SPDE LIMITS OF MANY-SERVER QUEUES. By Haya Kaspi and Kavita Ramanan Technion, Israel and Brown University, USA

Transcription

1 Submied o he Annals of Applied Probabiliy SPDE LIMITS OF MANY-SERVER QUEUES By Haya Kaspi and Kavia Ramanan Technion, Israel and Brown Universiy, USA This papers sudies a queueing sysem in which cusomers wih independen and idenically disribued service imes arrive o a queue wih N servers and ener service in he order of arrival. The sae of he sysem is represened by a process ha describes he oal number of cusomers in he sysem, and a measure-valued process ha keeps rack of he ages of cusomers in service, leading o a Markovian descripion of he dynamics. Under suiable assumpions, a funcional cenral limi heorem is esablished for he sequence of (cenered and scaled sae processes as N, he number of servers, goes o infiniy. The limi process describing he oal number in sysem is shown o be an Iô diffusion wih a consan diffusion coefficien ha is insensiive o he service disribuion beyond is mean. In addiion, he limi of he sequence of (cenered and scaled age processes is shown o be a diffusion aking values in a Hilber space and is characerized as he unique soluion of a sochasic parial differenial equaion ha is coupled wih he Iô diffusion describing he limiing number in sysem. Furhermore, he limi processes are shown o be semimaringales and o possess a srong Markov propery. CONTENTS 1 Inroducion Descripion of he model Fluid limi Cerain maringale measures and heir sochasic inegrals Main resuls Represenaion of he sysem dynamics Coninuiy properies Convergence resuls Proofs of main heorems A Properies of he maringale measure sequence B Ramificaions of assumpions on he service disribuion C Proof of he represenaion formula D Momen esimaes E Proof of consisency Parially suppored by he Israel Science Foundaion under Gran 81/9, he Milford Bohm Chair Gran and he US-Israel Binaional Science Foundaion under Gran BSF Parially suppored by he Naional Science Foundaion under Grans CMMI (formerly and CMMI (formerly , as well as he US-Israel Binaional Science Foundaion under Gran BSF AMS 2 subjec classificaions: Primary. 6K25, 6F17, 6H15; Secondary. 9B22, 68M2 Keywords and phrases: Many-server queues, GI/G/N queue, fluid limis, diffusion limis, sochasic parial differenial equaions, measure-valued processes, Io diffusion, Halfin-Whi regime 1

2 2 H. KASPI AND K. RAMANAN References Auhor s addresses Inroducion Background, moivaion and resuls.. Many-server queues consiue a fundamenal model in queueing heory and are ypically harder o analyze han single-server queues. The main objecive of his paper is o esablish useful funcional cenral limi heorems for many-server queues in he asympoic regime in which N, he number of servers, ends o infiniy and λ, he mean arrival rae in he sysem wih N servers, scales as λ = λn β N for some λ > and β (,. For many-server queues wih Poisson arrivals, his scaling was considered more han half a cenury ago by Erlang 8] and hereafer by Jagerman 17] for a loss sysem wih exponenial service imes, bu i was no unil he influenial work of Halfin and Whi 14] ha a heavy raffic limi heorem was esablished. As a resul, he asympoic regime in which λ = 1, β > is ofen referred o as he Halfin Whi regime. In conras o convenional heavy raffic scalings, in he Halfin Whi regime, he limiing saionary probabiliy of a posiive wai is non-rivial (i.e., i lies sricly beween zero and one, which beer capures he behavior of many sysems found in applicaions. Due o he simulaneous high uilizaion and good qualiy of service (as capured by a posiive probabiliy of no wai, his asympoic regime is someimes also referred o as he Qualiy-and-Efficiency-Driven regime. Under he assumpion of renewal arrivals, exponenial service imes, normalized o have uni mean, and limi mean arrival rae λ = 1, Halfin and Whi 14] showed ha he limi of he sequence of processes represening he (appropriaely cenered and scaled number of cusomers in he sysem is a diffusion process wih a consan diffusion coefficien and a sae-dependen drif ha is linear and resoring (resembling an Ornsein Uhlenbeck process below zero and consan above zero. Under he condiion β >, which ensures ha each N-server queue is sable, his characerizaion of he limi process was used in 14] o esablish approximaions o he saionary probabiliy of posiive wai in a queue wih N servers. However, in many applicaions, saisical evidence suggess ha i would be more appropriae o model he service imes as being non-exponenial (see, e.g., he sudy of real call cener daa in Brown e al. 4] ha indicaes ha he service imes are lognormally disribued. A naural goal is hen o undersand he behavior of many-server queues in his scaling regime when he service disribuion is no exponenially disribued. Specifically, in addiion o esablishing a limi heorem, he aim is o obain a racable represenaion of he limi process ha makes i amenable o compuaion, so ha he limi could be used o shed insigh ino performance measures of ineres for an N-server queue. When he service imes are no exponenially disribued, any Markovian represenaion mus keep rack of he residual service imes or he ages of cusomers in service. This implies ha he dimension of any finie-dimensional represenaion of he sae mus grow wih he number of servers (he dimension mus be a leas N +1 for an N-server sysem, which poses a challenge for obaining limi heorems as N. Insead, in his work a common infinie-dimensional sae space is used for all N-server sysems. Specifically, he sae of he N-server queue is represened by a non-negaive, ineger-valued process X ha records he oal number of cusomers in sysem, as well as a measure-valued

