PERFORMANCE OF MULTICLASS MARKOVIAN QUEUEING NETWORKS VIA PIECEWISE LINEAR LYAPUNOV FUNCTIONS 1

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1 The Annals of Appled Probablty 2001, Vol. 11, No. 4, PERFORMANCE OF MULTICLASS MARKOVIAN QUEUEING NETWORKS VIA PIECEWISE LINEAR LYAPUNOV FUNCTIONS 1 By Dmtrs Bertsmas, Davd Gamarnk and John N. Tstskls Massachusetts Insttute of Technology, IBM T. J. Watson Research Center and Massachusetts Insttute of Technology We study the dstrbuton of steady-state queue lengths n multclass queueng networks under a stable polcy. We propose a general methodology based on Lyapunov functons for the performance analyss of nfnte state Markov chans and apply t specfcally to Markovan multclass queueng networks. We establsh a deeper connecton between stablty and performance of such networks by showng that f there exst lnear or pecewse lnear Lyapunov functons that show stablty, then these Lyapunov functons can be used to establsh geometrc-type lower and upper bounds on the tal probabltes, and thus bounds on the expectaton of the queue lengths. As an example of our results, for a reentrant lne queueng network wth two processng statons operatng under a work-conservng polcy, we 1 show that EL =O 1 ρ 2, where L s the total number of customers n the system, and ρ s the maxmal actual or vrtual traffc ntensty n the network. In a Markovan settng, ths extends a recent result by Da and Vande Vate, whch states that a reentrant lne queueng network wth two statons s globally stable f ρ < 1. We also present several results on the performance of multclass queueng networks operatng under general Markovan and, n partcular, prorty polces. The results n ths paper are the frst that establsh explct geometrc-type upper and lower bounds on tal probabltes of queue lengths for networks of such generalty. Prevous results provde numercal bounds and only on the expectaton, not the dstrbuton, of queue lengths. 1. Introducton. Queueng networks are used to model manufacturng, communcaton and computer systems, and much recent research has focused on networks wth multple customer classes. In multclass queueng networks, the customers served at the same staton have n general dfferent servce requrements and follow dfferent paths through the network. Such networks are used to model, for example, wafer fabrcaton facltes, n whch there s a sngle stream of jobs arrvng nto a producton floor. Jobs follow a determnstcroute and revst the same staton multple tmes (see Fgure 1). Multclass queueng networks of ths type are called reentrant lne queueng networks (see [17, 9]). Receved March 2000; revsed February Research supported n part by NSF Grants DMI , ACI , ARO Grant DAAL G-0115 and by the Sngapore-MIT allance. AMS 2000 subject classfcaton. 60K20. Key words and phrases. Queueng, networks, bounds, Lyapunov functons. 1384

2 PERFORMANCE OF MQNETs 1385 Fg. 1. Re-entrant lne queueng network. The focus of ths paper s performance analyss of multclass queueng networks. Specfcally, we are nterested n estmatng the steady-state queue lengths n the network, when nterarrval and servce tmes are exponentally dstrbuted, assumng a stable schedulng polcy s used, whch brngs the system to steady-state. The performance of queueng networks s largely an open research area. Some of the earler and classcal results nclude product form probablty dstrbutons for Jackson and BCMP-type networks (see [13]). It was realzed, however, that the presence of multple classes does not allow, n general, for a product form dstrbuton even f the nterarrval and servce tmes have exponental dstrbutons and the frst-n-frst-out polcy s used. Several papers ([4, 18, 19, 15]) have analyzed the performance of multclass queueng networks usng quadratc Lyapunov functons. A certan lnear program s constructed, whch provdes numercal bounds on the achevable performance regon. The performance results obtaned usng quadratc Lyapunov functons were later analyzed and extended n a smpler and more ntutve way, usng conservaton laws ([3]). The performance analyss of multclass queueng networks s hghly nontrval, snce t s at least as hard as the stablty problem for whch no general condtons are avalable. It s known that the natural load condton ρ σ < 1 for each staton σ s necessary, but not suffcent, for stablty; a varety of counterexamples have been constructed n [23, 21, 5, 24, 7]. Suffcent condtons for stablty have been found usng Lyapunov functons [9, 10]. Furthermore, flud models were found to be a very useful tool for stablty analyss. Da s theorem [6] shows that the stablty of a flud model mples stablty of a correspondng stochastc model. A complete characterzaton of flud networks wth two statons whch are stable under any work-conservng polcy ( globally stable ) was obtaned by Bertsmas, Gamarnk and Tstskls [2] and subsequently by Da and Vande Vate [8]. The second work used a very ntutve noton of vrtual statons to explan nstablty n networks wth two statons. Both works ([2] and [8]) prove that the exstence of a pecewse lnear

3 1386 D. BERTSIMAS, D. GAMARNIK AND J. N. TSITSIKLIS Lyapunov functon s both necessary and suffcent for global stablty of flud networks wth two statons Our results. The goal of ths paper s to turn some of the stablty analyss tools nto useful performance analyss tools. We wll show how lnear and pecewse lnear Lyapunov functons and vrtual statons can be used to obtan upper and lower bounds on the steady-state queue lengths. For many examples consdered n ths paper, the upper bounds are fnte f and only f the network s stable. Our contrbutons are summarzed as follows. We start n Secton 3 wth an analyss of countably nfnte Markov chans. We show that f there exsts a Lyapunov functon provng the stablty of the Markov chan, then certan computable upper and lower bounds hold on the steady-state queue length probablty dstrbuton as well as on ts expectaton. We then apply ths methodology, n Sectons 4 and 5, to the performance analyss of multclass queueng networks wth exponentally dstrbuted nterarrval and servce tmes. Specfcally: 1. We show how lnear and pecewse lnear Lyapunov functons can be used to obtan lower and upper bounds, respectvely, on the steady-state queue lengths. The lower bounds we obtan are explct, whle the upper bounds are numercal and depend on the solutons of a certan lnear program whch generates the Lyapunov functon. Such lnear programs were ntroduced n [9] and [10] for the purposes of stablty analyss. 2. We use the noton of a vrtual staton, ntroduced n [8]. Ths showed that n networks wth two statons, some prorty polces lead to certan groups V of customer classes, called vrtual statons, whch cannot be served smultaneously. As a result, f the correspondng vrtual traffc ntensty ρv V ρ s bgger than one, then the network s unstable. Here ρ stands for the naturally defned load factor of class. We prove a matchng performance result: for networks wth two statons, f V s a vrtual staton, wth the correspondng vrtual traffc ntensty ρv, and f prorty s gven to the classes n V over the classes not n V then P {L 12 } ( ) ρv m m 2 ρv EL ρv 41 ρv where L s the total number of customers n the network. These lower bounds are extended to networks wth more than two statons. 3. It was also proved n [8] that queueng networks wth two statons are globally stable f the maxmum of all actual and vrtual traffc ntenstes, denoted by ρ, s less than 1 for the orgnal network and for a certan set of subnetworks. Also f ρ > 1, then the correspondng flud network s not globally stable. Whether ths holds true for stochastc Markovan networks s not known.

