nl. Before we proceed, let us notce that property (3.9) (along wth all the propertes descrbed n Lemma 3.2) holds wth f and τ replaced by every prelmt par of functons f nl and τ nl, for any tme t when the servce of a customer starts or, trvally, when the (flud) system s empty. It s easy to see that the set of such tme ponts s asymptotcally dense as n l,nthe sense that the followng property holds. For any n l and t 0, we defne s nl t to be the frst tme nstant n t at whch ether a new servce of a customer starts or the system s empty. Then (3.3) lm n l s nl t =t t 0 Indeed, f ths were not true, there must exst ε>0 such that for all suffcently large n l, nowhere n t t + ε does a new servce start or the system become empty. Ths would mean that for all suffcently large n l a sngle customer s served n that nterval, whch mples that for all suffcently large n l at least one of the functons f n l has a jump of sze at least ε n the nterval [0, t]. Ths contradcts the u.o.c. convergence f n l f for every Ths proves property (3.3). For notatonal convenence, let us denote by V the completed work mappng whch maps each f I+ N 0 nto the functon Vf + defned by the rght-hand sde (RHS) of (3.9). It s well known that Vf for any f I+ N 0, and the mappng V s contnuous on the elements f N. We now prove property (3.9). Usng the fact that the RHS of (3.9) s equal to Vf and s therefore a contnuous functon, and usng property (3.3), we nfer that (3.9) holds at every tme t where all τ are contnuous. Then usng rght-contnuty of all τ, and agan the contnuty of the RHS of (3.9), we get (3.9) for all t 0. We ntroduce some more notaton to prove the remanng propertes. For a nondecreasng functon h I + a tme t 0 wll be called a rght pont of growth of h f h t <h ξ for all ξ>t The proof of property (3.20) s by contradcton. Suppose, at some fxed tme t the flud system s empty, but τ t <tfor some.ift s not a rght pont of growth of any of the functons f, then t s easy to see that for any ε>0, for all suffcently large n l, there exsts a tme x nl t t + ε when the prelmt system wth ndex n l must be empty, whch means τ n l x nl =x nl. Ths would mply τ n l t + ε t, and therefore τ t + ε t. By rght-contnuty of each τ we get τ t t, a contradcton. Now suppose t s a rght pont of growth of at least one functon f. Notce that for every f τ t = f t due to the conservaton law and the

14 4 A. L. STOLYAR AND K. RAMANAN fact that the (flud) system s empty at tme t. From the latter observaton and the conservaton law we see that for at least one fxed k, and any ε>0, we have τ k t + ε >t[and τ k t + ε t + ε, of course]. On the other hand for any small δ > 0, there exsts δ > 0 such that for any ξ t t + δ, τ ξ τ t +δ. Ths means that the sequence of prelmt systems s such that for all large n l, there exsts a tme y nl t t + ε such that the servce of a type k customer (that arrved after tme t) starts, and yet τ n l y nl τ t +2δ. Snce for a fxed small δ > 0 (and the correspondng δ>0), we can choose ε 0 δ arbtrarly small, ths leads to a contradcton wth the LWDF schedulng rule, namely wth the property (3.0) sayng that f (n a prelmt system wth ndex n l ) a type k customer s chosen for servce at tme ξ, then r n l k ξ =max r n l ξ. Ths completes the proof of property (3.20). We now prove property (3.22). Frst, let us observe that f at tme t the flud system s empty; that s, we are n the condtons of statement (c)() of the theorem, then (3.20) holds and therefore (3.22) holds trvally. So we only need to consder the case when the flud system s nonempty at tme t, that s, f τ t = V f t < f t Then t follows from the defnton of the operator V that t s a rght pont of growth of V f. Ths n turn mples that for at least one m τ m t + ε > ˆτ m t for any ε>0; whch means that τ m t ˆτ m t by the rght-contnuty of τ m. Thus we get the exstence of m such that τ m t = ˆτ m t. Next, we observe that ths m can be chosen such that ˆr m t =max ˆr t = ˆr t [If ths were not true, we would get a contradcton to the LWDF schedulng rule smlar to the one we got n the proof of property (3.20).] We show below that for every = N, (3.32) r t ˆr t Indeed, for any ε>0, for all large n l there exsts a tme pont y nl t t + ε such that the servce of a class m customer starts at y nl, and therefore r n l m y nl =r nl y nl whch mples that for any, τ n l y nl t r nl y nl α Snce ε>0 can be arbtrarly small, we see that for any δ>0, τ t + δ lm nf n l Snce δ>0 s arbtrary, we get whch proves (3.32). τ n l t + δ t ˆr t α τ t =lm δ 0 τ t + δ t ˆr t α