3 SPDE LIMITS OF MANY-SERVER QUEUES 3 process ν ha keeps rack of he ages of cusomers in service, where he age is he ime elapsed since enry ino service. This represenaion, which provides a Markovian descripion of he dynamics, was firs inroduced in Kaspi and Ramanan 21], where i was shown ha he fluid-scaled sequence (X, ν /N converges almos surely o a cerain deerminisic process (X, ν, referred o as he fluid limi. Flucuaions around he fluid limi can be capured by he diffusion-scaled sae sequence {( X, ν } N N, which is obained by cenering he fluid-scaled sae (X, ν /N around he fluid limi (X, ν and muliplying he difference by N. In he presen work, under suiable assumpions, in each of he cases when he fluid limi is subcriical, criical or supercriical (which, roughly speaking, correspond o he cases λ < 1, λ = 1 or λ > 1, i is shown in Theorem 2 and Theorem 3 ha he diffusionscaled sae sequence {( X, ν } N N converges weakly o a càdlàg sochasic process ( X, ν. Moreover, he X componen of he limi is shown (in Corollary 5.1 o be a real-valued Iô diffusion wih a consan diffusion coefficien ha is insensiive o he service disribuion beyond is mean, and whose drif is an adaped process ha is a funcional of ν. In paricular, alhough X is non-markovian, i admis a fairly racable represenaion. The proof of his represenaion relies on an asympoic independence resul for he cenered arrival and deparure processes (see Proposiion 8.4, which may be of independen ineres. As for he age process, alhough he ν are (signed Radon measure valued processes, he limi ν lies ouside his space. A key challenge was o idenify a suiable space in which o esablish convergence wihou imposing overly resricive assumpions on he service disribuion G. Under condiions ha include a large class of service disribuions relevan in applicaions such as phase-ype, lognormal, logisic and (for a cerain class of parameers Erlang and Pareo disribuions, i is shown ha ν converges weakly o ν in he space of H 2 -valued càdlàg processes, where H 2 is he dual of a cerain Hilber space H 2. In addiion, in Theorem 5(a he ν componen of he limi is characerized as he unique soluion o a sochasic parial differenial equaion ha is coupled wih he Iô diffusion X. Furhermore, i is also shown in Theorem 4 ha ( X, ν is a semimaringale wih an explici decomposiion and in Theorem 5(b ha ( X, ν, along wih an appended sae, is a srong Markov process. The proof of he srong Markov propery relies on a consisency propery (see Lemma 9.3 and Appendix E, which shows ha he assumpions ha are imposed on he iniial sae are also saisfied by he sae a any posiive ime Relaion o prior work.. To dae, he mos general resuls on process level convergence in he Halfin Whi regime were obained in a nice pair of papers by Reed 27] and Puhalskii and Reed 26]. Under he assumpions ha λ = 1, he iniial residual service imes of cusomers in service are independen and idenically disribued (i.i.d. and aken from he equilibrium fluid disribuion, and he oal (fluid scaled number in sysem converges o 1, a heavy raffic limi heorem for he sequence of processes { X } N N was esablished by Reed 27] wih only a finie mean condiion on he service disribuion. This resul was exended by Puhalskii and Reed 26] o allow for more general, possibly inhomogeneous arrival processes as well as more general condiions on he residual service imes of cusomers in service a he iniial ime. In his seing, convergence of finie-dimensional disribuions was esablished in 26], and srenghened o process level convergence when

4 4 H. KASPI AND K. RAMANAN he service disribuion is coninuous. In boh papers, he limi is characerized as he unique soluion o a cerain sochasic convoluion equaion. Oher previous works had also exended he Halfin Whi process level resul for specific classes of service disribuions. Noeworhy amongs hem is he paper by Puhalskii and Reiman 25], in which phase-ype service disribuions are considered and he limi is characerized as a mulidimensional diffusion, where each dimension corresponds o a differen phase of he service disribuion. Whi 32] also esablished a process level resul for a many-server queue wih finie waiing room and a service disribuion ha is a mixure of an exponenial random variable and a poin mass a zero. In addiion o he process level resuls described above, under he sabiliy assumpion β >, resuls on he asympoics of seady sae disribuions in he Halfin Whi regime have been obained by Jelenkovic, Mandelbaum and Momçilović 18] for deerminisic service imes and by Gamarnik and Momçilović 12] for service imes ha are laice-valued wih finie suppor. Our work serves o complemen he above menioned resuls, wih he focus being on esablishing racabiliy of he limi process under assumpions on he service disribuion ha are saisfied by a large class of service disribuions of ineres. Whereas in all he above papers only he number in sysem is considered, we esablish convergence for a more general sae process, which implies he convergence of a large class of funcionals of he process and no jus he number in sysem. As a special case, we can recover he resuls of Halfin and Whi 14] and Puhalskii and Reiman 25] and, for he smaller class of service disribuions ha we consider, Reed 27]. The Markovian represenaion of he sae, hough infinie-dimensional, leads o an inuiive characerizaion of he dynamics, which faciliaes he incorporaion of more general feaures ino he model. For example, his framework was exended by Kang and Ramanan o include abandonmens in 19] and 2]. In he subcriical case he resuls of his paper also provide a characerizaion of he diffusion limi of he well-sudied infinie-server queue. The laer is easier o analyze due o he absence of a queue and he consequen lack of ineracion beween hose in service and hose waiing in queue. A few represenaive works on diffusion limis of he number in sysem in he infinie-server queue include Iglehar 15] and Glynn and Whi 13], where he limi process is characerized as an Ornsein Uhlenbeck process, and Krichagina and Puhalskii 22], who provide an alernaive represenaion of he limi in erms of he socalled Kiefer process. More recenly, a funcional cenral limi heorem in he space of disribuion-valued processes was esablished for he M/G/ queue by Decreusefond and Moyal 7]. In conras o he infinie-dimensional Markovian represenaion in erms of residual service imes used in Decreusefond and Moyal 7], he Markovian represenaion in erms of he age process ha is used here allows us o associae some naural maringales ha faciliae he analysis Ouline of he paper. Secion 2 conains a precise mahemaical descripion of he model and he sae descripor used, as well as he defining dynamical equaions. A deerminisic analog of he model, described by dynamical equaions ha are referred o as he fluid equaions, is inroduced in Secion 3. Secion 3 also recapiulaes he relevan funcional srong law of large numbers limi resuls esablished in 21]. In Secion 4 a sequence of maringales ha are obained as compensaed deparure processes and play an imporan role in he analysis is inroduced, and he associaed scaled maringale