4 PERFORMANCE OF MQNETs 1387 We show that ρ s a fundamental performance parameter. For reentrant lne networks wth two statons, we show that f ρ < 1, then the followng upper bound holds under any work-conservng polcy: EL C 1 ρ 2 where C s some constant, expressed explctly n terms of the parameters of the network. An mportant mplcaton of ths result s that the performance regon (the set of vectors of expected queue lengths obtaned under dfferent work-conservng schedulng polces) s bounded f and only f the correspondng flud network s globally stable. Our results show a deeper connecton between stablty and performance of multclass queueng networks. Also the results n ths paper are the frst ones that use lnear and pecewse lnear Lyapunov functons for performance analyss. Prevous methods for performance analyss have used quadratc Lyapunov functons, whch have certan lmtatons. In partcular, an example of a globally stable queueng network wth two statons was constructed n [10] for whch the quadratc Lyapunov functon method leads to an nfnte (nconclusve) upper bound, yet a pecewse lnear Lyapunov functon gves a fnte upper bound. The methods developed here, on the other hand, match the sharpest known stablty condton ρ < 1. The second lmtaton of quadratc Lyapunov functons s that the bounds constructed are n most cases only numercal and hold only for the expectatons of queue lengths. In contrast, we provde bounds on the dstrbuton of steady-state queue lengths, provng exponental decay of the tal probabltes. 2. Queueng model and assumptons Multclass Markovan queueng network. We consder a network consstng of J sngle server statons, whch are denoted by σ j j = 1 2J. The network ncludes I types of customers, where customers of type = 1, 2I arrve to the network from an exogenous source. The arrval process correspondng to type s assumed to be an ndependent Posson process wth rate λ. Let =λ 1 λ I denote the vector of arrval rates and let λ mn = mn λ. Wthout loss of generalty, we assume that λ mn > 0. Smlarly, we defne λ max = max λ. Customers of type go through J stages, each of whch corresponds to a servce completon on a partcular staton. We denote these statons by σ 1 σ 2 σ J. The processng tme of a type customer at staton σ k k = 1 2J, s assumed to be exponentally dstrbuted wth rate µ k and s ndependent from the processng tmes of all other stages of ths type, from the processng tmes of the other types and from the nterarrval tmes. We let =µ k 1 I 1 k J denote the vector of servce rates. Customers of type recevng servce at staton σ k are called class k customers. Let N = I =1 J be the total number of classes. For

5 1388 D. BERTSIMAS, D. GAMARNIK AND J. N. TSITSIKLIS convenence, we wll also dentfy every staton σ j wth the set of classes assocated wth ths staton. Let C k = j f class k customers are served at staton σ j.fork J we let C k = 0. Let C denote the correspondng I J max matrx, where J max = max J. The matrx C defnes the topology of the network. We assume that the buffers at each staton have nfnte capacty and no customers renege from the queue before recevng servce. A queueng network of the form just descrbed s called a Markovan multclass queueng network wth determnstc routng or a multtype queueng network. Throughout the paper we consder only networks of ths type. The parameters C consttute the prmary parameters of the network and we denote the network by C. For each class k, weletρ k = λ /µ k be the nomnal load of ths class. For each staton σ j j= 1 2J, we defne the nomnal load (traffc ntensty) as (1) ρ σj ρ k k σ j The evoluton of a queueng network s fully specfed only when a schedulng dscplne and an ntal state s gven. The schedulng dscplne (polcy) descrbes whch customers (f any) are served at any moment at each staton. Wthn each class, the customers are served n frst-n-frst-out (FIFO) fashon. Therefore, the servce dscplne only specfes whch customer type s served at any gven moment. We wll assume throughout the paper that the schedulng polces mplemented are Markovan, namely, schedulng decsons are purely a functon of the system state, whch n our case s the vector of all queue lengths. We also allow preempton. For example, preemptve prorty polces are Markovan. Many mportant polces are not Markovan, for example FIFO or head-of-the-lne-processor-sharng. We leave these polces out from the dscusson n ths paper, although we beleve that the results hold for them as well. We wll be consderng mostly work-conservng polces: each processng staton s requred to work on some customer, f there are any present at ths staton. Any preemptve polcy w satsfyng the Markovan assumpton can be descrbed by a functon w Z N + 0 1N where for any q Z N +, the k component of the vector wq s 1 f staton σ j whch contans class k works on a customer of class k, and s zero, otherwse. Of course, w k q =1 only f q k > 0, and for each staton σ j, (2) w k q 1 k σ j Gven a multclass queueng network C and some schedulng polcy, we let Qt =Q k t 1 I 1 k J denote the vector of queue lengths at tme t. Our focus s on estmatng the dstrbuton of the random vector Qt n steady-state. A necessary condton for the exstence of a steady-state s the

6 PERFORMANCE OF MQNETs 1389 load condton (3) ρ σj < 1 for each j = 1 2J Embedded Markov chans and unformzaton. Instead of analyzng the contnuous tme process Qt, we wll buld a dscrete tme analogue, whch has the same steady-state behavor usng the standard method of unformzaton (see [20]). We rescale the parameters, so that λ + k µ k = 1 and consder a superposton of I + I =1 J Posson processes wth rates λ µ k, respectvely. The arrvals of the frst I Posson processes correspond to external arrvals nto the network. The arrvals of the process wth rate µ k correspond to servce completons of class k customers, f a server actually worked on a class k customer, or they correspond to magnary servce completons of an magnary customer, f the server was dle or worked on customers from other classes. Let τ s s= 1 2 be the sequence of event tmes for ths superposton of Posson processes. Then, as a result of ths constructon and the Markovan polcy assumpton, Qτ s s a dscrete tme Markov chan wth the same steady-state dstrbuton as Qt (assumng t exsts). We can specfy the transtons of the Markov chan Qτ s as follows. For each class k let e k be an N-dmensonal unt vector wth the k component equal to 1 and all other components equal to zero. We adopt the conventon e 0 = e J +1 = 0 for each. The followng proposton holds. Proposton 1. Gven a multclass queueng network C rescaled so that λ + k µ k = 1 and gven a Markovan polcy w, the transton probabltes of the correspondng embedded Markov chan Qτ s s= 0 1 2are gven by Qτ s +e 1 wth probablty λ, Qτ Qτ s+1 = s e k +e k+1 wth probablty µ k w k Qτ s, (4) Qτ s wth probablty k 1 w k Qτ s. kµ Proof. Note that the change Qτ s+1 Qτ s of the embedded Markov chan corresponds to ether an arrval of a type customer, or to an actual servce completon of a class k customer and transton to the next stage k + 1. The frst event has a probablty λ and corresponds to a change e 1. The second event has probablty µ k w k Qτ s and corresponds to a change e k+1 e k. Defnton 1. A schedulng polcy w s defned to be stable f the correspondng embedded Markov chan Qτ s s = 1 2, admts a statonary probablty dstrbuton π = πw satsfyng (5) EQ k τ s < k

7 1390 D. BERTSIMAS, D. GAMARNIK AND J. N. TSITSIKLIS A queueng network s defned to be globally stable f every work-conservng Markovan polcy s stable. If a statonary dstrbuton π of Qτ s exsts, then by unformzaton and by aperodcty of our contnuous tme Markov chan, lm PQt =q =P { (6) π Qτs =q } t Thus, for performance analyss purposes, we may concentrate on the embedded chan Qτ s. Throughout the paper we use standard notatons O,, n the followng sense. If functons fsgs when s s 0 for some s 0 +, then g = Of g = f means that for some fxed constant c>0, gs cfs gs cfs for suffcently large s. If both g = Of, f = Og, then we wll wrte g = f. 3. InfnteMarkov chans and Lyapunov functons. In ths secton, we develop a general technque for steady-state analyss of nfnte Markov chans wth countably many states usng Lyapunov functons. Let Xtt = 0 1 2, be a dscrete tme, dscrete state Markov chan whch takes values n some countable set. The transtons occur at nteger tmes t = For any two vector x x, let px x denote the transton probabltes px x P { Xt + 1 =x Xt =x } If a statonary probablty dstrbuton π on the state space exsts, t satsfes πx =1 and for all x, (7) x πx = x πx px x The exstence of a statonary dstrbuton s usually establshed by constructng a certan Lyapunov functon. For a survey of Lyapunov methods for stablty analyss of Markov chans, see [22]. We now ntroduce the defntons of Lyapunov and lower Lyapunov functons. The goal s to use Lyapunov functons for the performance analyss of Markov chans, assumng a pror that the Markov chan s stable. The noton of a lower Lyapunov functon s ntroduced exclusvely as a means of gettng the lower bounds on the statonary dstrbuton of a Markov chan. In subsequent sectons, we apply the results here to the embedded Markov chan of a multclass queueng network. Defnton 2. A nonnegatve functon +