15 LARGEST WEIGHTED DELAY FIRST SCHEDULING 5 It remans to show that (3.33) and (3.34) f r t ˆr t then r t ˆr t f r t < ˆr t then r t =r t Property (3.33) for m s trval because (3.23) mples r t r m t = ˆr t To prove (3.33) for >m, we wll need the followng property of a prelmt system (wth ndex n l ) whch follows from the dscrete system jump rule (3.2). Suppose for some fxed and m >m, and for a fxed t 0, there s a class customer whch arrved at some tme ξ<t, has not started servce by tme t, and r n l t > t ξ /α m. Then for any tme n t, τ n l m η ξ mples τ n l t ξ η t α Returnng to the proof of (3.33) for >m, we observe that f r t ˆr t for some >m, then for any small ε>0, and for all suffcently large n l and η t, τ n l m η t α m ˆr t ε mples τ n l η t α ˆr t 2ε Let us fx a small ε>0. Then we can choose δ>0 small enough so that for all large n l, whch means that that s, τ n l m t + δ t α m ˆr t ε τ n l t + δ t α ˆr t 2ε r n l t + δ ˆr t 2ε Ths n turn mples r t ˆr t 2ε. Snce ε>0s arbtrary, we get r t ˆr t, and the proof of (3.33) s complete. To prove (3.34) we frst observe that r t < ˆr t =r m t s only possble f >m. Argung very smlarly to the proof of (3.33), we can see that for a suffcently small fxed ε>0, the prelmt system for all large n l must be such that τ n l s constant n the nterval t ε t + ε [because, n ths nterval, there always wll be type m customers whch accordng to the LWDF schedulng rule must be served before τ n l can move forward]. Ths means that τ s constant n t ε t + ε, and therefore r s contnuous at t. Ths completes the proof of property (3.22). α m

16 6 A. L. STOLYAR AND K. RAMANAN We now show that (3.2) holds. Frst notce that r t <r j t s only possble f >jand there s nonzero unfnshed work n the flud system. The proof then uses almost exactly the same argument as that used above to prove (3.34). Fnally (3.25) follows drectly from (3.22). Proof of Proposton 2. Gven f N, suppose there exst two dstnct τ and τ satsfyng all the lsted condtons, and defne the process = τ τ. Then we clam that the followng two observatons are true. Observaton For any t 0, f N t > 0 then there exsts ε>0 such that the dfference N s nonncreasng n the nterval t t + ε. Observaton 2 For any t 0, f N t > 0, then (3.35) 0 N t N t Lkewse, for any t 0, f N t 0, then (3.36) N t N t 0 We now show by contradcton that the two observatons mply that N must be zero everywhere, and relegate the proofs of the observatons to the end. Note that N 0 =0. Suppose the functon N s postve somewhere on [0, ), then there must exst a>0 and t 0 such that ether N t or N t s equal to a = sup N ξ > 0 0 ξ t and for all ξ<t (3.37) N ξ <a Now Observaton 2 mples that one cannot have N t <a, snce ths would mply that N t =a>0, whch contradcts (3.36) f N t < 0, and contradcts (3.35) f 0 N t <a. Thus we nfer that (3.38) N t = lm N s =a s t Along wth (3.37), ths mples that there exsts δ>0 and b<asuch that N t δ =b, and N ξ > 0 for ξ t δ t. From Observatons and 2 t follows that N t s nonncreasng on t δ t. Indeed, let s = sup u t δ t N s nonncreasng on t δ u Then s>t δ by Observaton, and by Observaton 2 N s nonncreasng on t δ s [snce N cannot jump up at s]. From ths we deduce that s = t, snce f s<tthe fact that N s > 0 along wth Observaton contradcts the defnton of s as the supremum. Ths mples that N t b, whch leads to a contradcton due to (3.38). Thus we have establshed that N must be nonpostve everywhere on 0. By symmetry (nterchangng the roles of τ and τ ), t s clear that n fact N 0 or, equvalently, for all t 0, (3.39) τ N t =τn t

