Stability analysis of parallel server systems under longest queue first

Transcription

1 Mah Meh Oper Res (2011) 74: DOI /s ORIGINAL ARTICLE Sabiliy analysis of parallel server sysems under longes queue firs Golshid Baharian Tolga Tezcan Received: 15 November 2010 / Acceped: 10 June 2011 / Published online: 21 July 2011 Springer-Verlag 2011 Absrac We consider he sabiliy of parallel server sysems under he longes queue firs (LQF) rule. We show ha when he underlying graph of a parallel server sysem is a ree, he sandard nominal raffic condiion is sufficien for he sabiliy of ha sysem under LQF when inerarrival and service imes have general disribuions. Then we consider a special parallel server sysem, which is known as he X-model, whose underlying graph is no a ree. We provide addiional drif condiions for he sabiliy and ransience of hese queueing sysems wih exponenial inerarrival and service imes. Drif condiions depend in general on he saionary disribuion of an induced Markov chain ha is derived from he underlying queueing sysem. We illusrae our resuls wih examples and simulaion experimens. We also demonsrae ha he sabiliy of he LQF depends on he ie-breaking rule used and ha i can be unsable even under arbirary low loads. Keywords Sabiliy Longes queue firs Parallel server sysems Fluid model 1Inroducion Scheduling policies have an immense effec on he performance of queueing sysems. Policies which use minimal informaion abou he sae and he parameers Research suppored by NSF Gran CMMI G. Baharian Indusrial and Enerprise Sysems Eng., Universiy of Illinois a Urbana-Champaign, Urbana, IL, USA gbahari2@illinois.edu T. Tezcan (B) Simon Graduae School of Business, Universiy of Rocheser, Rocheser, NY, USA olga.ezcan@simon.rocheser.edu

2 258 G. Baharian, T. Tezcan of he underlying sysem when making scheduling decisions have been exensively researched. Since in many applicaions arrival raes (and also someimes processing imes) canno be deermined accuraely in advance, policies ha do no need arrival rae informaion are especially valuable for applicaions. One such policy, known as he longes queue firs (LQF) policy, has araced a lo of aenion in applicaions and in he lieraure. Under LQF an idling server serves nex a cusomer from he longes queue among hose queues ha i can serve. Therefore, only real ime queue lengh informaion is needed o implemen i. I is well known ha he nominal raffic condiion is no sufficien for he sabiliy of a queuing sysem. Despie he populariy and desirable properies of LQF, necessary and sufficien condiions for he sabiliy of queuing sysems under LQF have no been esablished. In Dimakis and Walrand (2006) andkumar e al. (2002), sufficien condiions for he sabiliy of LQF are proved in a cerain class of sysems. However, i is no clear if hese condiions are necessary and if hey hold for more general sysems. Inconclusiveness of he lieraure on he sabiliy of LQF is somewha surprising in spie of is populariy and he availabiliy of he fluid model ools ha have been developed in recen lieraure, see Dai (1999), Bramson (2008)andhereferencesherein. One of he reasons, as we illusrae in his paper, urns ou o be he fac ha he sandard fluid limi approach is no sufficien in his case o idenify he sabiliy condiions. In his paper we focus on parallel server sysems and firs consider he case when he underlying graph of such a sysem is a ree. Using he sandard fluid model framework and a represenaion presened in Bell and Williams (2005) for parallel server sysems wih a ree srucure, we show ha he nominal raffic condiion is necessary and sufficien for he sabiliy of hese sysems when inerarrival and service imes have general disribuions. We nex consider a special parallel server sysem known as he X-model in he lieraure, see Fig. 1a, whose underlying graph is no a ree. For hese sysems we prove necessary and sufficien condiions (in addiion o he usual raffic condiion), which we refer o as drif condiions, for sabiliy, when inerarrival and service imes are exponenial, based on he saionary disribuion of an induced Markov chain. The induced Markov chain is derived from he Markov chain model of he underlying queueing sysem. In his sense our analysis is similar o ha in Fayolle e al. (1995). By using he saionary disribuion of he induced Markov chain we manage o characerize he amoun of ime each server spends on each cusomer class on he average, which canno be found using he sandard fluid model approach. This is reminiscen of he averaging principle sudied in Coffman e al. (1995), Perry and Whi (2010), Hun and Kurz (1994). For illusraive purposes, we also show via simulaions ha he drif condiion may also be used o characerize he sabiliy region of oher sysems by considering XN-model sysems, see Fig. 1b. To prove he sufficiency of he drif condiion, we show ha if he sysem is unsable hen i behaves exacly like he induced Markov chain. Using his fac and he drif condiion we prove ha queue lenghs mus visi a cerain sae infiniely ofen, conradicing insabiliy. To prove he necessiy of he drif condiion, we use a resul in Meyn (1995)basedonfluidmodels.Theconnecion beween hesabiliyofhefluid model and he sabiliy of he associaed queueing sysem is well known (Dai 1995). However i is also shown ha he converse is no necessarily rue, see Bramson (2008) and he references here in. We show ha every fluid model soluion saring wih a