5 SPDE LIMITS OF MANY-SERVER QUEUES 5 measures M, N N, are shown o be orhogonal. An associaed sequence of sochasic convoluion inegrals Ĥ,, which arise in he represenaion of he dynamics, is also defined. The main resuls and heir corollaries are saed in Secion 5, and heir proofs are presened in Secion 9. The proofs rely on resuls obained in Secions 6, 7 and 8. Secion 6 conains a succinc characerizaion of he dynamics and esablishes a represenaion (see Proposiion 6.4 for ν, he diffusion-scaled age process in he N-server sysem, in erms of Ĥ, anoher sochasic convoluion process K and he iniial daa. In Secion 7, i is shown ha he processes K, X and ν can be obained as a coninuous mapping of he iniial daa sequence and he process Ĥ. Secion 8 is devoed o esablishing he join convergence of he maringale measure sequence { M } N N and he associaed sequence {Ĥ } N N of sochasic convoluion inegrals, ogeher wih he sequence of cenered arrival processes and iniial condiions (see Corollary 8.7. To mainain he flow of he exposiion, some supporing resuls are relegaed o he Appendix. Firs, in Secion 1.4 we inroduce some common noaion and erminology used in he paper Noaion and erminology. As usual, le Z + denoe he se of non-negaive inegers, N denoe he se of naural numbers or, equivalenly, sricly posiive inegers and R denoe he se of real numbers. For a, b R, le a b and a b, respecively, denoe he maximum and minimum of a and b. The shor-hand noaions a + and a will someimes also be used for a and (a, respecively. Given B R, I B denoes he indicaor funcion of he se B (ha is, I B (x = 1 if x B and I B (x = oherwise Funcion spaces. Given any meric space E, B(E denoes he Borel ses of E (wih opology compaible wih he meric on E, and C(E denoes he space of coninuous real-valued funcions defined on E. Also, C b (E is he subse of bounded funcions in C(E and C c (E represens he subse of funcions in C(E ha have compac suppor in E. Le D E, denoe he space of E-valued càdlàg funcions defined on,, and le supp(ϕ denoe he suppor of a funcion ϕ. When E is a domain (open conneced subse or closure of a domain in R n, equipped wih he usual Euclidean meric and Lebesgue measure, le AC(E represen he space of absoluely coninuous funcions (in he sense of Carahéodory defined on E, and le AC b (E denoe he subse of bounded funcions in AC(E. We will mosly be ineresed in he case when E =, L and E =, L,, for some L (, ]. To disinguish hese cases, f will be used o denoe generic funcions on, L and ϕ o denoe generic funcions on, L,. By some abuse of noaion, given f on, L, i will also be reaed as a funcion on, L, ha is consan in he second variable. For eiher choice of E, le C 1 (E and C (E, respecively, represen he space of real-valued, once coninuously differeniable and infiniely differeniable funcions on E, le C 1 c(e be he subspace of funcions in C 1 (E ha have compac suppor and C 1 b (E he subspace of funcions in C1 (E ha, ogeher wih heir firs derivaives, are bounded. Here, a funcion on E is said o be differeniable if i is he resricion o E of a differeniable funcion on some open neighborhood of E. Recall ha given T < and a coninuous funcion f C, T ], he modulus of coninuiy w f ( of f is defined by (1.1 w f (δ =. sup f( f(s, δ >. s,,t ]: s <δ

6 6 H. KASPI AND K. RAMANAN For any ϕ AC(, L,, he parial derivaives ϕ x and ϕ s are well defined as locally inegrable funcions on, L,. The space C 1,1 (, L, is defined as he subse of funcions in AC(, L, for which ϕ x +ϕ s, he direcional derivaive in he (1, 1 direcion, is coninuous. Moreover, C 1,1 b (, L, and Cc 1,1 (, L, denoe he subse of funcions ϕ in C 1,1 (, L, such ha boh ϕ and ϕ x + ϕ s are bounded or, respecively, have compac suppor. Le I, denoe he space of non-decreasing funcions f D R, wih f( =. For L, ], L α, L, α 1, and L, L represen, respecively, he spaces of measurable funcions f on, L such ha,l f(x α dx < and he space of essenially bounded funcions (wih respec o Lebesgue measure on, L. Also, L α loc, L represens he corresponding space in which he associaed propery holds only locally, ha is, on every compac (i.e., closed and bounded inerval in, L. The consan funcions f 1 and f on, L will be represened by he symbols 1 and, respecively. Given any càdlàg, real-valued funcion.. f defined on, L, we define f T = sups,t ] f(s for every T < L, and le f = sup s,l f(s, which could possibly equal infiniy. As usual, for f D R, and >, le f( = lim u f(u denoe he lef limi of f a, wih he convenion ha f( = f(, and also le f( = f( f( denoe he jump of f a. For f, g C c, and n =, 1, 2,..., consider he weighed inner produc defined by (1.2 f, g Hn. = n and he associaed norm m= d m f dx m (x dm g dx m (x(1 + x 2 n dx, (1.3 f Hn. = ( f, f Hn 1/2. I is clear from he definiion ha he norm Hn is Hilberian (i.e., i saisfies f + g 2 H n f g 2 H n = 2 f 2 H n + 2 g 2 H n. For each n N, le H n be he (weighed Sobolev space obained as he compleion of C c, wih respec o he norm Hn. I is easy o verify ha each H n is a complee separable Hilber space and, for m < n, H n H m. Now, for HS m, n N, m < n, Hm is said o be HS weaker han Hn, and denoed Hm < Hn, if he injecion map from H n o H m is a Hilber Schmid (or, equivalenly, quasi-nuclear operaor (see, e.g., page 6 of 2] or page 33 of 31]. I follows from Theorems 3.6 and 3.7 of 2] ha for all m, n N, (1.4 n > m Hm HS < Hn. Now, le H n be he dual of H n, where he dual norm H n ν 2 H n = ν(e nk 2, ν H n, k=1 is given by where {e nk, k = 1,..., } is a complee orhonormal sysem in H n. Moreover, le S be he Schwarz space of rapidly decreasing funcions, namely he space of C funcions on,, for which he following semi-norms are finie: f β,γ = sup xβ dγ f dx γ (x, β N, γ N. x,