17 LARGEST WEIGHTED DELAY FIRST SCHEDULING 7 It s easy to verfy that the same argument can now be used nductvely to establsh that τ t =τ t for = N N 2, whch establshes the theorem. It only remans to prove the two observatons. Proof of Observaton. (3.40) or equvalently (3.4) Suppose there exsts t 0 such that τ N t <τ N t r N t >r N t By the conservaton law (3.9) t follows that N N f τ t = f τ t = Thus f τ t <τ t for every = N, then there exsts ε>0 such that f τ t +u =f τ t for all = N and u 0 ε. However, snce by (3.25) there exsts some for whch τ t = ˆτ t, ths would contradct the defnton of ˆτ. Thus there must exst j<nsuch that (3.42) = r j t r j t The orderng property (3.23) then dctates that (3.43) r N t r j t From the last four dsplays and the defnton of r we conclude that (3.44) r N t <r t r t Observaton then follows from (3.25), (3.2) and the fact that τ N s nondecreasng. Proof of Observaton 2. If both τ N and τn are contnuous at tme t, then Observaton 2 s trval. The rest of the proof reles on the followng two basc propertes of the backlog process, whch were establshed n Theorem 3.3. Frst, f τn jumps, then due to (3.22), (3.25) and the orderng property (3.23), t follows that (3.45) r N t =r N t = =r t Second, snce by (3.25) there always exsts one component such that τ t s a rght pont of growth for f, one can never have τ t <τ t for every = N, as ths would mply that N N f τ t < f τ t = whch volates the conservaton law (3.9). These propertes clearly also hold wth τ s nterchanged wth τ s. =

18 8 A. L. STOLYAR AND K. RAMANAN We now proceed wth the proof of (3.35). Frst consder the case when τ N jumps at tme t. Then τ N must jump at t as well because f not, by the orderng property, (3.45) and the assumpton that N t 0, t follows that r t =r 2 t = =r N t <r N t r N t = r N t r N t r t Ths mples that for every = N, τ t >τ t whch s mpossble as argued above. So τ N must also jump at t, whch mples that (3.45) also holds wth r replaced by r. Ifr N t <r N t, then once agan the last dsplay holds, and f r N t <r N t then the last dsplay holds wth > replaced by <. Snce both these cases contradct the conservaton law, we deduce that N t =0, whch proves (3.35). Now suppose τn does not jump at t, but τ N does. Then snce τ N s nondecreasng, N t N t. Moreover, f τ N t >τn t then r t =r 2 t = r N t <r N t < r t whch leads to a contradcton as above. Ths mples that N t 0, whch establshes (3.35). The property (3.36) follows from symmetry. Ths concludes the proof of Observaton 2, and therefore of the proposton. Havng establshed Theorem 3.3, we can now defne the operators A and R for nput flows n N. Defnton 3.4. Gven f N, let Af be equal to the unque τ descrbed n the statement of Theorem 3.3. [Note that by Theorem 3.3, τ = Af f and only f τ satsfes (3.7) (3.25).] In addton, we defne the operator R N + 0 n the natural way by settng for t 0, Rf t = max t Af t α Remark 3.5. () Note that n Secton 3.2 we defned the operators A and R on + N, and n ths secton we defned the operators on the doman N. Snce + N N s the zero functon, for whch both defntons can be seen to concde, the operators A and R are well defned on + N N. () It s easy to verfy from the defnton that the operators A and R are scalable [see defnton (.6)]. We end ths secton wth a result that s essentally a corollary of Theorem 3.3. Even though t s not used n the proofs of our man results, t s mportant snce the result can be used drectly to derve the LDP (va the contracton prncple [4, 2]) for the maxmal weghted delay process r for a system wth random flud nput flows.