8 PERFORMANCE OF MQNETs 1391 s sad to be a Lyapunov functon f there exst some γ>0 and B 0, such that for any t = 1 2 and any x, wth x >B, (8) E [ Xt + 1Xt =x ] x γ Also a nonnegatve functon (9) + s sad to be a lower Lyapunov functon f there exsts some γ>0, such that for any t = 1 2 and any x, wth x > 0, E [ Xt + 1Xt =x ] x γ Remarks. () We refer to the terms γ and B as drft and excepton parameters, respectvely. () We could also ntroduce an excepton parameter B for the lower Lyapunov functon, but t s not requred for the examples n ths paper. We assume that the Markov chan Xt s postve recurrent, and we denote by π the correspondng statonary dstrbuton. Namely, πx s the steadystate probablty P π Xt =x that the chan s n a certan state x. Also, we denote by E π the expectaton wth respect to the probablty dstrbuton π. For a gven functon +, let (10) ν max sup x x x x px x >0 and (11) ν mn nf x x x x px x >0x<x Namely, ν max s the largest possble change of the functon durng an arbtrary transton, and ν mn s the smallest possble ncrease of the functon. Also let (12) p max = sup x x x<x px x and (13) p mn = nf x x x<x px x Namely, p max and p mn are tght upper and lower bounds on the probablty that the value of s ncreasng durng an arbtrary transton. In ths paper, we wll be nterested n Lyapunov functons wth fnte ν max, and lower Lyapunov functons wth postve ν mn and p mn. We need the followng lemma, the proof of whch can be found n Appendx A.

9 1392 D. BERTSIMAS, D. GAMARNIK AND J. N. TSITSIKLIS Lemma 1. Consder a Markov chan Xt wth statonary probablty dstrbuton π, and suppose that s a Lyapunov functon wth drft γ and excepton parameter B, such that E π Xt s fnte. Then, for any (possbly negatve) c B ν max, (14) P π { c + νmax <Xt } p maxν max p max ν max + γ P { π c νmax <Xt } where ν max and p max are defned n (10) and (12), respectvely. Also, f s a lower Lyapunov functon wth drft γ, such that E π Xt s fnte, then, for any c 0, { { } p P π c Xt mn ν mn p mn ν mn + 2γ P π c 1 } (15) 2 ν mn Xt where ν mn and p mn are defned n (11) and (13), respectvely. Ths lemma shows how one can obtan a smple recurrence relaton on the tal probabltes P π c<xt. We use ths recurrence n the proof of the followng result. Theorem 1. Consder a Markov chan Xt wth a statonary probablty dstrbuton π such that E π Xt <. () If there exsts a Lyapunov functon wth drft γ>0, and excepton parameter B 0, then for any m = 0 1 2, (16) P π { Xt >B+ 2νmax m } ( ) pmax ν m+1 max p max ν max + γ As a result, E π Xt B + 2p maxν max 2 (17) γ () If there exsts a lower Lyapunov functon wth drft γ>0, then for any m = 0 1 2, (18) P π { Xt 1/2νmn m } ( ) 1/2pmn ν m mn 1/2p mn ν mn + γ As a result, (19) E π Xt p mnν mn 2 4γ Remark. The bounds (16), (17) and (18), (19) are meanngful only f ν max < (the Lyapunov functon has unformly bounded jumps) and ν mn p mn > 0, respectvely.

10 PERFORMANCE OF MQNETs 1393 Proof. In order to prove (16), we let c = B ν max. By applyng Lemma 1, we obtan P π B<Xt p maxν max p max ν max + γ P { π B 2νmax <Xt } p maxν max p max ν max + γ We contnue smlarly, usng c = B + ν max c = B + 3ν max c = B + 5ν max. By applyng agan Lemma 1, we obtan the needed upper bound on the tal dstrbuton. In order to prove (17), note that E π Xt B P π Xt B+ B+2ν max m+1 m=0 { P π B+2νmax m<xt B+2ν max m+1 } = B P π Xt B { +B P π B+2νmax m<xt B+2ν max m+1 } However, m=0 { +2ν max m+1p π B+2νmax m<xt B+2ν max m+1 } m=0 { m + 1P π B + 2νmax m<xt B + 2ν max m + 1 } m=0 = { P π B + 2νmax m<xt } m=0 Applyng the bounds from (16), we obtan E π Xt B + 2ν max m=0 ( pmax ν max p max ν max + γ ) m+1 = B + 2p maxν max 2 γ To prove (18), let c = ν mn /2c = ν mn c = 3ν mn /2. Then, by applyng Lemma 1, we obtan { P π Xt 1/2νmn m } ( ) 1/2pmn ν m mn P 1/2p mn ν mn + γ π Xt 0 ( ) 1/2pmn ν m = mn 1/2p mn ν mn + γ

11 1394 D. BERTSIMAS, D. GAMARNIK AND J. N. TSITSIKLIS In order to prove (19), we have E π Xt 1 2 { ν mn mp π 1/2νmn m Xt < 1/2ν mn m + 1 } m=0 = 1 2 ν mn From (18) we obtan { P π 1/2νmn m Xt } m=1 E π Xt 1 2 ν mn ( ) 1/2pmn ν m mn = p mnν mn 2 1/2p mn ν mn + γ 4γ m=1 Ths completes the proof of the theorem. As mentoned above, the steady-state behavor of Markovan queueng networks s equvalent to the steady-state behavor of the embedded Markov chan. Applyng Theorem 1, we can analyze the performance of Markovan queueng networks by constructng sutable Lyapunov functons. Ths s the subject of the followng sectons. 4. Lower bounds on queue lengths usng lnear lower Lyapunov functons. In ths secton, we use lnear lower Lyapunov functons to fnd closed form lower bounds on the dstrbuton and expectaton of steady-state queue lengths, whch hold when an arbtrary stable Markovan schedulng polcy s mplemented. Gven a stable schedulng polcy w, let π = πw denote the correspondng statonary dstrbuton (of the queueng network and ts embedded Markov chan). We wll show that [ ] ( ) J ( ) 1 1 E π Q k t = = 1 ρ k σj 1 ρ j=1 where ρ σj s the traffcntensty at staton σ j and ρ = max 1 j J ρ σj. We wll also derve lower bounds on the dstrbuton and expected queue lengths whch hold specfcally when prorty polces are mplemented, by usng the noton of a vrtual staton. Fnally, we wll apply these results to some examples Closed form lower bounds for arbtrary work-conservng polces. Recall that under any Markovan schedulng polcy, the transtons of the unformzed embedded Markov chan are gven by Proposton 1. For each staton σ j,we now construct a lower Lyapunov functon. For any class k, let ρ σ j+ k = (20) ρ k k k σ j k k