7 SPDE LIMITS OF MANY-SERVER QUEUES 7 Le S be he opological dual of S, which is he Schwarz space of empered disribuions. I is well known ha S and S are separable, nuclear Fréche spaces. Moreover, he projecive limi of he spaces H n, n N, coincides wih S, and he dual space saisfies S = n= H n; see Theorem 3.8, relaion (3.39 and he commen a he end of Secion 3.1 of 2]. For ν S and f S and likewise, for ν H n and f H n, le ν(f denoe he dualiy pairing. For n = 1, 2 and f for which he corresponding firs or second (weak derivaives are well defined, someimes he noaion f = f (1 and f = f (2 will also be used. The usual L 2 norm is given by ( 1/2 f L 2 = f(x dx 2. R I follows immediaely from he definiion of he norms n given above ha (1.5 f L 2 = f H, f 2 L 2 + f 2 L 2 f 2 H 1, f 2 L 2 + f 2 L 2 + f 2 L 2 f 2 H 2. Moreover, for any f H 1, i is easy o deduce he norm inequaliies (1.6 f( 2 f H1, f 2 f H1, which will be used in he sequel. Indeed, if f H 1 hen f is absoluely coninuous and boh f and f lie in L 2. Therefore, here exiss a real-valued sequence {x n } wih x n and f(x n as n. Since for each n N, we have f 2 (x n f 2 ( = 2 x n f(uf (u du, applying he Cauchy Schwarz inequaliy and hen aking limis as n, we obain f( 2 2 f L 2 f L 2 2 f 2 H 1, where he las inequaliy follows from he second inequaliy in (1.5. This yields he firs inequaliy in (1.6. The second inequaliy in (1.6 can be inferred in a similar manner, applying he Cauchy Schwarz inequaliy o he relaion f 2 (x = f 2 ( + 2 x f(uf (u du and using he firs inequaliy in (1.6. Finally, le D, denoe he usual space of es funcions, namely he space C c, equipped wih he following noion of convergence: f n f in D, if he funcions f n are suppored in a common compac se and, for every m N, d m f n /dx m d m f/dx m uniformly. Also, le D, denoe is dual, he space of disribuions. I is well known (see, e.g., Theorem 3.9 of 2] ha boh D, and D, are nuclear spaces Measure spaces. The space of Radon measures on a meric space E, endowed wih he Borel σ-algebra, is denoed by M(E. M F (E is he subspace of finie measures in M(E and M N (E is he subspace of posiive measures wih oal mass less han or equal o N. Noe ha hen M 1 (E is he space of sub-probabiliy measures. For any Borel measurable funcion f : E R ha is inegrable wih respec o ξ M(E, he shor-hand noaion f, ξ =. E f(x ξ(dx will be used. Recall ha a Radon measure on E is one ha assigns finie measure o every relaively compac subse of E. By idenifying a Radon measure µ M(E wih he mapping on C c (E defined by f f, µ, a Radon measure on E can be equivalenly defined as a linear mapping from C c (E ino R such ha for every compac se K E, here exiss L K < such ha f, µ L K f f C c (E wih supp(f K.

8 8 H. KASPI AND K. RAMANAN The space M F (E is equipped wih he weak opology generaed by ses of he form {µ : f 1, µ µ o < ε 1,..., f n, µ µ o < ε n }, for µ o M F (E, n N, f i C b (E and ε i >, i = 1,..., n]. Also, recall ha a sequence {µ n } n N in M F (E converges o µ M F (E in he weak opology (denoed µ n w µ if for every f Cb (E, f, µ n f, µ as n. The symbol δ x will be used o denoe he measure wih uni mass a he poin x, and he symbol will be used o denoe he idenically zero Radon measure. When E is an inerval, say, L, for conciseness he noaion M, L will be used insead of M(, L. Also, for ease of noaion, given ξ M, L and an inerval (a, b, L, ξ(a, b and ξ(a will be used o denoe ξ((a, b and ξ({a}, respecively Sochasic processes. Given a Polish space V, D V, T ] and D V, denoe he spaces of V-valued, càdlàg funcions on, T ] and,, respecively, endowed wih he usual Skorokhod J 1 -opology; see 3] for deails on his opology. Then D V, T ] and D V, are also Polish spaces. We will be ineresed in V-valued sochasic processes, especially in he cases when V = R, V = M F, L for some L, V = S, L and V = H n, L for n = 1, 2, and producs of hese spaces. These are random elemens ha are defined on a probabiliy space (Ω, F, P and ake values in D V,, equipped wih he Borel σ-algebra (generaed by open ses under he Skorokhod J 1 -opology. A sequence {Z } N N of càdlàg, V-valued processes, wih Z defined on he probabiliy space (Ω, F, P, is said o converge in disribuion o a càdlàg V-valued process Z defined on (Ω, F, P if and only if for every bounded, coninuous funcional F : D V, R, ] lim n E F (Z = E F (Z], where E and E are he expecaion operaors wih respec o he probabiliy measures P and P, respecively. Convergence in disribuion of Z o Z will be denoed by Z Z. 2. Descripion of he model. The many-server model under consideraion is inroduced in Secion 2.1. The sae descripor and he dynamical equaions ha describe he evoluion of he sae are presened in Secion The N-server model. Consider a sysem wih N servers, in which arriving cusomers are served in a non-idling, firs-come-firs-serve (FCFS manner, ha is, a newly arriving cusomer immediaely eners service if here are any idle servers, or if all servers are busy, hen he cusomer joins he back of he queue and he cusomer a he head of he queue (if one is presen eners service as soon as a server becomes free. Our resuls are no sensiive o he exac mechanism used o assign an arriving cusomer o an idle server as long as he non-idling condiion is saisfied. Cusomers are assumed o be infiniely paien; ha is, hey wai in queue ill hey receive service. Servers are non-preempive and serve a cusomer o compleion before saring service of a new cusomer. Le E denoe he cumulaive arrival process, wih E ( represening he oal number of cusomers ha arrive ino he sysem in he ime inerval, ], and le he service requiremens be given by he i.i.d. sequence {v i, i = N +1, N +2,...,, 1,...}, wih common cumulaive disribuion funcion G. Le X ( represen he number of cusomers in he sysem a