19 LARGEST WEIGHTED DELAY FIRST SCHEDULING 9 Theorem 3.6. The mappng A N I N + 0 s contnuous n the product topology nduced by the convergence. Proof. Let τ = Af and τ k (3.46) Consder f N and let f k f k N, be a sequence such that f k f as k = Af k. We need to prove that τ k τ If (3.46) were false, then we could choose a subsequence f k l such that (3.47) τ K l τ τ For each τ k l, by the defnton of A, there exsts a sequence f k n l f kl n + N n= 2 such that f kl n f k l and lm Af kl n = τ k l n where recall that here A s defned on dscrete nput paths va equatons (3.) and (3.2). Then usng Cantor s dagonal procedure (smlarly to the way t s done n [24]), we can construct a sequence f kl m n m m= 2 such that f k l m n m f but Af k l m n m τ τ However, ths contradcts the defnton of τ as the unque lmt of Af m for any f m f. 4. Most lkely paths for large statonary delays under the LWDF dscplne. It s well known that large devatons technques can be used to convert the problem of characterzng asymptotc lmts of probabltes nto that of solvng varatonal problems. The optmal solutons of the varatonal problems shed nsght nto the most probable way n whch rare events occur. In Secton 6 we show that the optmal decay rate J of the tal of the statonary maxmal weghted delay process assocated wth the LWDF dscplne (see Theorem 6.8) s gven by the followng varatonal problem: (4.) where for every s 0, (4.2) J = nf J s f s>0 f a N Rf s J s f = N Js f =0 and Js s defned n (2.). In ths secton we characterze the optmal path f of the varatonal problem that acheves the cost J. We use ths characterzaton n Secton 6 to prove the LD lower bound (2.6) of our man result Theorem 2.2. In the next three lemmas we show that the nfmzaton n (4.) can wthout

20 20 A. L. STOLYAR AND K. RAMANAN loss of generalty be restrcted to successvely smpler sets. In Theorem 4.4 we show that the nfmzaton s actually attaned on a smple element (whose defnton s gven below), and characterze J n terms of a fnte-dmensonal optmzaton problem. Lemma 4.. (4.3) subject to and (4.4) Let J be as defned n 4. Then J = J s f N f t >t = nf s>0 f N a Rf s = fort 0 s Proof. As stated n Remark 3.5, the operator R s scalable. Thus gven any path f a N such that Rf s =c> for some s>0, there exsts a scaled down path of no greater cost for whch equalty holds. Specfcally, consder the path f = Ɣ c f. Then f a N, R f s/c = by the scalablty of R, and J s/c f = N = s/c 0 L f u du = c N = s 0 L f u du = c J s f Thus we can restrct the nfmzaton n (4.) to the set of paths for whch Rf s =. Now gven any path f a N wth Rf s =, we can wthout any ncrease n cost replace t by the ncrements of f after the queue s empty for the last tme. More precsely, replace f by the path f defned by { f f t + s t = for t 0 s s, f s s +λ t s + s for t s s for = N, where { s = max ξ s [ ( N )] } f ξ =ξ + nf f η η 0 0 η ξ = and recall that λ s the mean nput rate for flow that was ntroduced n Assumpton 2.. It can be easly verfed that Rf s s = and J s s f J s f, whch establshes (4.3). Now consder any f a N such that Rf T = for some T > 0 and condton (4.4) s satsfed wth s = T. Let r = Rf. For = N defne (4.5) T = sup t T r t =r t Due to the orderng property, (4.6) r t r 2 t r N t for t 0

21 LARGEST WEIGHTED DELAY FIRST SCHEDULING 2 proved n Theorem 3.3, these tme nstants are well defned and satsfy T N+ = 0 T N T N T = T The property (3.22) that r t =r t f r jumps at t n fact shows that the supremum n (4.5) s attaned, and therefore that an equvalent defnton of T s (4.7) T = mn t T τ t =τ T We defne the pecewse lnearzaton f of f as follows. Let τ = Af. Then for j = N,weset t τ j T + f j τ j T + τ j T t f j τ j T + τ j T τ j T + (4.8) f j t = f t τ j T + τ j T j f j τ j T j + λ j t τ j T j f t τ j T j Lemma 4.2. Consder any f N a such that Rf T = for some T>0 and condton 4 4 s satsfed wth s = T. Let f be ts pecewse lnearzaton defned above. Moreover, defne τ = Af and r = Rf. Then the followng propertes hold: () For every j N+, (4.9) τ j T =τ j T and for each N such that T + functons < T, and t T + T, the (4.0) r t = =r t =r t are equal and lnear wth dervatve (4.) () Furthermore (4.2) γ = r T r T + T T + J T f J T f > 0 Proof. By constructon for each j N (4.3) f j τ j T = f j τ j T for = N + j and the functon f j s lnear n each nterval τ j T + τ j T for = N j. Gven j N, let the process τj I + N 0 be such that (4.4) τj T =τ j T for = N + N j