12 PERFORMANCE OF MQNETs 1395 In words, ρ σ j+ k s the sum of traffcntenstes of classes of type startng from stage k onward whch are processed on staton σ j. Let j Q = k ρ σ j+ k Q λ k Proposton 2. Let w be an arbtrary Markovan polcy. Then, j s a lower Lyapunov functon wth drft γ j = 1 ρ σj and p mn = λ ν mn ρ σj /λ max. (21) Proof. Usng Proposton 1, we have E j Qτ s+1 Qτ s = j Qτ s + Note from (20) that I =1 ρ σ j+ 1 λ = λ + k I I =1 λ ρ σj+ 1 λ µ k w k Qτ s ( ρ σ j+ k+1 ρσ j+ ) k /λ ρ k =1 k k σ j k 1 = ρ σj Observe that for any staton σ j, µ k w k Qτ s ( ρ σj+ k+1 ρσ j+ ) k /λ = ( µ k w k Qτ s ρ ) k λ k σ j k σ j = w k Qτ s 1 k σ j where the last nequalty follows from the feasblty constrant (2) for the polcy w. Also note that ρ σ j+ k+1 ρσ j+ k = 0 when k σ j. Combnng wth (21) we obtan E [ j Qτ s+1 Qτ s ] j Qτ s ρ σj 1 Ths proves that j s a lower Lyapunov functon. We now bound the parameters ν mn and p mn. From Proposton 1, f a transton of the Markov chan Qτ s corresponds to a servce completon n the class k, then the correspondng change n the value of the Lyapunov functon j s ρ σ j+ k /λ + ρ σ j+ k+1 /λ whch by defnton s nonpostve. Therefore, the value of the Lyapunov functon can ncrease only at the arrval tmes and, as a result, p mn = I =1 λ.at an arrval of a type customer, the value of the Lyapunov functon ncreases by ρ σj /λ. Therefore ν mn ρ σj /λ max.

13 1396 D. BERTSIMAS, D. GAMARNIK AND J. N. TSITSIKLIS We now are ready to state the man result of ths secton. Theorem 2. Consder a multclass queueng network C operatng under an arbtrary stable Markovan polcy. The followng lower bounds hold on the steady-state number of customers n the network: for each j = 1 2J, and m = 0 1 2, { ρ σ j+ k P Q λ k t ρ } ( σ ρσj ) m j m 2λ max 2 ρ σj and k [ E k ρ σ ] j+ k Q λ k t ρ 2 σ j 4λ max 1 ρ σj The result follows by applyng Proposton 2 and Theorem 1. Remarks. () The bounds hold whether we have rescaled the parameters to λ + k µ k = 1 or not, snce ρ σj, and the rato λ /λ max are nsenstve to rescalng. () The bounds hold whether the polcy used s work-conservng or not. The bounds of Theorem 2 are smplfed when the multclass queueng network has a reentrant lne structure, namely, I = 1. In ths case, all customers follow the same route n the network. We denote by Q k t the queue length at the kth stage n the network. The parameters ρ k ρ σ j+ k ρ k and ρ σ j+ k are denoted smply by. The lower bounds on the queue lengths are smplfed as follows. Corollary 1. Gven a reentrant lne-type queueng network C, operatng under any stable Markovan polcy, the followng lower bounds hold on the number of customers n the network n steady-state. For each j = 1, 2J, and m = 0 1 2, { P ρ σ j+ k Q k t ρ } ( σ j 2 m ρσj ) m 2 ρ σj and k [ E k ] ρ σ j+ k Q k t ρ 2 σ j 41 ρ σj 4.2. Closed form lower bounds under a prorty polcy. In ths secton, we derve lower bounds on the tal probabltes and the expected number of customers n a multclass queueng network operatng under a prorty polcy w θ that s descrbed by a permutaton θ of the set of classes k 1 I 1 k J. For two classes k k assocated wth the same staton σ j, we say that class k has a hgher prorty than class k f θ k <θ k.

14 PERFORMANCE OF MQNETs 1397 A correspondng prorty polcy w θ can be descrbed as follows: for each state q Z N + w θq s an N-dmensonal bnary vector whose components satsfy w θ kq =1 f and only f q k > 0 and q k = 0, whenever k σ j, where j s such that k σ j, and θ k <θ k. In other words, the polcy w θ at each transton epoch τ s, selects wthn each staton σ j the hghest prorty class wth a postve number of customers and works on a customer from ths class. We thus assume that w θ s a preemptve resume prorty polcy. Clearly, preemptve prorty polces are Markovan. The lower bounds to be derved n ths secton are based on the concept of a vrtual staton and vrtual traffcntensty ntroduced n [8], where the vrtual staton concept s used for the stablty analyss. We wll show how vrtual statons characterze the performance of multclass queueng networks. The defntons below follow [8] very closely. Defnton 3. A collecton of classes e = k 1 k k 2, correspondng to a type- customer s defned to be an excurson f all these classes are from some staton σ j, but classes k 1 1 and k are not from staton σ j. Ths ncludes the possblty k 1 = 1ork 2 = J. The classes k 1 k 2 1 are called the frst classes of the excurson e and class k 2 s called the last class of the excurson e. We denote the sequence of all excursons correspondng to type by e 1, e 2 e R. Defnton 4. Gven a multclass queueng network C, suppose that a collecton of statons σ 1 σ 2 σ J wth sze = K, and nonempty collectons of classes V j σ j, j are selected. The set of classes V = j V j s defned to be a K-vrtual (or just a vrtual) staton f the followng condtons hold: () No classes of the frst excurson are n V e 1 V =, for each = 1 2I. () If the last class of some excurson e l s n V, then all the classes of ths excurson are n V, and f a frst class of the excurson e l s n V, then every frst class of e l s n V. Thus, a vrtual staton must have ether none of the classes, all of the classes, or all but the last class of each excurson. () If a class k s the frst class of an excurson e l wth l 1 [that s σ k 1 σ k], then class k V f and only f k 1 V. For example, Classes 2, 3 and 5 n the reentrant lne network n Fgure 2, consttute a 2-vrtual staton. The followng result was proved n [1] and [14]. Proposton 3. Suppose that a set of classes V = j V j forms a K- vrtual staton for some 1 2J and that w θ s a stable prorty polcy

15 1398 D. BERTSIMAS, D. GAMARNIK AND J. N. TSITSIKLIS Fg. 2. Classes 2, 3 and 5 consttute a vrtual staton n a reentrant lne network. whch gves prorty to classes n V over classes not n V. Namely, whenever k V and k V θ k <θ k. Then, the correspondng statonary dstrbuton π θ satsfes { } P πθ w θ k Qt K 1 = 1 k V Namely, n steady-state, at most K 1 of classes n V can receve servce smultaneously. We see, n partcular, that for networks wth two statons, f V s a 2-vrtual staton for a prorty polcy w θ, then, n steady-state, only one of the classes of V can receve servce at a tme. Thus, V acts as a staton sharng ts resources among ts classes. Ths justfes the name vrtual staton. Let V = j V j be a K-vrtual staton n a multclass queueng network wth determnstc routng. Smlar to Secton 4.1, we ntroduce (22) ρv ρ k (23) ρ V+ k k V k V k k ρ k 1 I 1 k J Proposton 4. Suppose w θ s a stable prorty polcy n a multclass queueng network C. Suppose also that the set of classes V = j V j forms a K-vrtual staton, and w θ gves prorty to classes n V over classes not n V. Then, for the correspondng embedded Markov chan Qτ s the functon Q = k ρ V+ k Q λ k