9 SPDE LIMITS OF MANY-SERVER QUEUES 9 ime. Then, due o he non-idling condiion, he number of cusomers in service a ime is equal o X ( N. The sequence {v i, i = X ( N + 1,..., } represens he service requiremens of cusomers already in service a ime zero, ordered according o heir ages a ime zero, where he age of a cusomer ha has enered service is defined o be he minimum of he amoun of ime elapsed since he cusomer enered service and he service ime, as defined explicily in (2.5 below. In paricular, v is he service ime of he cusomer who has spen he leas ime in service amongs hose in service a ime zero. On he oher hand, for i N, v i represens he service requiremen of he ih cusomer o ener service afer ime. Consider he càdlàg process R (2.1 R E (s =. inf E defined by { u > s : E (u > E (s } s, s,. Noe ha R E (s represens he ime, a s, o he nex arrival. The following mild assumpions will be imposed hroughou, wihou explici menion. E is a càdlàg non-decreasing pure jump process wih E ( = and almos surely, E ( < and E ( E ( {, 1} for,. The process R E is Markovian wih respec o he usual augmenaion of is own naural filraion; see, for example, page 1 of 3] for an explici consrucion of he filraion. The cumulaive arrival process is independen of he i.i.d. sequence of service requiremens {v j, j = N + 1,..., }. Moreover, given σ(r E (, he σ-algebra generaed by R E (, he process {E (, > } is independen of X ( and he ages of he cusomers ha have enered service by ime zero. G has densiy g. Wihou loss of generaliy, we assume ha he mean service requiremen is 1: (2.2 (1 G(x dx = xg(x dx = 1.,, Also, he righ end of he suppor of he service disribuion is denoed by L. = sup{x, : G(x < 1}. Noe ha he exisence of a densiy for G implies, in paricular, ha G(+ =. Remark 2.1. The assumpions above are fairly general, allowing for a large class of arrival processes and service disribuions. When E is a renewal process, R E is simply he forward recurrence ime process, he second assumpion holds (see Proposiion V.1.5 of Asmussen 1] and he model corresponds o a GI/GI/N queueing sysem. However, he second assumpion holds more generally such as, for example, when E is an inhomogeneous Poisson process; see, for example, Lemma II.2.2 of Asmussen 1]. The processes R E and E described above have rajecories in D R, ; see, for example, Appendix A of 19]. The sequence of processes {R E, E, X (, v i, i = N + 1,...,, 1,...} N N are all assumed o be defined on a common probabiliy space (Ω, F, P ha is large enough for he independence assumpions saed above o hold.

10 1 H. KASPI AND K. RAMANAN 2.2. Sae descripor and dynamical equaions. As in he sudy of he funcional srong law of large numbers limi for his model, which was carried ou in 21], he sae of he sysem will be represened by he vecor of processes (R E, X, ν, where R E deermines he cumulaive arrival process via (2.1, X ( Z + represens he oal number of cusomers in sysem (including hose in service and hose waiing in queue a ime and ν is a discree, non-negaive finie measure on, L ha has a uni mass a he age of each cusomer in service a ime. Here, he age a j of he jh cusomer is (for each realizaion a piecewise linear funcion ha is zero unil he cusomer eners service, hen increases linearly while in service (represening he ime elapsed since service began and hen remains consan (equal o is service requiremen afer he cusomer complees service and depars he sysem. In order o fully describe he sae dynamics, i will be convenien o inroduce he following auxiliary processes: he cumulaive deparure process D, where D ( is he number of cusomers ha have depared he sysem in he inerval, ]; he process K, where K ( represens he cumulaive number of cusomers ha have enered service in he inerval, ]. A simple mass balance on he whole sysem shows ha (2.3 D (. = X ( X ( + E (,,. Likewise, recalling ha 1, ν = ν, L represens he oal number of cusomers in service, an analogous mass balance on he number of cusomers in service yields he relaion (2.4 K ( =. 1, ν 1, ν + D (,,. For j N, le θ j. = inf{s : K (s j}, wih he usual convenion ha he infimum of an empy se is infiniy. Noe ha θ j denoes he ime of enry ino service of he jh cusomer o ener service afer ime. In addiion, for j = (X ( N + 1, X ( N,..., se θ j = a j ( o be he amoun of ime ha he jh cusomer in service a ime has already been in service. Then, for, and j = (X ( N + 1,...,, 1,..., he age process is given explicily by (2.5 a j ( = { v j θ j ] if θ j < v j, oherwise. Due o he FCFS naure of he service, K ( is also he highes index of any cusomer ha has enered service, and (2.5 implies ha for j > K (, θ j > and a j ( =. The measure ν represening he age disribuion a ime can hen be expressed as (2.6 ν = K ( j= 1,ν +1 δ a j ( I {a j (<v j },