22 22 A. L. STOLYAR AND K. RAMANAN τ j s lnear n each nterval T + T for = N j, and for t T j T, (4.5) τ j t =τ j T j =τ j T We now show that τ defned above satsfes condtons (3.7) (3.25), whch by Defnton 3.4 automatcally establshes that τ = Af Property (3.7) follows trvally from the correspondng property for τ. Let r t = t τ t /α for t 0 and = N. We clam that for t T + T, (4.6) r t = =r t =r t and f j>and T j <T, then for t T + T, r j t <r t The above two propertes clearly hold for t = T + and t = T, and so by the lnearty of r j t also holds for all t T + T. Ths establshes propertes (3.20) (3.25). For every = N, the fact that f j s nondecreasng and τ j T T, shows that N N N f j T f j τ j T = f j τ j T = T j= j= j= Thus the lnearty of τ j on T + T dctates that (4.7) N j= f j t t for t T + T and (4.8) N j= f j τ j t = t Propertes (4.7) and (4.8) verfy the conservaton law (3.9). Fnally, we note that (4.2) follows from the defntons of f and J and the fact that the functons L, = N are convex. Consder f a N as defned n Lemma 4.2, and let f be ts pecewse lnearzaton as defned above. For k N such that T k+ <T k, and k, let d k = f τ T k and let γ be as n (4.). Then for k such that T k+ <T k the unt cost c k of rasng the maxmal weghted delay wthn the tme nterval T k+ T k s r T k r T k+ = N = τ T k r T k r T k+ τ T k+ L f s ds k τ T k τ T k+ L d k =

23 LARGEST WEIGHTED DELAY FIRST SCHEDULING 23 where we have used the fact that f s =λ for s>τ T and L λ =0. Usng (4.0) and the last dsplay, we see that the unt cost c k s gven by α c k = γ γ k L d k f T k+ <T k, (4.9) k k otherwse (by conventon). Note that c k represents the unt cost of rasng r n the tme nterval T k+ T k, n whch only the frst k flows are served. If j s such that c j s the mnmum among all the unt costs, then t s ntutvely belevable that f can be replaced by another trajectory f 0, such that only the frst j classes are served whle the maxmal weghted delay attans level, and the cost of f 0 does not exceed that of f. Ths s the content of Lemma 4.3, and s a consequence of the fact that the parameters γ j d j k, and α j that characterze the pecewse lnearzatons f satsfy certan useful relatons derved below. Defne r = Rf. Then for j such that T j+ <T j, from defntons (4.), (4.9) and (4.7) t follows that for t T j+ T j, and ṙ j+ t < ṙ j t =γ j τ j+ t =0 Ths mples that ẇ j+ t =, or equvalently ṙ j+ t =/α j+, whch when substtuted to the last but one dsplay yelds (4.20) /α j+ <γ j whch n turn means that for all j, (4.2) /α + <γ j Obvously for every = N and t 0 such that the dervatve τ t exsts, (4.22) τ t 0 whch mples that for all j, (4.23) α γ j Fnally from (4.0), (4.), and the conservaton law (3.9), we see that n each nterval T j+ T j such that T j+ <T j, (4.24) α γ j d j = j We now state and prove Lemma 4.3.

24 24 A. L. STOLYAR AND K. RAMANAN Lemma 4.3. Gven f a N as defned n Lemma 4 2, let f be ts pecewse lnearzaton and let c k be the assocated unt costs defned n 4 9. Let j = arg mn k= N c k (where the maxmum ndex s chosen n case of a te), let γ = γ j and x = d j for = N. Defne the trajectory f0 by λ t for >j t 0, (4.25) f 0 t = x t for j t 0 T 0, f 0 T0 +λ t T 0 for j t T0, where T 0 = /γ and for = j, T 0 = α γ T 0. Then τ 0 = Af 0 and r 0 = Rf 0 satsfy (4.26) τ 0 t =0 for t 0 T0 and > j (4.27) r 0 t = =r0 j t =r0 t =γt for t 0 T 0 (4.28) r 0 T 0 = and J T 0 f 0 J T f J T f Proof. By Defnton 3.4 t suffces to show that any τ 0 satsfyng propertes (4.26) and (4.27) satsfes condtons (3.7) (3.25). From (4.2), (4.24) and (4.23) t follows that j (4.29) x > (4.30) = α j+ <γ = j x j α x α j wth /α N+ = 0 conventon. Now f τ 0 satsfes (4.26) and (4.27), then due to the above relatons we obtan for t 0 T 0, (4.3) and ṙ 0 t =/α <γ=ṙ 0 t for >j (4.32) ṙ 0 t =γ =ṙ0 t and τ 0 t 0 for >j whch establshes (3.7) and (3.20) (3.25). Moreover the conservaton law (3.9) holds snce N f j 0 τ0 t = j x τ 0 t = (4.33) x α γ = = = where the last equalty follows from (4.30). Ths establshes that τ 0 = Af 0. Fnally, to prove (4.28) we wrte J T 0 f 0 =c j c j 0 r T r T + N N = 0 c r T r T + J T f J T f