3 Sabiliy analysis of parallel server sysems 259 (a) X-model Fig. 1 Two parallel server sysems (b) XN-model posiive amoun of fluid is increasing afer a cerain ime, before which i is bounded from below. This implies insabiliy by Theorem 3.2 in Meyn (1995). Our analysis parallels he analysis in Tezcan (2010)whereparallelserversysemsundersaicprioriy policies are analyzed and i can be exended o cover he case where inerarrival and service imes have phase-ype disribuions as in ha paper. In addiion, i can be viewed as a mehod o characerize fluid limis in more deail han he sandard fluid model approach. Our analysis should also be usable in oher sysems when his is necessary. Oher echniques have also been used o obain beer characerizaion of fluid limis for various sysems in Foss and Kovalevskii (1999) and Solyar and Ramakrishnan (1999). To he bes of our knowledge, he sabiliy of LQF is sudied in only wo papers. In Dimakis and Walrand (2006), he sabiliy of he LQF policy for a generalized swich model is considered and new sufficien condiions for LQF o be hroughpu opimal are idenified. I is claimed ha deerminisic fluid analogs are no adequae o analyze he sabiliy of LQF since variabiliy in sochasic arrivals may affec sabiliy under LQF. Diffusion-scaled properies of he sample pahs wih he fluid limi framework are combined o esablish sufficien condiions for he sabiliy of hese sysems. In Kumar e al. (2002) heissueofraesabiliyofsablemarriage(sm)algorihmsin Inpu-queued (IQ) swiches (which is similar o LQF in cerain seings) in a discree ime framework, using fluid model echniques, is addressed. Necessary condiions on he average rae of arrival for cerain classes of sysems for sabiliy are obained. The sysems we consider have differen feaures han hose in Dimakis and Walrand (2006) and Kumar e al. (2002)andsoiisnopossibleocompareourresulsdirecly.Also,we esablish under some addiional condiions ha he drif condiion is necessary for sabiliy. In a recen paper (Solyar and Yudovina 2010) parallel server sysems wiharee srucure is considered in a many-server asympoic regime under he longes-queue frees-server rule. Under his policy, i is proved ha he many-server fluid limi (which is differen from he fluid limis we consider here) of he sysem may be unsable. The res of his paper is organized as follows. In Sec. 2,wepresenhegeneralparallel-server queueing sysem model along wih is corresponding fluid model equaions.

4 260 G. Baharian, T. Tezcan We also illusrae why in general he fluid model equaions are no sufficien o analyze he sabiliy of LQF in hese sysems. We prove he sabiliy of parallel server sysems wih a ree srucure under LQF in Sec. 3.InSec.4 we sudy he sabiliy of X-model sysems under LQF. We presen examples and numerical resuls in Sec. 5. 2Thequeueingmodel,sabiliyandfluidmodels We consider a parallel queueing sysem ha consiss of several cusomer classes and servers. Le J ={1,...,J} and I ={1,...,I } denoe he se of servers and cusomer classes, respecively, of his generic queueing sysem. Servers can possibly serve more han one cusomer class, and jobs (or cusomers) are ordered wihin he queue according o heir arrival imes wih he earlies arrival being a he head of he line. The se of servers ha can serve cusomer class i cusomers and he se of queues ha server j can handle are denoed by S(i) and C( j),respecively. When a class i cusomer arrives o he sysem, i proceeds o service if here is an available server in se S(i).Eachcusomereneringhesysemrequiresasingleservice before leaving he sysem, hence here is no re-rouing. In addiion, once a cusomer has commenced service a a server, i remains here unil is service is complee, i.e., preempions are no allowed. Service of a class i cusomer by a given server j is called aprocessingaciviy denoed by {ij}. Leλ i and µ ij denoe he arrival rae o class i and he service rae of a class i cusomer by server j,respecively. We use X i () and Q i () o denoe he number of class i cusomers in he sysem and in he queue a ime, respecively,andsex i = (X i () : 0) and Q i = (Q i () : 0). Also,leZ j () C( j) {0} denoe he saus of server j a ime ; Z j () = i, ifserver j is serving a class i cusomer and Z j () = 0ifserver j is idle a ime.inaddiion,let ij () and Y j () denoe he oal ime server j spends working on class i cusomers and he oal ime i idles up o ime. Le A i () denoe he number of arrivals by ime. WedenoebyV ij (n) he oal service ime of he firs n class i cusomers who receive service by server j. We assume ha he inerarrival and service imes are given by sequences of i.i.d. random variables s {u i (n) : n 1} and {v ij (n) : n 1}, respecively,fori I, j S(i). Le S ij () denoe he number of class i service compleions by server j,ifheserverworks on class i cusomers for ime unis, ha is; S ij () = sup{n 0 : V ij (n) }. The number of class i service compleions by server j by ime is given by S ij (T ij ()). We le X = (X i, A i, S ij, Z j, T ij, Y j, j J, i I) denoe he parallel queueing sysem and noe ha i saisfies he following equaions for 0; X i () = X i (0) + A i () Q i () = X i () j S(i) j S(i) S ij (T ij ()), for all i I (2.1) 1 { Z j () = i }, for all i I (2.2)

5 Sabiliy analysis of parallel server sysems 261 T ij () + Y j () =, for all j J (2.3) i C( j) I {Z j (s )=i} 0 T ij () = ds, for all j J and i I (2.4) Y j () can only increase when Q i () = 0, for all j J. (2.5) i C( j) The conrol policy ha we focus on is commonly referred o as he LQF policy. Under his policy, a server, say j, ha has jus finished service, serves he longes waiing cusomer in he longes queue among hose in C( j). When he queue lenghs are equal a a decision insan, a ie breaking rule mus be used, and we mainly focus on saic-prioriy ype ie-breaking rules. If all queues are empy, he server remains idle unil a new cusomer arrives. How arriving cusomers are roued o available serves does no have an effec on sabiliy (however i migh affec he oher performance merics). Since we assume ha he conrol policy is LQF, he following equaion is added o he above se of equaions: B ij ( 2 ) B ij ( 1 ) = 0 for j J, i C( j), if Q i (s) <max k C( j) Q k (s) for all s [ 1, 2 ] and 1 2,whereB ij () is he number of class i cusomers whose service sared by server j before ime. We laer consider wo special parallel server sysems, he X- and XN-models, in his paper. An X-model consiss of 2 cusomer classes and 2 servers, hence I ={1, 2} and J ={1, 2} and each server can serve boh cusomer classes, S(1) = S(2) = {1, 2}, C(1) = C(2) ={1, 2}, seefig.1a. We also consider a model wih 3 cusomer classes and 3 servers, I ={1, 2, 3} and J ={1, 2, 3}, whichwereferoas he XN-model, see Fig. 1b. The firs wo servers can serve class 1 and 2 cusomers and he hird server can serve class 2 and 3 cusomers, hence S(1) ={1, 2}, S(2) = {1, 2, 3}, S(3) ={3}, C(1) = C(2) ={1, 2}, andc(3) ={2, 3}. The main goal of his paper is o analyze he sabiliy of LQF in a parallel server sysem. When inerarrival and service imes have general disribuions, a queueing sysem is said o be sable if X is posiive Harris recurren, see Dai (1999). When X can be modeled as a coninuous ime Markov chain, (posiive) Harris recurrence is equivalen o (posiive) recurrence. In his case, he queueing sysem is said o be sable if X is recurren and unsable if X is ransien. The nominal raffic condiion can be formulaed using a linear program which is commonly known as he saic planning problem (SPP). In our seing he SPP is given by min ρ s.. j S(i) µ ij x ij = λ i, for i I,