11 SPDE LIMITS OF MANY-SERVER QUEUES 11 where δ x represens he Dirac mass a he poin x. The non-idling condiion, which sipulaes ha here be no idle servers when here are more han N cusomers in he sysem, is expressed via he relaion (2.7 N 1, ν = N X (] +,,. Noe ha (2.3, (2.4 and (2.7, ogeher wih he elemenary ideniy x x = x, imply he relaion (2.8 K ( = X ( N X ( N + D (,,. Clearly 1, ν N for every, because he maximum number of cusomers in service a any given ime is bounded by he number of servers. In addiion, if he suppor of ν lies in, L, hen i follows from (2.5 and (2.6 ha ν akes values in M F, L for every,. Thus, he sae of he sysem is represened by he càdlàg process, X, ν, which akes values in, N M F, L. For an explici consrucion of he sae ha also shows ha he sae and auxiliary processes are well defined and càdlàg; see Lemma A.1 of 19]. The resuls obained in his paper are independen of he paricular rule used o assign cusomers o saions, bu for echnical purposes i (R E will be convenien o also inroduce he addiional saion process σ. = (σ j, j { N + 1,..., } N. For each,, if cusomer j has already enered service by ime, hen σ j ( is equal o he index i {1,..., N} of he saion a which cusomer j receives/received service and σ j ( =. oherwise. Finally, for,, le F be he σ-algbera generaed by {R E (s, a j (s, σ j (s, j { N,..., } N, s, ]}, and le {F, } denoe he associaed righ coninuous filraion ha is compleed (wih respec o P so ha i saisfies he usual condiions. Then i is easy o verify ha (R E, X, ν is {F }-adaped; see, for example, Secion 2.2 of 21]. In fac, as shown in Lemma B.1 of 19], {(R (, X (, ν E, F, } is a srong Markov process. For fuure purposes, we inroduce some sandard Markov process noaion (see 3] associaed wih he Markov process (R E, X, ν = {(R E (, X (, ν, }. Le G, be he σ-algebra generaed by {(R E (s, X (s, ν s, s, ]}, and le G, = σ( G,. For (r, k, µ, N M F, L, le P r,k,µ be he law of he Markov process (R E, X, ν wih iniial condiion (R E (, X (, ν = (r, k, µ. Also, le { G, } be he usual augmenaion of he filraion {G,, } (as carried ou, e.g., in page 25 of 3] and le G = s> G s,, be he associaed righ-coninuous filraion. Noe ha for every, G F and ha {(R E, X, ν, P r,k,µ, (r, k, µ, N M F, L} is a srong Markov family. Remark 2.2. The assumed Markov propery of R E wih respec o he compleed, righ coninuous version of is naural filraion ogeher wih he independence properies of E assumed in Secion 2.1 imply ha for any,, given σ{r (} he E

12 12 H. KASPI AND K. RAMANAN fuure arrivals process {E (s, s > } is independen of F, and hence of G because G F. 3. Fluid limi. In his secion we recall he funcional srong law of large numbers limi or, equivalenly, fluid limi obained in 21]. The iniial daa describing he sysem consiss of E, he cumulaive arrivals afer zero, X (, he number in sysem a ime zero, and ν, he age disribuion of cusomers in service a ime zero. The iniial daa belongs o he following space: (3.1 I N. = { (f, x, µ I,, M N, L : N 1, µ = N x] +}, where we recall ha I, is he subse of non-decreasing funcions f D,, wih f( =. When N = 1, I 1 will be denoed simply by I. Assume ha I N is equipped wih he produc opology. Consider he fluid scaled versions of he processes H = E, X, K, D and measures H = ν defined by (3.2 H. = H N, and le R E ( =. R E (E (,,, for N N. Observe ha he fluid-scaled iniial daa (E, X (, ν lies in I. The srong law of large numbers resuls in 21] were obained under Assumpions 1 and 2 below. Assumpion 1. There exiss (E, x, ν I such ha almos surely, as N, (E, X (, ν (E, x, ν in I, and moreover, as N, EX (] x and EE (] E( for every,. Nex, recall ha G has densiy g, and le h denoe is hazard rae, (3.3 h(x =. g(x, x, L. 1 G(x Observe ha h is auomaically locally inegrable on, L because for every a b < L, (3.4 b a h(x dx = ln(1 G(a ln(1 G(b <. However, h is no inegrable on, L. In paricular, when L <, h is unbounded on (l, L for every l < L. Assumpion 2. A leas one of he following wo properies holds: (a L = and here exiss l < such ha h is bounded on (l, ;

13 SPDE LIMITS OF MANY-SERVER QUEUES 13 (b here exiss l < L such ha h is lower-semiconinuous on (l, L. A succinc descripion of he dynamics of he N-server sysem in erms of cerain inegral equaions is provided in Proposiion 6.1; see also Theorem 5.1 of 21]. The deerminisic analog of hese equaions, he so-called fluid equaions, is inroduced below. Definiion 3.1. (Fluid Equaions The càdlàg funcion (X, ν defined on, and aking values in, M 1, L is said o solve he fluid equaions associaed wih (E, x, ν I if X( = x and for every,, (3.5 h, ν s ds < and he following relaions are saisfied: for every ϕ C 1,1 c (, L,, (3.6 (3.7 and ϕ(,, ν = ϕ(,, ν + ϕ s (, s + ϕ x (, s, ν s ds h( ϕ(, s, ν s ds + ϕ(, s dk(s, X( = X( + E( h, ν s ds,,] ( , ν = 1 X(] +, where K is a non-decreasing process ha saisfies (3.9 K( = 1, ν 1, ν + h, ν s ds,,. We now recall he resul esablished in 21] (see Theorems 3.5 and 3.7 herein, which shows ha under Assumpions 1 and 2, he fluid equaions uniquely characerize he funcional srong law of large numbers or mean-field limi of he N-server sysem, in he asympoic regime where he number of servers and arrival raes boh end o infiniy. Theorem 1 (Kaspi-Ramanan 21]. Suppose Assumpions 1 and 2 are saisfied and (E, x, ν I is he limi of he iniial daa as saed in Assumpion 1. Then here exiss a unique soluion (X, ν o he associaed fluid equaions (3.5 (3.8 and, as N, (X, ν converges almos surely o (X, ν. Moreover, (X, ν saisfies for every f C b (,, 1 G(x + f(x ν (dx = f(x + ν (dx,l,l 1 G(x +,] f( s(1 G( sdk(s, where K is a non-decreasing process ha saisfies he relaion (3.9. Furhermore, if E is coninuous, hen (X, ν and K are also coninuous.