25 LARGEST WEIGHTED DELAY FIRST SCHEDULING 25 In summary, together the last three lemmas n ths secton show that gven any f a N for whch the maxmal weghted delay exceeds one at some fnte tme, we can fnd a smple element of no greater cost for whch the maxmal weghted delay attans the level. Moreover, Lemma 4.3 also shows that the backlog τ 0 = Af 0 assocated wth any such smple element f 0 has the smple structure dctated by (4.26) and (4.27). Ths motvates the followng defnton. Defnton. A smple element f 0 s a functon defned by (4.25) for some parameters j x j N x +, j whch satsfy the constrants (4.29) and (4.30). We conclude ths secton wth Theorem 4.4, whch shows that the varatonal problem (4.) s mnmzed by a smple element. Theorem 4.4. The nfmum J of the varatonal problem defned n 4 s the soluton to the followng fnte-dmensonal optmzaton problem: j (4.34) J = mn α j x x j γ γ L x = subject to j j N x > 0 x > and j = <γ= x α j j+ = α x α j where α N+ = by conventon. Moreover, the nfmum n 4 s attaned on smple elements f 0 assocated wth parameters j x x j that solve Remark. The structure of a smple element f 0 s llustrated by Fgure. (The pcture s for the case N = 3 and j = 2.) For a subset of flows of the form j for some j j N, the nput rate has a value x >λ n the tme nterval 0 T α where T = /γ; and after tme T α the rate swtches to the mean λ. The remanng flows j + N (f any) always have the mean nput rate λ. The flows from the latter subset are not served at all. Proof of Theorem 4.4. Wthout loss of generalty assume that all α j are dstnct. We proved n Lemma 4.3 that any element f a N such that r T = Rf T for some T< can be replaced by a smple element f 0 such that the cost of r 0 = Rf 0 reachng does not exceed the cost of r = Rf to reach. Moreover there s a one-to-one correspondence between smple elements and parameters j x j N x + j satsfyng (4.29) and (4.30). Furthermore, the unt cost of ncrease of r for a smple element s gven by c = c x x j j = j (4.35) α γ γ L x = =

26 26 A. L. STOLYAR AND K. RAMANAN Fg.. The structure of the smple element f 0 N = 3 j= 2. Thus, to show that the nfmum of the varatonal problem (4.) s attaned on a smple element t suffces to show that there exsts j N and x j + satsfyng (4.29) and (4.30) whch mnmze the cost c defned n (4.35). We clam that for every vector x = x x N such that (4.36) x 0 for = N and (4.37) N x > = there exst a unque j = j x and correspondng γ = γ x such that (4.29) and (4.30) are satsfed. To prove the clam, we rewrte (4.30) as α j j x <z = α j+ j α α j x and notce that z = z j satsfes (a) z j < { for j = mn k k } x > (b) z j α j α j+ mples z j +

27 LARGEST WEIGHTED DELAY FIRST SCHEDULING 27 (c) z j > α j α j+ mples z j + > Now for each x descrbed by (4.36) and (4.37), we defne (4.38) c x = c x x j j wth j = j x and the RHS defned n (4.35). Let us denote (4.39) c = nf x + N c x N= x > and consder a sequence x l l= 2 such that (4.40) lm l c x l =c It s easy to see from (4.35) and (4.37) that γ x l remans unformly bounded away from both 0 and nfnty, that s, there exst ε ε 2 such that for l = (4.4) 0 <ε <γ x l <ε 2 < Let us choose a subsequence (we wll keep the same notaton x l for t) such that γ x l γ > 0 2 j x l = j s fxed Then t s easy to see that for any j x l must stay bounded away from nfnty. Let us choose a further subsequence whch we contnue to denote by x l such that (4.42) x l x for = j We see that condtons (4.29) and (4.30) are satsfed for x = j, where { j j f γ > /α = j+, j + f γ = /αj+, and also x j The proof s complete. = x j+ = λ j f j = j + c x x j j =c