16 PERFORMANCE OF MQNETs 1399 s a lower Lyapunov functon wth drft K 1 ρvp mn = λ and ν mn ρv/λ max. Proof. From Proposton 1 n Secton 2, we have EQτ s+1 Qτ s Qτ s I = λ ρ V+ 1 + µ λ k w θ k Qτ s 1 ( ρ V+ k+1 =1 λ ρv+ k) k where we assume that ρ V+ J +1 = 0. Note that { ρ V+ k+1 ρv+ k = ρ k f k V, 0 f k V. Therefore, EQτ s+1 Qτ s Qτ s = ρ V+ 1 k V w θ k Qτ s From Proposton 3 and from ρ V+ 1 = ρv we obtan that the drft s γ = K 1 ρv. We obtan the expressons for p mn and ν mn as n the proof of Proposton 2. A corollary of ths result s the transence (nstablty) of a prorty polcy w θ f for some vrtual staton V, wehaveρv >K 1. Ths nstablty result was proven n [1] and [14] under more general assumptons, nterarrval and servce tmes have a general (as opposed to exponental) dstrbuton. We now derve a matchng performance result, when ρv <K 1. The followng theorem s the man result of ths secton. Theorem 3. Suppose we are gven a multclass queueng network C, and a set of classes V that forms a K-vrtual staton. If a stable prorty polcy w θ gves prorty to classes n V over the classes outsde V, then the followng lower bounds hold on the steady-state dstrbuton and expectaton of the number of customers n the network. For each j = 1 2J, and m = 0 1 2, { } ρ V+ k P Q k t ρv ( ) ρv m m 2λ max 2K 1 ρv and k λ [ E ρ V+ k λ k ] Q k t ρ 2 V 4λ max K 1 ρv The proof s smlar to the one of Theorem 2. The lower bounds of Theorem 3 are also smplfed when the network s reentrant lne-type.

17 1400 D. BERTSIMAS, D. GAMARNIK AND J. N. TSITSIKLIS Fg. 3. Lu Kumar network. Corollary 2. Suppose that C s a reentrant lne-type queueng network and that a set of classes V s a K-vrtual staton. If a stable prorty polcy w θ gves prorty to classes n V over classes outsde V, then the followng lower bound holds on the number of customers n the network n steady-state. For each m = 0 1 2, { P k ρ V+ k Q kt ρv } ( 2 m ρv 2K 1 ρv ) m and [ E k ] ρ V+ k Q kt ρ 2 V 4K 1 ρv 4.3. Examples. In ths secton, we demonstrate the usage of the technques developed n the prevous sectons on two specfc networks. The Lu Kumar network. Consder the network n Fgure 3. Ths reentrant lne network s descrbed by the followng parameters: =λ =µ 1 µ 2 µ 3 µ 4 ρ = λ/µ = ρ σ1 = ρ 1 + ρ 4 ρ σ2 = ρ 2 + ρ 3 We have ρ σ 1+ 1 = ρ 1 + ρ 4 and ρ σ 1+ = ρ 4 for = Smlarly, ρ σ 2+ = ρ 2 + ρ 3 for = 1 2ρ σ 2+ 3 = ρ 3 and ρ σ 2+ 4 = 0. Proposton 5. In the Lu Kumar network of Fgure 3, the followng lower bounds on a statonary probablty dstrbuton π hold, under any stable schedulng polcy: { P π 2 ρ } ( ) 1 + ρ 4 Q 1 t+ρ 4 Q 2 t+ρ 4 Q 3 t+ρ 4 Q 4 t ρ1 + ρ m m 4 ρ 1 + ρ 4 2 ρ 1 ρ 4 { P π 2 ρ } ( ) 2 + ρ 3 Q 1 t+ρ 2 + ρ 3 Q 2 t+ρ 3 Q 3 t ρ2 + ρ m m 3 ρ 2 + ρ 3 2 ρ 3 ρ 2

18 PERFORMANCE OF MQNETs 1401 for all m = Also [ E π ρ1 + ρ 4 Q 1 t+ρ 4 Q 2 t+ρ 4 Q 3 t+ρ 4 Q 4 t ] 1 ρ 1 + ρ ρ 1 ρ 4 [ E π ρ2 + ρ 3 Q 1 t+ρ 2 + ρ 3 Q 2 t+ρ 3 Q 3 t ] 1 ρ 2 + ρ ρ 2 ρ 3 If, n addton, the network operates under prorty polcy w θ wth prorty rule θ4 <θ1θ2 <θ3, then { P πθ 2 ρ } 2 + ρ 4 Q 1 t+ρ 2 + ρ 4 Q 2 t+ρ 4 Q 3 t+ρ 4 Q 4 t m ρ 2 + ρ 4 ( ) ρ2 + ρ m 4 2 ρ 2 ρ 4 for all m = 0 1 2, and [ E πθ ρ2 + ρ 4 Q 1 t+ρ 2 + ρ 4 Q 2 t+ρ 4 Q 3 t+ρ 4 Q 4 t ] (24) 1 ρ 2 + ρ ρ 2 ρ 4 The frst part of the proposton s obtaned by applyng Corollary 1 to statons σ 1 and σ 2, the second part s obtaned by applyng Corollary 2 to the vrtual staton V =2 4. A 3-staton, 6-class reentrant lne. Consder the reentrant lne queueng network wth sx classes and three statons descrbed n Fgure 1. Ths network was consdered n [7], where the authors ntroduce the prorty rule θ wth θ4 <θ1θ2 <θ5θ6 <θ3 and show that the set V =2 4 6 forms a 3-vrtual staton. Applyng Corollary 2 we obtan the followng result. Proposton 6. Consder the network n Fgure 1, under the prorty rule θ4 <θ1θ2 <θ5θ6 <θ3. Suppose n addton that the polcy s stable. Then, for the correspondng statonary probablty dstrbuton π θ, { P πθ 2 ρ 2 + ρ 4 + ρ 6 Q 1 t+ρ 2 + ρ 4 + ρ 6 Q 2 t+ρ 4 + ρ 6 Q 3 t ρ 2 + ρ 4 + ρ ρ 4 + ρ 6 Q 4 t+ρ 6 Q 5 t+ρ 6 Q 6 t ρ 2 + ρ 4 + ρ 6 for each m = Also, } ( ) ρ2 + ρ m 4 + ρ m 6 4 ρ 2 ρ 4 ρ 6 [ E πθ ρ2 + ρ 4 + ρ 6 Q 1 t+ρ 2 + ρ 4 + ρ 6 Q 2 t+ρ 4 + ρ 6 Q 3 t +ρ 4 + ρ 6 Q 4 t+ρ 6 Q 5 t+ρ 6 Q 6 t ] 1 ρ 2 + ρ 4 + ρ ρ 2 ρ 4 ρ 6