6 262 G. Baharian, T. Tezcan i C( j) x ij ρ, for j J, x ij 0 for j J, i C( j), where x ij can be inerpreed as he long-run proporion of ime server j spends serving aclassi cusomer. The objecive of his LP is o minimize he nominal uilizaion of he busies server. Le ρ denoe he opimal objecive funcion value of he SPP. I is well-known ha ρ 1isanecessarycondiionforsabiliy(Dai 1999), herefore he sabiliy analysis is ineresing only when his condiion holds. 2.1 Fluid models and sabiliy of queueing sysems Fluid models are proved o be a powerful ool in he analysis of sabiliy in queuing sysems, see Dai (1999) and Bramson (2008). In his secion, we provide he basics of fluid models ha we laer use in he proof of our resuls. They consis of a se of deerminisic equaions obained from he queueing nework equaions based on srong law of large numbers. The fluid model of a parallel server sysem under LQF consiss of he following equaions, see Dai (1999) formoredeails. Q i () = Q i (0) + λ i i C( j) j S(i) µ ij T ij () for all i I, (2.6) T ij () + Ȳ j () = for all j J, (2.7) Ȳ j () can only increase when i C( j) Q i () = 0. (2.8) Also, X i () = Q i () for all i I and 0. Under LQF he following addiional policy-specific fluid model equaion also holds; i M j T ij () = 1 for all j J, (2.9) where M j = { i : i C( j), Q i () = max k C( j) Q k () }.Wenoehaallhefluid model soluions are differeniable a.e. (see Dai 1999). Equaion (2.6) saeshahe fluid level in queue i a ime is equal o he iniial fluid level plus he oal number of arrivals o class i minus he oal number of deparures from class i. Also,Eq.(2.7) indicaes ha a server can eiher be idle or busy. Equaion (2.8)isanon-idlingpolicy which will no allow he servers o be idle when cusomers are presen in he queue. Finally, Eq. (2.9)saeshaaserverwillonlyservicehosecusomershaarewaiing in he longes queue. The sabiliy of a queueing sysem is closely conneced wih he sabiliy of is fluid model. To explain he deails, we nex inroduce some erminology. A fluid model is said o be sable if here exiss δ>0suchhaforanyfluidmodelsoluionwih

7 Sabiliy analysis of parallel server sysems 263 Q(0) =1, Q() =0for δ,where is he sum norm. I is shown in Dai (1995) under some general condiions ha if he fluid model of a queueing sysem is sable hen he underlying queueing sysem is Harris recurren. I is also possible o show he insabiliy of a queueing sysem by showing ha he corresponding fluid model is unsable in a cerain sense. A fluid model is said o be weakly unsable if here exiss 0 > 0suchha Q() >0for 0,foranyfluidmodelsoluionwih Q(0) =0. Afluidmodelissaidobeunsableifhereexissɛ>0suchha Q() ɛ, for 0,for any fluid model soluion wih Q(0) = 1. The underlying queueing sysem is unsable if he fluid model is unsable (see Meyn 1995) orweaklyunsable(see Dai 1996). To prove insabiliy, we show ha he fluid model is unsable, in he laer sense, when he drif condiion does no hold. 2.2 Analysis wih he radiional fluid model Before we proceed wih our analysis, we show ha he radiional fluid model approach is no sufficien o esablish sabiliy/insabiliy of LQF in parallel server sysems in general. We noe ha he sabiliy or insabiliy of he underlying queueing sysem is only implied by he sabiliy/insabiliy of all he fluid model soluions. However, he soluions of he fluid model Eqs. (2.6) (2.9)presen boh behavior as we illusrae nex. Consider he X-model. The fluid model equaions of his sysem under LQF consis of (2.6) (2.9). We can rewrie (2.9) as follows: T 11 () = T 12 () = 1 T 21 () = T 22 () = 1 if Q 1 () > Q 2 (), if Q 1 () < Q 2 (), which implies ha if he number of cusomers (in erms of fluid level) in one queue is sricly greaer han he oher one, boh servers will serve he cusomer class wih he longer queue. In order o prove sabiliy using fluid limis one needs o show ha he fluid model is sable. We noe ha he deparure rae from a queue in he fluid model is given by j S(i) µ ij T ij (),see(2.6). Therefore o show sabiliy one needs o deal wih T ij s, bu he fluid model equaions do no provide any informaion abou hem when Q 1 () = Q 2 ().Tobemorespecific,oncehequeuelenghsbecomeequaland are posiive, he only informaion abou T ij s wih which he fluid model equaions provide is T 11 () + T 21 () = 1, T 12 () + T 22 () = 1, wih T ij s being non-negaive, where f () = df()/d for any funcion f ha is differeniable a. To furher illusrae his issue, consider an X-model wih λ 1 = 0.9,λ 2 = 0.9,µ 11 = µ 22 = 1andµ 21 = µ 12 = 0.1. Assume ha Q 1 (0) = Q 2 (0) = 0.5. The following soluions boh saisfy he fluid model equaions as long as Q i () >0; for 0