14 14 H. KASPI AND K. RAMANAN Remark 3.2. In his conex, he unique soluion (X, ν o he fluid equaions will also be referred o as he fluid limi. The fluid limi is said o be criical if X( = 1 for all,. In addiion, i is said o be subcriical (respecively, supercriical if for every T,, sup,t ] X( < 1 (respecively, inf,t ] X( > 1. Alhough, in general, he fluid limi may no say in one regime for all and may insead experience periods of subcriicaliy, criicaliy and supercriicaliy, for many naural choices of iniial daa, such as eiher saring empy, ha is, (x, ν = (,, or saring on he so-called invarian manifold of he fluid limi, he fluid limi does belong o one of hese hree caegories. Specifically, if ν is he invarian fluid age measure, defined o be (3.1 ν (dx = (1 G(x dx, x, L, hen i follows from Remark 3.8 and Theorem 3.9 of 21] ha he fluid limi associaed wih he iniial daa (1, 1, ν is criical, he fluid limi associaed wih he iniial daa (1, a, ν for some a > 1 is supercriical, and if he suppor of G is,, hen he fluid limi associaed wih he iniial daa (λ1,, is subcriical whenever λ 1. A complee characerizaion of he invarian manifold of he fluid in he presence of abandonmens can be found in 2]. 4. Cerain maringale measures and heir sochasic inegrals. We now inroduce some quaniies ha arise in he proof of he funcional cenral limi heorem of he sae process. The sequence of maringales obained by compensaing he deparure processes in each of he N-server sysems played an imporan role in esablishing he fluid limi resul in 21]. Whereas under he fluid scaling he limi of his sequence converges weakly o zero, under he diffusion scaling considered here, i converges o a non-rivial limi. This limi can be described in erms of a corresponding maringale measure, which is inroduced in Secion 4.1. In Secion 4.2 cerain sochasic convoluion inegrals wih respec o hese maringale measures are inroduced, which arise in he represenaion formula for he cenered age process in he N-server sysem (see Proposiion 6.4. Finally, he associaed limi quaniies are defined in Secion 4.3. The reader is referred o Chaper 2 of 31] for basic definiions of maringale measures and heir sochasic inegrals A Maringale Measure Sequence. Throughou his secion, le (E, x, ν be an I N -valued random elemen represening he iniial daa of he N-server sysem, and le (R E, X, ν be he associaed sae process described in Secion 2.2. For any measurable funcion ϕ on, L,, consider he sequence of processes {Q ϕ } N N defined by (4.1 Q ϕ ( =. s,] K ( j= 1,ν +1 I da j da j (s >, d d ϕ(a (s+= j (s, s for N N and,, where K and a j are, respecively, he cumulaive enry ino service process and he age process of cusomer j as defined by he relaions (2.4

15 SPDE LIMITS OF MANY-SERVER QUEUES 15 and (2.5. Noe from (2.5 ha he jh cusomer compleed service (and hence depared he sysem a ime s if and only if da j (s > d This is equivalen o he condiion a j and da j (s+ =. d (s = v j, and hus ϕ(a (s, s can in fac be replaced by ϕ(v j, s in (4.1. Subsiuing ϕ = 1 in (4.1, i is clear ha Q 1 is equal o D, he cumulaive deparure process of (2.3. Moreover, for N N and every bounded measurable funcion ϕ on, L,, consider he process A ϕ defined by ( (4.2 A ϕ (. = and se,l (4.3 M ϕ ϕ(x, sh(x ν s (dx. = Q ϕ A ϕ. ds, j,, I was shown in Corollary 5.5 of 21] ha for all funcions ϕ C b (, L,, A ϕ he {F }-compensaor of Q ϕ, and M ϕ is a càdlàg {F }-maringale. Remark 4.1. In fac, M ϕ is a càdlàg {F }-maringale for all ϕ in he larger class of bounded and measurable funcions on, L,. Indeed, since he mapping (ω, s (ω, s on Ω, is coninuous and {F }-adaped, i is {F }-predicable. (a j Therefore, when ϕ is measurable, he mapping (ω, s ϕ(a j (s, ω, s from Ω, o R is also {F N }-predicable. The asserion hen follows from essenially he same argumen as ha used in Lemma 5.9 of 21]. From he proof of Lemma 5.9 of 21] i follows ha for any bounded and measurable ϕ on, L,, he {F }-predicable quadraic variaion of M ϕ akes he form ( (4.4 M ϕ = A ( = ϕ 2 (x, sh(x ν ϕ 2 s (dx ds,,l for,. Now, for B B, L and,, define (4.5 M (B =. M ( = Q ( A (. I B Le B, L denoe he algebra generaed by he inervals, x], x, L. I is easy o verify ha M = {M (B, F,, B B, L} is a maringale measure (for compleeness, a proof is provided in Lemma A.1 of he Appendix. We now show ha M is in fac an orhogonal maringale measure (see page 288 of Walsh 31] for a definiion. Essenially, he orhogonaliy propery holds because almos surely, no wo deparures occur a he same ime. In Lemma 4.2 below, we firs sae a sligh generalizaion of his I B I B is

16 16 H. KASPI AND K. RAMANAN laer propery, which is also used in Secion 8.2 o esablish an asympoic independence resul. Given r, s,, le D,r (s denoe he cumulaive number of deparures during (r, r + s] of cusomers ha enered service a or before r. In wha follows, recall ha he noaion f( = f( f( is used o denoe he jump of a funcion f a. Lemma 4.2. For every N N, P almos surely, (4.6 D ( 1,,, and (4.7 s, E (r + s D,r (s =, r,. The proof of he lemma is relegaed o Secion A.2 and he orhogonaliy propery is now esablished. Corollary 4.3. For each N N, he maringale measure M = {M (B, F ;, B B, L} is orhogonal and has covariance funcional Q (B, B. = M (B, M ( B = A I B B( (4.8 ( = h(x ν s (dx ds for B, B B, L. B B Proof. In order o show ha he maringale measure M is orhogonal, i suffices o show ha for every B, B B, L such ha B B =, he maringales {M (B; } and {M ( B; } are orhogonal or, in oher words, ha (4.9 B B = M (B, M ( B. Here,, represens he {F }-predicable quadraic covariaion beween he wo maringales. Fix wo ses B, B B, L wih B B =. By (4.1, (4.3 and Remark 4.1, i are maringales ha are compensaed sums of jumps, where he jumps occur a deparure imes of cusomers whose ages lie in he ses B and B, respecively. Since, by (4.6 of Lemma 4.2, here are almos surely no wo deparures ha occur a he same ime, he se of jump poins of M (B and M ( B are almos surely disjoin. By Theorem 4.52 of Chaper 1 of Jacod and Shiryaev 16], i hen follows ha he maringales are orhogonal and (4.9 holds. The relaion (4.8 follows on combining (4.9 wih (4.4 and he biaddiiviy of he covariance funcional. follows ha M (B = M I B and M ( B = M I B The orhogonaliy propery esablished in Corollary 4.3 allows us o define sochasic inegrals wih respec o he maringale measure M. The sochasic inegral is defined for a large class of so-called predicable inegrands saisfying a suiable inegrabiliy propery (see page 292 of Walsh 31] which, since EA 1 (T ] < by Lemma 5.6 of 21]