19 1402 D. BERTSIMAS, D. GAMARNIK AND J. N. TSITSIKLIS Observe that the lower bound on the expected number of customers n the network above has a sngularty at ρ 2 + ρ 4 + ρ 6 = 2. Ths descrbes a heavytraffcbehavor not observed before n the lterature. 5. Upper bounds on queue lengths usng pecewse lnear Lyapunov functons. The man focus of ths secton s n dervng upper bounds on the steady-state queue lengths n multclass queueng networks by means of a pecewse lnear Lyapunov functon. Gven that a certan lnear program has a feasble soluton, we construct a Lyapunov functon for the embedded Markov chan Qτ s of the queueng network. In ths way we obtan fnte bounds on the tal dstrbuton and the expectaton of the queue lengths n the network, operatng under any Markovan work-conservng polcy Arbtrary work-conservng polces. Consder a multclass Markovan queueng network C. Down and Meyn [10] (concentratng only on reentrant lne case I = 1) showed that f the followng lnear program has a feasble soluton wth strctly postve γ, then any work-conservng polcy s stable (global stablty): (25) (26) (27) GLPdm L j 1 λ +L j k+1 µ k L j k µ k +V j γ L j k+1 µ k L j k µ k V j 1 L j k J 1 Lj k j j k σ j 1 j J k σ j k / σ j (28) L j k V j 0 Specfcally, usng the technque of [12], Down and Meyn [10] proved that f GLP[dm] has a feasble soluton wth postve γ, then a random perturbaton of the followng pecewse lnear functon: (29) x max 1 j J L j x where L j =L j k, s a Lyapunov functon wth drft γ and some (unknown, but fnte) excepton parameter B. For networks wth two statons the lnear program GLP[dm] takes the followng form (we denote t by LP[dm]): (30) L 1 1 λ + L 1 k+1 µ k L 1 k µ k + V γ k σ 1 (31) (32) (33) (34) (35) (36) L 1 k+1 µ k L 1 k µ k V L 2 1 λ + L 2 k+1 µ k L 2 k µ k + W γ L 2 k+1 µ k L 2 µ k W L 1 k L2 k L 1 k L2 k LVWγ 0 k σ 2 k σ 2 k σ 1 k σ 1 k σ 2

20 PERFORMANCE OF MQNETs 1403 The pecewse lnear functon (29) s very close to beng a Lyapunov functon of the embedded Markov chan Qτ s. It satsfes the drft condton (8) for all x Z N +, except near the ntersectons of hyperplanes, namely, near the sets x Z N + L x = L jxj = 1 2J j. The smoothng random perturbaton used by Down and Meyn solves ths techncal dffculty. We now use the same smoothng operaton as n [10] but, unlke [10], our dervaton s explct and a closed form estmate of the excepton parameter B wll be gven. In fact, we reestablsh the results obtaned n [10]. Agan we rescale tme so that I λ + µ k = 1 =1 k Let L 1 L 2 L J γ be any feasble soluton of GLP[dm] wth γ>0. Let { j L max max L k} k j For all j = 1 2J,welet { O j = z =z 1 1 z 1 2 z I JI N + [ z k L j k + J 1 2J γ Lj k + 1 ] (37) 2 γ for k σ j [ z k L j k Lj k + 1 } ] 2J γ for k σ j Consder the unform probablty densty functon p j z on the set O j. We wll denote by Z j a random varable wth dstrbuton p j z and denote by z j a sample pont from the set O j. For any z 1 z 2 z J O 1 O J and any x Z N + let (38) 0 z 1 z 2 z J x = max 1 j J z j x and let [ s x =E u 0 Z 1 Z 2 Z J x ] (39) = 0 z 1 z 2 z J xp 1 z 1 p J z J dz 1 dz 2 dz J We use a subscrpt u to emphasze the unform dstrbuton u. We next show that the modfed functon s s a Lyapunov functon. Proposton 7. Let L 1 L 2 L J γ be any feasble soluton of GLP[dm] wth γ>0. Then for any Markovan work-conservng polcy w s s a Lyapunov functon of the embedded Markov chan Qτ s wth drft equal to 1 γ and excepton parameter 4 B = 16NJ 2 J 1L max + γ 3 γ 2 Also ν max L max +1/2γ.

21 1404 D. BERTSIMAS, D. GAMARNIK AND J. N. TSITSIKLIS For the proof, see Appendx B. We now apply Theorem 1 to obtan the followng result. Theorem 4. Gven a multclass queueng network C, wth parameters rescaled so that I =1 λ + k µ k = 1, suppose that the correspondng lnear program GLP[dm] has a feasble soluton L 1, L 2, L J γ wth postve γ. Then the followng upper bound holds on the statonary dstrbuton π correspondng to any stable work-conservng Markovan polcy w: { L j P Qt B } ( ) π 2L max +1/2γ m Lmax +1/2γ m L max +3/4γ for all m = 0 1 2, and all j = 1 2J, where { j } L max = max L k 1 j J 1 I 1 k J and Also B = 16NJ 2 J 1L max + γ 3 γ 2 E π L j Qt 16NJ 2 J 1L max + γ 3 + 8L max +1/2γ 2 γ 2 γ for all j = 1 2J. Proof. The bounds are a drect corollary of Proposton 7, Theorem 1, equaton (38) and the fact 0 z 1 z 2 z J x z j x L j x for all z j O j j= 1 2J. We also use p max 1. Remark. It s known (see [2, 8]) that a flud network wth two statons s globally stable f and only f the lnear program LP[dm] has a feasble soluton wth postve γ. Therefore, for networks wth two statons, the bounds are fnte f and only f the correspondng flud network s globally stable Upper bounds for networks wth two statons. In ths secton, we provde explct performance bounds for queueng networks wth two statons. We wll consder only reentrant lne queueng networks. The reference to the type s thus omtted. The Posson arrval rate s denoted by λ. An explct and tght characterzaton of global stablty of flud networks wth two statons s gven n [8]. Specfcally, t s proved that a flud queueng network wth two statons s globally stable f and only f the maxmal of all the real and vrtual traffcntenstes ρ (to be defned below) s smaller than 1. From ths result and Da s theorem [6] connectng flud and stochastc stablty, the condton ρ < 1 s also suffcent for global stablty of the stochastc network (wth arbtrary and not necessarly exponental servce dstrbuton). In ths secton, we

22 PERFORMANCE OF MQNETs 1405 derve a matchng performance result: whenever ρ < 1, we construct a fnte upper bound on the tal probabltes and the expectaton of queue lengths n the network. We show that ρ s a fundamental performance parameter of the network. In partcular, we prove that under any work-conservng polcy w, the correspondng statonary dstrbuton π satsfes [ ] N E π Q t =1 ( = O 1 1 ρ 2 Followng [8], we ntroduce the defntons of separatng sets and recall the defnton of a 2-vrtual staton (Defnton 4 wth K = 2). In ths secton, we wll refer to a 2-vrtual staton as a vrtual staton as we only consder networks wth only two statons. Recall that a set of classes k 1 k 2 s defned to be an excurson f all of these classes belong to some staton σ j j= 1 2, but classes k 1 1k are not from staton σ j. Let e 1 e 2 e R denote the set of all excursons. We assume wthout loss of generalty that e 1 σ 1. For each excurson e r =k 1 k 2, the class k 2 s called the last class of excurson e r and s denoted by le r. The classes k 1 k 2 1 are called the frst classes of the excurson e r and are denoted by fe r. Defnton 5. A set of excursons S s defned to be a separatng set f t contans no consecutve excursons. Namely, e r S mples e r 1 e r+1 / S. We have assumed that the frst excurson e 1 belongs to the frst staton; that s, e 1 σ 1. A separatng set S s defned to be strctly separatng f t does not contan e 1. Two separatng sets consstng only of excursons n σ 1 or of excursons n σ 2 are called trval separatng sets. Each separatng set of excursons nduces a collecton VS consstng of the classes n excursons n S together wth the frst classes of excursons (other than e 1 ) whose mmedate predecessor s not n S. Thus, ( ) ( ) VS = e r fe r+1 e r S If S s n addton strctly separatng, we refer to VS as a vrtual staton. It s not hard to see that f S s a strctly separatng set then VS s a vrtual staton as defned by Defnton 4. We now ntroduce some addtonal notatons. Let S be a separatng set and let us choose an excurson e r. Denote ρvsσ 1 ρ k (40) ρvse r σ 1 ρe r σ 1 e r / S k σ 1 VS k σ 1 VS k>le r k σ 1 k<le r ρ k ) ρ k