8 264 G. Baharian, T. Tezcan 1. T 11 () = 0.5, T 21 () = 0.5, T 12 () = 0.5, T 22 () = 0.5, 2. T 11 () = 0.9, T 21 () = 0.1, T 12 () = 0.1, T 22 () = 0.9. In he second soluion he deparure rae from each queue (ha is equal o 0.91) is greaer han he arrival rae o each queue. Therefore, Q 1 and Q 2 become zero afer afinieimeinhiscase,saisfyinghecondiionsforasablefluidmodelsoluion. On he oher hand, in he firs soluion, he deparures raes from queues 1 and 2 are equal o 0.55, hence he queue lenghs are increasing. Thus, he fluid model has a leas wo soluions wih differen sabiliy behaviors and so we canno esablish he sabiliy of he X-model only using (2.6) (2.9) due o he fac ha he fluid model does no deermine wha happens wih T ij s when Q 1 () = Q 2 (). We will show ha he rue values of T ij s can be found using he saionary disribuion of he induced Markov chain. 3Sabiliyofparallelserversysemswihareesrucure In Sec. 2.2,wehavedemonsraedhaheradiionalfluidmodelanalysiscannobe used o esablish he sabiliy of LQF in parallel server sysems in general. However, when he srucure of he queueing sysem saisfies a cerain condiion, i is possible o prove sabiliy under LQF using his approach. In his secion we provide he deails. Consider he graph G in which servers and queues form he nodes, and edges beween nodes are given by aciviies. Noe ha he graph of any queueing sysem is specified by he se of queues, servers and aciviies. Hence, given a se of queues, A, servers, B,and aciviies,c,we someimes wrie G(A, B, C) o denoe he underlying graph when we wan o make his dependence explici. A graph G is said o be a ree if i has no cycles, i.e., if for each pair of nodes, here is a mos one pah (consising of edges) joining ha pair, see, for example, he N-model sudied in Bell and Williams (2001) and Tezcan and Dai (2010). We assume ha he disribuions of inerarrival and service imes are unbounded and spread ou, see Dai (1995). This assumpion holds, for example, if hey are assumed o have phase-ype disribuions (or more specifically exponenial disribuions). The main resul of his secion follows. Theorem 3.1 Consider a parallel server sysem under LQF whose underlying graph G is a ree. If he SPP has an opimal soluion wih ρ < 1, henheparallelserver sysem is sable, i.e., X is posiive Harris recurren. The proof of Theorem 3.1 follows from he fluid model analysis using he Lyapunov funcion f : R I R defined by f ( Q()) = max i Q i (). (3.1) Clearly f is a valid Lyapunov funcion, see Definiion in Dai (1999), hence in order o prove he sabiliy of he underlying sysem, i is enough o prove by Lemma in Dai (1999) ha

9 Sabiliy analysis of parallel server sysems 265 f ( Q()) < 0, (3.2) whenever f ( Q()) > 0, for any regular poin of f.(a poin is said o be a regular poin of a funcion f if f is differeniable a ha poin.) We proceed o prove (3.2) nex. Proposiion 3.2 Le X denoe a fluid model soluion for a parallel server sysem whose underlying graph is a ree and assume ha he SPP has an opimal soluion wih ρ < 1.Thenforanyregularpoinof f definedby(3.1),hereexissɛ>0 such ha if f ( Q()) > 0. f ( Q()) < ɛ, (3.3) Proof Consider he fluid model soluion of a parallel server sysem under LQF and assume ha he SPP has an opimal soluion wih ρ < 1. Also assume f ( Q()) > 0 for some regular poin of f and ha he underlying graph G is a ree. Le A ={k I : Q k () = max i I Q i ()} denoe he se of indices ha correspond o he queues wih maximum lengh a ime and B := { j J : C( j) A = }denoe all he servers ha can serve hose queues. By (2.9), T ij () = 0, for all j B and i / A.Hence,wecanfocusonhesysemhaconsissofqueues in A,serversinB and aciviies in C = {{ij}:i A, j B },andproveha Q i () <0foralli A.Weassumewihoulossofgeneraliyhaheunderlying graph of his sysem G(A, B, C ) is conneced, as oherwise he analysis below can be carried ou for each disjoin componen. Wih a sligh abuse of noaion, we use G insead of G(A, B, C ) for noaional simpliciy. To faciliae he proof we need o firs represen he graph G in a special ree-like srucure as in Bell and Williams (2005). To ha end, we sar wih one of he queues ha has more han one server linked o i, say class i,andsupposeha i is he roo node (level 1). Now we add all he servers j S(i ) o he nex level below i (level 2) using aciviies {i j} as links. Then we place all he queues ha can be served by hese servers in level 3 (below level 2) using appropriae links, and so on, unil no more queues or servers are lef. Because G is a ree, hen each node in his srucure can have a mos one paren node; a node i is conneced in a level wih a lower index. In addiion, we use he erm branch o indicae a subree whose opmos node is locaed in level 2. No link can go ou from one branch o he oher since i forms a cycle; herefore, he branches in his srucure are separaed and no conneced o each oher because G is a ree. Suppose ha S(i ) =mor equivalenly he ree has m branches. We prove he resul only for a sysem where all he servers can serve more han one cusomer class. The proof for he queueing sysems no saisfying his propery follows in a similar fashion. Under his assumpion, he lowes (also referred o as he erminal) level of

10 266 G. Baharian, T. Tezcan he ree consiss of only queues. We sar he argumen from he lowes level (say level L)ofanarbirarybranchandproveheresulbyconradicion. Assume on he conrary ha on a regular poin of f f ( Q()) 0, (3.4) which implies Q i () 0foralli A by Lemma in Dai (1999). We show below ha Q i () <0, hence a conradicion. For any regular poin of f,bylemma2.8.6indai (1999), f () = Q i () for all i A where Q i () = λ i j S(i) µ ij T ij (). Fix a branch and consider a class i L in he lowes level of his branch. Denoe by j L 1 is paren node and noe ha i is unique, see Fig. 2 for an illusraion of he branches. By (3.4), Q il () = λ il j S(i L ) µ il j T il j() = λ il µ il j L 1 T il j L 1 () 0, (3.5) where he second equaliy follows from he way he queues are indexed and he inequaliy follows from (3.4). Fig. 2 Srucure of he firs branch L-3 L-2 L-1 L