28 28 A. L. STOLYAR AND K. RAMANAN 5. Optmalty of LWDF for a system wth flud nputs. In ths secton we show that n a system wth flud nputs, the LWDF dscplne s optmal n the class of work-conservng dscplnes. Namely, LWDF maxmzes the (mnmal) cost of r t reachng level. The result s very smple, and although t s not used drectly to prove the lower bound (2.6) n our man result Theorem 2.2, t provdes the key ntuton wthout nvokng all the techncal machnery requred to prove t n full generalty. New defntons ntroduced n ths secton are confned to ths secton only, and not used n the rest of the paper. Consder the class of operators A G (wth G beng a queueng dscplne defned for a flud nput system only) such that each operator A G maps a flud nput f N nto a vrtual backlog path τ G = A G f I+ N 0, that satsfes for t 0, τ G t t for = N and the work conservaton condton ( f τ G t ) [ = t + Let us denote mn 0 ξ t ( )] f ξ ξ 0 (5.) J A G = nf Ĵ s f s>0 f N R G f s where Ĵs f s an arbtrary nonnegatve cost functon, and R G f t = max t τ G t α Lemma 5.. Suppose there exsts an operator A 0 from the class defned above, and T 0 > 0 such that the mnmum cost J A 0 s attaned on the path f 0 N such that R 0 f 0 T 0 =, f 0 t >t for t 0 T0 all functons f 0 are strctly ncreasng n 0 T0, and for some subset K 2 N, T 0 τ 0 T0 =α τ 0 T0 =0 for K for K Then for any operator A G n the class, J A G J A 0 Proof. Consder the mnmal cost element f 0, and let τ 0 = A 0 f 0 and r 0 = R 0 f 0. Consder any other operator A G n the class, and let τ G = A G f 0 and r G = R G f 0. We now show that r G T 0 rrespectve of the partcular

29 LARGEST WEIGHTED DELAY FIRST SCHEDULING 29 dscplne G chosen. By assumpton, each functon f 0 ncreasng n 0 T 0 and satsfes the nequalty f 0 t t for t 0 T0 Then by the work conservaton condton, f 0 ( τ G T 0 ) N f 0 ( τ G T 0 ) N = T 0 = K = = s contnuous, strctly f 0 ( τ 0 T 0 ) = f 0 ( τ 0 T 0 ) K where the last equalty follows from the defnton of f 0. Snce each f 0 s nonnegatve and strctly ncreasng on 0 T 0, we conclude that for at least one K t must be that Therefore, τ G T0 τ 0 T0 r G T 0 r G T0 r 0 T0 =r 0 T 0 = In other words, for any operator A G the mnmal cost of r G reachng level sat most J A 0. If we replace the cost functon Ĵs by the cost functon J s defned n (4.2), then t s clear from Theorem 4.4 and Lemma 4.3 that the condtons of Lemma 5. are satsfed wth A beng the operator A assocated wth the LWDF dscplne and K = 2 j for some j N. If the class of queueng dscplnes were restrcted to dscplnes for whch the same extenson procedure carred out n Theorem 3.3 can be used to obtaned a unque and well-defned operator A G on the flud nputs (as s the case for LWDF, FIFO, LIFO, processor sharng, generalzed processor sharng, prorty, and most other non-pathologcal dscplnes), then we could use Lemma 5. n the proof of Theorem 2.2 () drectly. However, our class, and therefore the result of Theorem 2.2, s more general. As a consequence the proof of the lower bound n Theorem 2.2 s more nvolved, even though ts key dea s the smple argument used n the proof of Lemma Rgorous statement and Proof of Theorem 2.2. In ths secton we make precse the statement of the man result gven n Theorem 2.2 and then prove t. In Secton 6. we ntroduce the statonary state process of the system, whch s ndependent of the partcular work-conservng dscplne used. Ths, along wth the operator R G defned earler, s then used n Secton 6.2 to rgorously defne the statonary maxmal weghted delay ˆr G 0 assocated wth any queueng dscplne G and stochastc nput flows f that satsfy Assumpton 2.. Snce we allow for arbtrary dscplnes G, whch are not necessarly even scalable, ˆr G 0 need not always be measurable. Ths necesstates the use of the nner measure n the statement of the lower bound n Theorem 2.2. For the LWDF dscplne however, the statonary process ˆr 0