23 1406 D. BERTSIMAS, D. GAMARNIK AND J. N. TSITSIKLIS Smlarly, we ntroduce ρvsσ 2 ρvse r σ 2 and ρe r σ 2. Let ρ denote the maxmal actual or vrtual traffc ntensty: { } (41) ρ max maxρvse r ρ σ1 ρ σ2 S e r where ρvse r =ρvse r σ 1 1 ρe r σ 2 +ρvse r σ 2 (42) 1 ρe r σ 1 +ρe r σ 1 +ρe r σ 2 ρe r σ 1 ρe r σ 2 Da and Vande Vate [8] proved that f ρ < 1, then a two-staton queueng network s stable under all work-conservng polces (globally stable). Also, f there exsts a vrtual staton VS such that ρvs > 1 then there exsts an unstable prorty polcy. Our goal n ths secton s to derve closed form upper bounds on the steadystate number of customers n the network, n terms of the parameter ρ.an outlne of our approach s as follows. We consder a certan modfcaton of the lnear program LP[dm] from Secton 5.1. We use the results n [8] to show that f ρ < 1, then ths modfed lnear program has a feasble soluton wth postve γ and the result of Theorem 4 becomes applcable. In addton, by analyzng the lnear program we obtan explct bounds on the soluton varables and specfcally on the drft γ. The latter allows us to obtan the explct dependence of the drft on the maxmal traffc ntensty ρ. We consder now the followng lnear program consdered n [8] (Equatons (4.11) (4.15), (5.1), (5.2) n [8]): (43) λ x µ k x k + λε 0 k σ j j= 1 2 σ j (44) x x 0 for any excurson e σ 2 (45) (46) (47) (48) σ 1 >le σ 2 >le x σ 2 le σ 1 le x 0 x k + ε = 1 k σ 1 x k + ε = β k σ 2 x ε 0 for any excurson e σ 1 We denote ths lnear program by LP[dv]. Note that β could be treated as a varable n the lnear program above. But nstead, as n [8], we wll treat t as a parameter. Note also that constrants (43), (46), (47) and (48) of LP[dv] mply (49) (50) x k ρ k x k βρ k k σ 1 k σ 2

24 PERFORMANCE OF MQNETs 1407 We now show that f LP[dv] has a feasble soluton wth postve ε, then LP[dm] also has a feasble soluton wth postve γ. (51) Proposton 8. L 1 k = Let xε be a feasble soluton to LP[dv]. Let also k σ 1 k k x k L 2 k = k σ 2 k k x k L j = ( L j 1 Lj N) j = 1 2 γ = λε Then L 1, L 2, γ, V = 0, W = 0 s a feasble soluton to LP[dm]. In partcular, f ε s postve then γ s also postve. Ths soluton satsfes L j k Lj k whenever k k. For the proof, see Appendx B. The connecton between the lnear program LP[dv] and ρ < 1 s establshed n [8] by usng network flow technques. Specfcally, Secton 5 of [8] shows that f there exsts a β such that (52) 1 ρvsσ 1 ρvsσ 2 >β> ρvs σ 1 1 ρvs σ 2 for every nontrval strctly separatng set S, and every nontrval separatng set S, then there exsts a feasble soluton ε = εβ > 0 to the lnear program LP[dv] wth (53) εβ mn { 1 ρ σ1 1 ρ σ2 β1 ρvs σ 2 ρvs σ 1 1 ρvsσ 1 βρvsσ 2 } (the mnmum s over all strctly separatng sets S and all separatng sets S). In the next lemma, whch s a slght modfcaton of the argument n Secton 6 of [8], we establsh the connecton between the lnear program LP[dv] and the condton ρ < 1. Lemma 2. Suppose ρ < 1. Then, there exsts a β and a feasble soluton xε of LP[dv] such that ε 1 ρ. For the proof, see Appendx B. We now have all the necessary tools to state and prove the man result of ths secton. Theorem 5. We consder a reentrant lne queueng network wth two statons σ 1 σ 2, arrval rate λ and servce rates µ 1 µ 2 µ N. Class 1 s assumed to belong to staton σ 1.Ifρ < 1, then the followng upper bounds hold on the

25 1408 D. BERTSIMAS, D. GAMARNIK AND J. N. TSITSIKLIS steady-state number of customers n the network: { N P ρ σ 1+ Q t B m 1 + ρ + 2ρ } N =1 ρ 1 =1 1 + N =1 ρ 1 (54) ( ρ + ρ N =1 ρ 1 ) m ρ + ρ N =1 ρ 1 and (55) { N P ρ σ 1+ =1 le 2 +1 ρσ 2+ for all m = 0 1 2, where Also (56) and (57) [ ] N E ρ σ 1+ Q t =1 In partcular, (58) [ N E Q t B m 1 + ρ + 2ρ } N =1 ρ n =1 ρ 1 ( ρ + ρ N =1 ρ 1 ρ σ 1+ = ρ + ρ N =1 ρ 1 B = le 2 +1 ρσ 1+ ) m 64Nρ N =1 ρ N =1 ρ 1 1 ρ 2 64Nρ N =1 ρ N =1 ρ 1 1 ρ + 21+ρ +2ρ N =1 ρ N =1 ρ 1 1 ρ ] Q t [ ] N E Q t =1 64Nρ N =1 ρ N =1 ρ 1 1 ρ ρ + 2ρ N =1 ρ N =1 ρ 1 1 ρ ( = O 1 1 ρ 2 Remarks. () The bounds are asymmetrcwth respect to the order of the statons. If class 1 belongs to Staton σ 2, the correspondng bounds are obtaned trvally by exchangng σ 1 and σ 2. () The condton ρ < 1 guarantees that ρ + ρ N =1 ρ ρ + ρ < 1 N =1 ρ 1 As a result, the bounds of the theorem are nontrval and, n partcular, are of the geometrctype. )

26 PERFORMANCE OF MQNETs 1409 For the proof of Theorem 5, see Appendx B. 6. Extensons and examples. We apply the results obtaned n the prevous secton to several specfc examples. Feedforward networks. We start wth a defnton. Defnton 6. A multclass queueng network C s defned to be feedforward (acyclc) f k σ j1 k + 1 σ j2 mples j 1 j 2. In words, customers vst the statons n nondecreasng order. The stablty of feedforward networks under the usual load condtons ρ σj < 1 was proved n [6] and [11]. Snce the stablty condtons for feedforward networks are gven explctly as ρ σj < 1, then t s natural to expect that performance bounds can be constructed, whch are fnte whenever the load condton ρ σj < 1 holds. In the next theorem we wll show that ths s the case. Theorem 6. Consder a feedforward multclass queueng network C operatng under an arbtrary work-conservng polcy π. Let ρ mn = mn k ρ k, ρ = max j ρ σj, and let ρ σ j+ k be defned by (20). The followng upper bounds hold on the steady-state number of customers n the network: { L j P Qt B } ( ρ 2ρ +1/2 λ mn 1 ρ m +1/2 λ mn 1 ρ ) m ρ +3/4 λ mn 1 ρ for all m = 0 1 2, and all j = 1 2J, where ( ) L j k = ρ j 1 mn ρ σ j+ J 1ρ k L j = L j k k (59) ( ) ρ J 1 λ mn = mn λ J 1ρ mn (60) B = 16NJ 2 J 1ρ + λ mn 1 ρ 3 λ 2 mn 1 ρ 2 Also, E L j Qt 16NJ 2 J 1ρ + λ mn 1 ρ 3 λ 2 mn 1 + 8ρ + λ mn 1 ρ 2 ρ 2 λ mn 1 ρ for all j = 1 2J. In partcular, [ ] N ( ) 1 E Q t = O 1 ρ =1 2 For the proof, see Appendx B.