11 Sabiliy analysis of parallel server sysems 267 Because SPP has an opimal soluion wih ρ < 1(andi L has only one paren node j L 1 ), we have λ il µ il j L 1 x i L j L 1 = 0whichimplies T il j L 1 () x i L j L 1. (3.6) Noe ha his inequaliy holds for all he queues i L whose paren node is j L 1. Also, because he SPP has an opimal soluion wih ρ < 1 i C( j L 1 ) x ij L 1 < 1. (3.7) Le i L 2 denoe node j L 1 s paren node and noe again ha i is unique because G is a ree. Because of our assumpion f ( Q()) > 0, we have i C( j L 1 ) T ijl 1 () = 1, (3.8) which along wih (3.6) and(3.7) impliesha T il 2 j L 1 () >x i L 2 j L 1. (3.9) This resul holds for any server j L 1 locaed in level L 1whichisconnecedo queue i L 2.Hence,by(3.4), we have for he paren node j L 3 of he queue i L 2 ha T il 2 j L 3 () <x i L 2 j L 3. Le j 2 denoe he roo node server of his branch. Coninuing in his fashion, one can prove ha for any queue i 3 whose paren node is j 2 we have T i3 j 2 () <x i 3 j 2, (3.10) which in urn implies ha T i j 2 () >x i j 2. The above resul holds for he opmos servers of all he oher m 1branches. Because S(i ) is he se of all he servers in level 2 or equivalenly he opmos servers of all he branches, i follows ha Q i () = λ i j S(i ) <λ i j S(i ) µ i j T i j() µ i j xi j = 0, (3.11) which conradics wih (3.4).

12 268 G. Baharian, T. Tezcan 4SabiliyanalysisofX-modelsysems In his secion we focus on X-model sysems under LQF. We noe ha he underlying graph of X-model sysems is no a ree. Wealsoresricouraenionohecase when inerarrival and service imes are exponenially disribued bu our resuls can be exended o he case when hey have phase-ype disribuions as in Tezcan (2010), see Remark 4.2. As illusraed in Sec. 2.2, he sandard fluid model approach is no sufficien o analyze he sabiliy of hese sysems, mainly because T ij s canno be deermined when wo or more queues are equal in he fluid model. The approach in his secion allows us o overcome his problem. 4.1 The induced Markov chain For our sabiliy analysis we need o consider a Markov chain ha is similar o bu differen from X. ThisnewMarkovchain,whichwereferoasheinducedMar- kov chain, is obained by assuming ha queues can assume negaive values (hence servers never idle) as we explain below. Alhough our focus is on X-model sysems, we give he descripion of he induced Markov chain for a general sysem. We laer illusrae ha induced Markov chains can also be used o idenify he sabiliy region of XN-model sysems. Le i = { i () : 0} and = { i : i = 1,...,I 1}. Inheinducedchain, i = { i () : 0} can be hough of as corresponding o he difference beween Q i+1 and Q 1 in he original sysem bu here is an imporan difference; we assume queues are never empy. The induced Markov chain Y consiss of saes and he server occupaion process Z ={ Z j ; j J } and i evolves as follows; when server j idles, i checks all he queues in C( j) and serves he queue ha maximizes i, for i C( j). Wihheexcepionofhecasewhen1 C( j) and i < 0forall i C( j),heniservesclass1.ifipicksajobfromqueuei = 1, we se he new sae i = i 1andifipicksajobfromqueue1,wesehenewsae i = i + 1for all i = 1,...,I 1. We also se Z j = i in he former case and Z j = 1forhelaer. Service compleion raes are he same wih he original sysem, ha is, server j serves aclassi cusomer wih rae µ ij.arrivalsalsochangehesaeofhesysem.incase of a class i = 1arrival,whichhappenswihraeλ i (as in he original sysem), we updae he sae o i = i + 1. If a class 1 cusomer arrives o he sysem, we updae he sae o i = i 1, for all i = 1,...,I 1. The ineresed reader can ake a look a Sec. 5.1 and Fig. 3 o see he definiion of he induced chain for X-model sysems. Le Y = {, Z} denoe he induced Markov chain wih sae space S Y = (N + ) I 1 I Z, where N + denoes he se of nonnegaive inegers. We assume ha ς ij = lim T ( ) T 0 1 Z j () = i d T (4.1)

13 Sabiliy analysis of parallel server sysems 269 Fig. 3 Induced Markov chain for X-model under saic allocaion ie breaking rule exiss for any pair (i, j), i I, j J,andforanyiniialsaea.s.andhahelimiis aconsan.noeha hisassumpion isauomaically saisfied ify is posiive recurren (Dai and Meyn 1995). The quaniies ς ij s are also conneced o he fluid limis, hey can be hough of as being equal o T ij () when queues are equal. Our sabiliy resul depends on he quaniies ς ij s, however he induced Markov chain Y does no have o be posiive recurren in general. For an X-model he induced Markov chain Y is always posiive recurren under a naural condiion. Lemma 4.1 For an X-model sysem, he induced Markov chain Y is posiive recurren if λ i <µ i1 + µ i2, for i = 1, 2. (4.2) Proof Consider an X-model and assume ha (4.2) holds.we use a fluid model approach o show ha Y is posiive recurren. Le denoe a fluid limi, defined as in Secion 2.3 of Dai (1999), of. IfollowsasinDai (1999) ha is differen-

14 270 G. Baharian, T. Tezcan iable almos everywhere. And similar o Theorem in Dai (1999) onecanshow ha saisfies he following equaions. () = (0) + λ 2 λ µ 1 j T 1 j () j=1 2 µ 2 j T 2 j (), (4.3) T 1 j () = 1, if () <0, and T 2 j () = 1, if () >0, (4.4) T ij is non-decreasing and T ij (0) = 0, for i, j = 1, 2. (4.5) Therefore, if () >0, hen by (4.2) and(4.3) (4.5) Similarly if () <0, hen () = λ 2 λ 1 () = λ 2 λ 1 + j=1 2 µ 2 j < 0. (4.6) j=1 2 µ 1 j > 0. (4.7) This gives he desired resul by Lemma and Theorem in Dai (1999). (I is obvious ha he fluid limis of Z are equal o 0). Remark 4.2 In he nex secion, we show ha he sabiliy region of an X-model can be specified in erms of sysem parameers and ς ij. Similar resuls also hold when inerarrival and service imes have phase-ype disribuions. In ha case he definiion of he induced Markov chain, Y, has o be modified as follows. In order o model Y as a Markov process, one needs o keep rack of he phase of he cusomer receiving service (if any) from server j, inaddiionoheclassofhacusomerandhephase of he inerarrival ime for each cusomer class. When inerarrival and service imes are exponenial, each service and inerarrival ime only have a single phase, herefore i is no necessary o keep rack of his informaion. j=1 4.2 Sabiliy and insabiliy condiions In his secion we presen our main resuls for he sabiliy of he X-model sysems. We define he drif d i for each class by d i = λ i j S(i) µ ij ς ij, i I. (4.8) Theorem 4.3 (Sabiliy) An X-model sysem under LQF is sable if he SPP has an opimal soluion wih ρ < 1 and d i < 0 for all i I.

15 Sabiliy analysis of parallel server sysems 271 Proof The idea behind he proof is similar o ha of Theorem 3.2 in Tezcan (2010). Consider an X-model sysem and noe ha (4.1) holds by Lemma 4.1. Also assume ha d i < 0foralli I. Weproveheresulbyconradicion.AssumehaX is ransien. Le τ 0 denoe he number of imes he process X ever visis one of he saes wih boh queues empy. Since he underlying Markov chain is irreducible, ransience implies P x (τ 0 = n) > 0, (4.9) for some n < and iniial sae x.(sinceheunderlyingmarkovchainisirreducible he same resul holds for any iniial sae.) Le s n denoe he ime of he nh visi o one of he saes wih all he queues empy and se i equal o 0 if he chain visis hese saes less han n imes. Also consider he induced Markov chain Y defined as in 4.1 wih he iniial sae i (0) = Q i (s n ) Q 1 (s n ), i = 1,...,I 1and Z(0) = Z(s n ) (4.10) and he same residual service and inerarrival imes wih he original process a ime s n.noehabecauses n is a sopping ime for X and X is a Markov chain, Y is a Markov chain and (4.1) sill holds wih given iniial service and inerarrival imes. Therefore on {τ 0 = n}, T ij () lim = ς ij, (4.11) because X(s n + ) and Y( ) evolve exacly he same way on {τ 0 (ω) = n}. By (2.1), for i I, Q i () = Q i(s n ) = Q i(s n ) j S(i) + A i() + λ i j S(i) ( Si (T ij ()) j S(i) S ij (T ij ()) µ ij T ij () µ ij T ij () ( Ai () + λ i ) ). (4.12) By he funcional srong law of large numbers, see, for example, Chen and Yao (2001), and he fac ha T ij ()/ 1, ( ) Ai () λ i 0and ( Si (T ij ()) ) T ij () µ ij 0, as, a.s. (4.13)

16 272 G. Baharian, T. Tezcan This implies by (4.8), (4.11), and (4.12) on{τ 0 (ω) = n} ha lim sup Q i () < 0, which obviously conradics wih he fac ha Q i () 0forall 0. Hence (4.9) canno hold if d i < 0foralli I. Nex we focus on he necessiy of he drif condiion. Theorem 4.4 (Insabiliy) An X-model sysem is unsable if d i > 0 for a leas one i {1, 2}. Proof Consider an X-model sysem. Noe ha (4.1) holds by Lemma4.1. Assume ha d i > 0foraleasonei I.FixT > 0andle{x r } denoe a sequence of iniial saes such ha x r as r.asindai (1995), here exiss a furher subsequence denoed again by r such ha 1 x r Qxr ( x r ) Q() 0, (4.14) T as r for Q ha is coninuous and differeniable a.e. In addiion, using sandard argumens in Dai (1995), one can show ha Q saisfies (2.6) (2.9) for [0, T ]. In order o prove insabiliy, we use Theorem 3.2 in Meyn (1995). We need o show ha Q() ɛ, (4.15) for all iniial condiions wih Q(0) =1, some ɛ>0, and T. Le {x r } denoe a sequence of iniial saes and Q denoe a fluid limi such ha (4.14) holdsand Q(0) =1. Because of uniform convergence in (4.14), lim r xr / x r =x, (4.16) for some x R + wih x =1. Le τ x = inf{ : Q1 x() = Qx 2 ()} and τ = inf{ : Q 1 () = Q 2 ()}.Thenby(4.14) and(4.16), τ xr x r τ a.s. as r. (4.17) This resul wih (4.16) and he fac ha Q saisfies (2.6) (2.9) for [0, T ] implies ha for all [0,τ]. Q i () ɛ, (4.18)

17 Sabiliy analysis of parallel server sysems 273 Consider he induced Markov chain Y r ={, Z} whose ransiion raes are given as in 4.1 which is independen from { X xr (), [0,τ xr ) }.Leheiniialsaederived from he sae of X xr a ime τ xr be as follows Z r (0) = Z xr ( τ xr ) and r (0) = Q xr 2 We define a new process X r wih ( ) ( ) τ xr Q1 xr τ xr. (4.19) { X r X x r (), [ ) 0,τ () = xr Y r (), [ τ xr, rt ]. (4.20) Because here are only finiely many possible iniial saes for each Z r j,by(4.1), lim sup T 0 T ( ) x r 0 1 Z r j () = i d x r ς ij = 0. (4.21) We show below ha X r has he same disribuion as X xr on [0, T ] for large enough r if d i > 0forsomei = 1, 2. This gives T ij () = ς ij, for τ. (4.22) This implies desired { resul by he fac ha d i > 0forsomei and (4.18). Le σ xr = inf : Q1 xr () + Qxr 2 }.Toprove () = 0 X r has he same disribuion as X xr on [0, T ] for large enough r i is enough o show ha lim inf r σ xr x r > T. (4.23) Nex, we prove (4.23). By (4.18) and(4.14), for r large enough σ xr >τ xr and lim inf r σ xr τ xr x r > ɛ, (4.24) for some ɛ >0. The original process Z xr ( ) is idenical o he process Z r ( τ xr ) on [ τ x r,σ xr ].Le lim inf r σ xr τ xr x r = σ. (4.25) We have σ>0by(4.24). Combining (4.21)wih(4.14)andhefachad i > 0gives lim inf r Q xr i ( σ x r x r T ) x r > d i (σ T ). (4.26)