27 1410 D. BERTSIMAS, D. GAMARNIK AND J. N. TSITSIKLIS Remark. The bound [ ] N E Q t =1 s an mprovement on the bound [ ] N E Q t =1 ( = O ( = O 1 1 ρ ρ J 0 obtaned n [15] usng quadratclyapunov functons. Here J 0 denotes the number of statons wth traffcntensty equal to ρ (the number of bottleneck statons). Note that t s possble to have J 0 = J. The Lu Kumar network. Consder the network n Fgure 3. The network s descrbed by the followng parameters: =λ =µ 1 µ 2 µ 3 µ 4 ρ = λ/µ = ρ σ1 = ρ 1 + ρ 4 ρ σ2 = ρ 2 + ρ 3 We have ρ σ 1+ 1 = ρ 1 + ρ 4 and ρ σ 1+ = ρ 4 for = The set of excursons n ths network s gven as e 1 = 1e 2 = 2 3, e 3 =4. Then ρ σ 1+ le 2 +1 = ρ 4. The only two nontrval separatng sets n ths network consst of the sngle excursons e 1 =1 and e 3 =4. The separatng set e 1 wth ts set of classes Ve 1 =1 has ρvse k =0for all k = For the separatng set e 3, wth ts set of classes Ve 3 =2 4, we have ρve 3 e 1 =ρ 2 + ρ 4 ) ) ρve 3 e 2 =ρ 1 + ρ 4 However, the second term s equal to ρ σ2 < 1. We conclude that ρ = maxρ 2 + ρ 4 ρ σ1 ρ σ2 Assume now, n addton, that ρ 2 ρ 3 and ρ 4 ρ 1. Then ρ = ρ 2 +ρ 4. Applyng now Theorem 5, we obtan the followng result. Proposton 9. In a Lu Kumar network satsfyng ρ 2 ρ 3 ρ 4 ρ 1 and ρ 2 + ρ 4 < 1, the followng upper bounds hold for any work-conservng polcy w and the correspondng statonary probablty dstrbuton π: P π {ρ 1 + ρ 4 Q 1 t+ρ 4 Q 2 t+ρ 4 Q 3 t+ρ 4 Q 4 t B m 1 + ρ 2 + ρ 4 + 2ρ 2 + ρ 4 4 =1 ρ =1 ρ 1 ( ρ 2 + ρ 4 +ρ 2 + ρ 4 4 =1 ρ ρ 2 + ρ 4 +ρ 2 + ρ 4 4 =1 ρ 1 } ) m

28 PERFORMANCE OF MQNETs 1411 and P π {ρ 4 ρ 2 + ρ 3 Q 1 t+ρ 4 ρ 2 + ρ 3 Q 2 t+ρ 4 ρ 3 Q 3 t B m 1 + ρ 2 + ρ 4 +ρ 2 + ρ 4 4 =1 ρ =1 ρ 1 ( ρ 2 + ρ 4 +ρ 2 + ρ 4 4 =1 ρ 1 ) m ρ 2 + ρ 4 +ρ 2 + ρ 4 4 =1 ρ 1 for all m = 0 1 2, where Also, and B = 256ρ 2 + ρ =1 ρ =1 ρ 1 1 ρ 2 ρ 4 2 E π [ ρ1 + ρ 4 Q 1 t+ρ 4 Q 2 t+ρ 4 Q 3 t+ρ 4 Q 4 t ] 256ρ 2 + ρ =1 ρ =1 ρ 1 1 ρ 2 ρ ρ 2 + ρ 4 + 2ρ 2 + ρ =1 ρ 1 1 ρ 2 ρ 4 } 4=1 ρ 1 2 [ E π ρ4 ρ 2 + ρ 3 Q 1 t+ρ 4 ρ 2 + ρ 3 Q 2 t+ρ 4 ρ 3 Q 3 t ] 256ρ 2 + ρ =1 ρ =1 ρ 1 1 ρ 2 ρ ρ 2 + ρ 4 + 2ρ 2 + ρ 4 4=1 ρ =1 ρ 1 1 ρ 2 ρ 4 Smlar bounds can be obtaned for the cases ρ 1 >ρ 4 or ρ 3 >ρ 2. Note that the result above mples [ ] 4 ( ) 1 E π Q t = O 1 ρ 2 ρ 4 2 =1 Contrast ths wth the lower bounds (24). 7. Conclusons. We have proposed a general methodology based on Lyapunov functons for the performance analyss of nfnte state Markov chans and appled t specfcally to multclass queueng networks wth exponentally dstrbuted nterarrval and servce tmes. We have proved that whenever some pecewse lnear Lyapunov functon s a wtness for the global stablty of the network, certan fnte upper bounds can be derved on the probablty dstrbuton and expectaton of queue lengths. Lower bounds are also constructed by means of lnear lower Lyapunov functons. Thus, for certan computable constants 0 <c 1 <c 2 < 1, we have constructed bounds of the form c m 1 PL m cm 2 2

29 1412 D. BERTSIMAS, D. GAMARNIK AND J. N. TSITSIKLIS wth L the total number of customers n the network. These bounds hold unformly under any work conservng polcy. The lower bounds are extended to prorty polces as well. Snce pecewse lnear Lyapunov functons provde an exact test for stablty of flud networks wth two statons, our bounds for two-staton networks are fnte f and only f the correspondng flud network s globally stable. Whether ths remans true for the orgnal stochastc network remans to be seen. For reentrant lne-type queueng networks wth two processng statons, the constants c 1 and c 2 can be expressed explctly n terms of traffc ntenstes (actual and vrtual) of the network. Closed form bounds were also constructed on the total expected number of customers n the network. In partcular, we have proved that ( EL =O 1 1 ρ 2 where ρ s a maxmal (actual or vrtual) traffc ntensty. It would be nterestng to strengthen ths result, perhaps by removng the exponent 2. The results obtaned here are the frst ones that establsh exponental upper and lower bounds on the dstrbuton of queue lengths n networks of such generalty. Prevous results on performance analyss of multclass queueng networks can n general acheve only numercal bounds and only on the expectaton of queue lengths. ) APPENDIX A The key to our analyss s a modfed Lyapunov func- Proof of Lemma 1. ton, defned by (61) x =maxc x for some c R +, and the correspondng equlbrum equaton (62) E π [ Xt ] = Eπ [ Xt + 1 ] We can rewrte (62) as (63) E π [ Xt + 1 Xt ] = 0 We frst prove (14). Let us fx c as n the statement of the lemma, and consder the functon x ntroduced n (61). Snce E π Xt s fnte and π s a statonary dstrbuton, we can rewrte (63) as (64) πx ( E Xt + 1 Xt =x x ) = 0 x