18 274 G. Baharian, T. Tezcan Because lim inf r Q xr i ( σ x r ) x r = 0, (4.27) and σ>0, we have (4.23) forr large enough by (4.26). 4.3 Generalizaion of he drif condiion In his secion we only focus on X-model sysems. I is desirable o esablish similar drif condiions for general parallel server sysems and we believe ha similar condiions are necessary and sufficien in general. However, here are wo issues ha need o be addressed. Firs, i is no clear how o compue ς ij s especially when Y is no posiive recurren (and i may be as difficul o show ha Y is posiive recurren as i is o show X is). When Y is no posiive recurren, one can esimae ς ij s using a simulaion and use hese esimaed values o compue esimaes for d i s. This should be more reliable han checking wheher X is sable wih simulaions since we only need o esimae ς ij s and do no need o deermine wheher he queue lenghs are diverging. The values of ς ij s also give an idea how much addiional capaciy is needed o make he sysem sable. The second (and more imporan) issue is ha our proof echnique canno be easily exended o cover more general cases. In he X-model sysem, servers will only idle if boh queues are empy. Therefore, if he sysem is ransien, queues will never be empy afer a finie amoun of ime (wih non-zero probabiliy). Hence, he sysem evolves similar o he induced Markov chain. This may no be he case for a general sysem. 5Examples In his secion our goal is o illusrae our main resuls, Theorems 4.3 and 4.4,wihsimulaion experimens. We focus on wo specific parallel server sysems, he X-model, see Fig. 1a andhexn-model,seefig.1b. 5.1 Sabiliy analysis of he X-model Firs, we specify he induced Markov chain Y obained for X-model sysems in deail. Consider he coninuous ime Markov chain Y = (, Z), where N can be considered as he difference beween queue lenghs; Q 2 Q 1 and Z = (Z 1, Z 2 ) wih Z i {1, 2}, fori = 1, 2. (We drop from he noaion for simpliciy.) Transiions from one sae o anoher occur as a resul of arrivals or service compleions. For wo saes Y (1) = ( (1), Z(1)) and Y (2) = ( (2), Z(2)) wih Z(n) = (Z 1 (n), Z 2 (n)), for n = 1, 2, he ransiion raes are given as follows: γ a (Y (1), Y (2)) = { λ1, if (2) = (1) 1, Z(2) = Z(1) λ 2, if (2) = (1) + 1, Z(2) = Z(1), (5.1)

19 Sabiliy analysis of parallel server sysems 275 and γ s (Y (1), Y (2)) = 2i=1 µ ij 1 { Z j (1) = i }, if (1) <0, (2) = (1) + 1, Z j (2) = 1, Z j (2) = Z j (1), j = 1, 2 2i=1 µ ij 1 { Z j (1) = i }, if (1) >0, (2) = (1) 1, Z j (2) = 2, Z j (2) = Z j (1), j = 1, 2 where j = 2if j = 1and j = 1if j = 2. Above, γ a and γ s are he ransiions corresponding o arrival and service compleion evens, respecively, for he induced Markov chain. If a cusomer arrives o class 1 or class 2, hen we se (2) = (1) 1 and (2) = (1) + 1, respecively. The firs erm given for γ s follows from he fac ha if (1) <0(herearemoreclass1cusomershanclass2cusomers)henwhen here is a service compleion, he server ha finishes service firs will be assigned a class 1 cusomer. The second erm follows similarly. There is one imporan case missing from he ransiion raes given above, wha if (1) = 0? Noe ha his is he case when queues are equal and he ransiion raes in his case depends on he ie-breaking rule used wih LQF. Consider he ie-breaking rule which allows he server o pick a cusomer from one of he classes wih equal probabiliy when boh queues are equal lengh. We refer o his rule as he fair selecion rule. In his case, γ s (Y (1), Y (2)) = i=1 µ ij 1 { Z j (1) = i }, if (1) = 0, (2) = (1) + 1, Z j (2) = 1, Z j (2) = Z j (1), j = 1, i=1 µ ij 1 { Z j (1) = i }, if (1) = 0, (2) = (1) 1, Z j (2) = 2, Z j (2) = Z j (1), j = 1, 2. Nex consider he ie-breaking rule, referred o as he saic allocaion rule, when server 1 serves class 1 cusomers and server 2 serves class 2 cusomers when = 0. In his case γ s (Y (1), Y (2)) = 2i=1 µ i1 1 {Z 1 (1) = i}, if (1) = 0, (2) = (1) + 1, Z 1 (2) = 1, Z 2 (2) = Z 2 (1) 2i=1 µ i2 1 {Z 2 (1) = i}, if (1) = 0, (2) = (1) 1, Z 2 (2) = 2, Z 1 (2) = Z 1 (1). Given a ie-breaking rule and he ransiion raes above, we se γ a (Y (1), Y (2)) = 0 and γ s (Y (1), Y (2)) = 0ifY (1) and Y (2) saisfy none of he condiions above. The ransiion raes for Y are given by γ a (Y (1), Y (2))+γ s (Y (1), Y (2)). In Fig. 3,wepresen he ransiion diagram of he induced Markov chain for X-model sysems under saic allocaion ie breaking rule. By Lemma 4.1, Y is posiive recurren assuming SPP has an opimal soluion wih ρ < 1. Clearly, he saionary disribuion of Y depends on he ransiion raes when

20 276 G. Baharian, T. Tezcan = 0, in urn, on he ie-breaking rule used wih LQF. Le π Y denoe he saionary disribuion of Y and denoe by π Y (Z j = i) he seady sae probabiliy ha he server j is serving a class i cusomer. (We omi he dependence on he ie-breaking rule for he noaional simpliciy in our noaion.) Using Theorems 4.3 and 4.4 and he fac ha Y is posiive recurren, we have ha an X-model sysem under LQF is posiive recurren if and is ransien if λ i <µ i1 π Y (Z 1 = i) + µ i2 π Y (Z 2 = i), for i = 1, 2, (5.2) λ i >µ i1 π Y (Z 1 = i) + µ i2 π Y (Z 2 = i), for i = 1or2. (5.3) When he SPP has an opimal soluion wih ρ < 1, i is easy o show ha d 1 = d 2, hence he drif condiion is eiher saisfied or violaed by boh classes. In addiion, i is no possible o find a closed-form expression for he saionary disribuion of Y.In our numerical experimens explained here and in he following secion, whenever we need o find he saionary disribuion of a recurren induced Markov chain Y,which has infiniely many saes, we firs limi i s by a large number. Then we increase his limi in an ieraive approach unil he maximum difference beween he saionary probabiliies in wo consecuive seps becomes small enough. Nex we presen he resuls of some numerical experimens. We begin wih illusraing ha he X-model sysems under LQF can be unsable even wih arbirarily low loads, i.e., SPP has an opimal soluion wih arbirarily small ρ.wealsoillusrae he effec of ie-breaking rules on he sabiliy of such a sysem. In he firs se of experimens, we se λ 1 = λ 2 = 1,µ 11 = µ 22 = 10, and µ 12 = µ 21 = 0.1. For his example ρ = 0.1 < 1. We use he fair selecion ie-breaking rule. Because Y is posiive recurren, using he saionary disribuion of Y we obain π Y (Z 1 = 1) = π Y (Z 2 = 2) = ; herefore d 1 = λ 1 µ 11 π Y (Z 1 = 1) µ 12 π Y (Z 2 = 1) = > 0, implying ha he sysem is unsable by Theorem 4.4. Nex, we experimen wih differen ie-breaking rules. In addiion o he fair selecion rule, we consider wo oher rules. We refer o he firs one as he saic-fas-service rule, which gives prioriy o cusomers wih shorer expeced service imes. The nex ie-breaking rule is referred o as he saic-slow-service rule in which an idling server services he cusomer wih larger mean service ime whenever here is a ie (we only use his rule for illusraive purposes, we do no claim i should be used in pracice). In hese se of experimens, we se λ 1 = λ 2 = 2,µ 11 = µ 22 = 10, and µ 12 = µ 21 = 1. We plo he number of cusomers in he firs queue in Fig. 4 for he hree above-menioned ie-breaking rules as a funcion of ime. In addiion, resuling d 1 s (see (4.8)) are presened in Table 1 (recall ha in his case d 1 = d 2 ). Observe ha he sysem is sable only under he saic-fas-service ie-breaking rule and no under oher rules as can be seen from Table 1.Simulaionexperimensverifiesourresul;seeFig.4,where

21 Sabiliy analysis of parallel server sysems Fair Selecion Saic Fas Service Saic Slow Service Q x 10 4 Time Fig. 4 Simulaion resuls for X-model under hree differen ie-breaking rules Table 1 Drif under 3 differen ie breaking rules Tie breaking rule π Y (Z 1 = 1) d 1 Fair selecion Saic-fas-service Saic-slow-service we plo he number of cusomers in he firs queue versus ime under hree differen ie-breaking rules. 5.2 Sabiliy analysis of he XN-model Nex we consider a slighly more complicaed sysem han he X-model, he XNmodel, see Fig. 1b. We noe ha Theorems 4.3 and 4.4 are only proved for X-model sysems. Our purpose in his secion is o illusrae ha drif condiion may be used o sudy he sabiliy of oher (more general) sysems as well. To sae he sabiliy condiion for he XN-model, we need o consider he induced Markov chain Y. TheMarkovchainY has five componens 1, 2, Z 1, Z 2 and Z 3. In his seing 1 can be hough of as he difference beween lengh of queues 2 and 1, and 2 as he difference beween queues 3 and 1. The ransiion raes can be wrien similarly o he X-model herefore we omi he deails. In he numerical experimens we consider, Y is ergodic and we use π Y o denoe is saionary disribuion. Then, inuiively, he XN-model is sable if λ i < j S(i) µ ij π Y (Z j = i), for i = 1, 2, 3. (5.4) Now we consider he XN-model wih cerain parameers and he fair selecion iebreaking rule (each queue has an equal probabiliy o be seleced in case of a ie). Le

22 278 G. Baharian, T. Tezcan Q x 10 4 Time Fig. 5 Simulaion resuls for he XN-model Q x 10 4 Time λ 1 = 2,λ 2 = 7,λ 3 = 1,µ 23 = 5,µ 11 = µ 22 = 10, and µ 12 = µ 21 = µ 33 = 1.85, and noe ha he opimal soluion for he SPP is ρ = 0.6 < 1. To check wheher his sysem saisfies he sufficien condiion for sabiliy under LQF, we verify he condiion (5.4). We have ha π Y (Z 1 = 1) = ,π Y (Z 2 = 2) = , and π Y (Z 3 = 3) = , so we obain d i = λ i 3 µ ij π Y (Z j = i) = 0.04 < 0, for i = 1, 2, 3. j=1 Hence, he XN-model is sable wih he given parameers. This resul is also verified by a simulaion experimen; he number of cusomers in he firs queue (he second and he hird queues presen similar behavior) is ploed in he lef secion of Fig. 5. Now, we change µ 12,µ 21,andµ 33 from 1.85 o 1.75 which resuls in ρ = 0.61 < 1. The number of cusomers in he firs queue in his new seing is illusraed in he righ secion of Fig. 5 which shows ha he sysem presens unsable behavior wih hese slighly differen parameers. In his sysem he saionary disribuion probabiliies are given by π Y (Z 1 = 1) = ,π Y (Z 2 = 2) = , and π Y (Z 3 = 3) = , wih d 1 = d 2 = d 3 = 0.035; hence, (5.4) doesnohold.weremindhereaderha Theorem 4.4 only covers he X-model, bu simulaions verify ha he drif condiion for insabiliy seems o hold for he XN-model as well in his case. References Bell SL, Williams RJ (2001) Dynamic scheduling of a sysem wih wo parallel servers in heavy raffic wih resource pooling: asympoic opimaliy of a hreshold policy. Ann Appl Probab 11: Bell SL, Williams RJ (2005) Dynamic scheduling of a parallel server sysem in heavy raffic wih complee resource pooling: asympoic opimaliy of a hreshold policy. Elecron J Probab 10: Bramson M (2008) Sabiliy of queueing neworks. Probab Surv 5: Chen H, Yao DD (2001) Fundamenals of queueing neworks : performance, asympoics, and opimizaion. Springer, New York Coffman EG, Puhalskii AA, Reiman MI (1995) Polling sysems wih zero swichover imes: a heavy-raffic averaging principle. Ann Appl Probab 5:

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