17 SPDE LIMITS OF MANY-SERVER QUEUES 17 and ν is a finie non-negaive measure, includes he class of deerminisic funcions in C b (, L,. Moreover, by Theorem 2.5 on page 295 of Walsh 31], i follows ha for all ϕ C b (, L,, he sochasic inegral {M (ϕ(b, {F };, B B, L} of ϕ wih respec o M is a càdlàg orhogonal maringale measure wih covariance funcional (4.1 M (ϕ(b, M ( ϕ( B = ( B B ϕ(x, s ϕ(x, sh(xν s (dx ds for bounded, coninuous ϕ, ϕ and B, B B, L. When B =, L, we will drop he dependence on B and simply wrie M (ϕ = M (ϕ(, L. Remark 4.4. For ϕ C b (, L,, he sochasic inegral M (ϕ admis a càdlàg version. Indeed, he càdlàg maringale M ϕ defined in (4.3 is a version of he sochasic inegral M (ϕ. I was shown in Lemma 5.9 of 21] ha M. = M N M in he space D MF,L,. Now, le M be he diffusion-scaled version of he process, (4.11 M. = M N. I is clear from he above discussion ha each M is an orhogonal maringale measure wih covariance funcional ( Q (B, B = h(x ν s (dx ds, B B and ha for any ϕ in a suiable class of funcions ha includes he space C b (, L,, he sochasic inegral M (ϕ is a well defined càdlàg, orhogonal {F } maringale measure. Moreover, for every ϕ, ϕ C b (, L, and,, ( (4.12 M (ϕ, M ( ϕ = ϕ(x, s ϕ(x, sh(x ν s (dx ds Some associaed sochasic convoluion inegrals. In Proposiion 6.4 i is shown ha he sochasic measure-valued process {ν, } ha describes he ages of cusomers in he N-server sysem admis a represenaion ha is similar o he represenaion (3.1 for is fluid counerpar {ν, }, excep ha i conains an addiional sochasic,l

18 18 H. KASPI AND K. RAMANAN erm involving a sochasic convoluion inegral wih respec o he maringale measure M, which is defined below. For N N, ϕ C b (, L, and,, define (4.13 H (ϕ. =,L,] ϕ(x + s, s 1 G(x + s M (dx, ds. 1 G(x For each,, he sochasic inegral wih respec o M in (4.13 is well defined because M is an orhogonal maringale measure and he funcion (x, s ϕ(x + s, s(1 G(x + s/(1 G(x lies in C b (, L, for all ϕ C b (, L,. The scaled version of his quaniy is hen defined in he obvious manner: for N N, ϕ C b (, L, and,, le (4.14 Ĥ (ϕ. N = H =,L,] 1 G(x + s ϕ(x + s, s M (dx, ds. 1 G(x 4.3. Relaed limi quaniies. We now define some addiional quaniies, which we subsequenly show (in Corollaries 8.3 and 8.7 o be limis of he sequences { M } N N and {Ĥ } N N. Fix a probabiliy space ( Ω, F, P and, on his space, le M = { M (B, B B, L,, } be a coninuous maringale measure wih he deerminisic covariance funcional (4.15 Q (B, B =. ( M(B, M( B = I B B(xh(xν s (dx ds,l for,. Thus, M is a whie noise. Le C M denoe he subse of coninuous funcions on, L, ha saisfies ( (4.16 ϕ 2 (x, sh(x ν s (dx ds <,,.,L Noe ha C M includes, in paricular, he space C b (, L,. For any ϕ C M and,, he sochasic inegral of ϕ wih respec o M on, L, ], denoed by (4.17 M (ϕ =. ϕ(x, s M(dx, ds,,l,] is well defined. In fac, for such ϕ, M(ϕ = { M (ϕ, } is a càdlàg, orhogonal maringale measure; see page 292 of Walsh 31] for he definiion. Moreover, because M is a coninuous maringale measure, M(ϕ has a version as a coninuous real-valued process. In fac, as Corollary 8.3 shows, M admis a version as a coninuous H 2 -valued process.

19 SPDE LIMITS OF MANY-SERVER QUEUES 19 Nex, for, and f C b, L, le Ĥ(f be he random variable given by he following convoluion inegral: (4.18 Ĥ (f =. 1 G(x + s f(x + s M(dx, ds. 1 G(x,L,] In order o express he convoluion inegrals in a more succinc fashion, consider he family of operaors {Ψ, } defined, for > and (x, s, L,, by (4.19 (Ψ f (x, s =. f(x + ( s + 1 G(x + ( s+, 1 G(x for bounded and measurable funcions f on, L. where recall ( s + = max( s,. Each operaor Ψ maps he space of bounded measurable funcions on, o he space of bounded measurable funcions on, L, and we also have (4.2 sup Ψ f f., The processes Ĥ and Ĥ, respecively, can hen be rewrien in erms of M and as follows: M (4.21 Ĥ (f = M (Ψ f, Ĥ (f = M (Ψ f,. I is shown in Lemma 8.6 and Corollary 8.7 ha if f is bounded and Hölder coninuous hen he real-valued sochasic process Ĥ (f = {Ĥ (f, } admis a càdlàg version and he process Ĥ(f = {Ĥ(f, } admis a coninuous version, and, moreover, ha Ĥ and Ĥ also admi versions as, respecively, càdlàg and coninuous H 2-valued processes. 5. Main resuls. The main resuls of he paper are saed in Secion 5.3. They rely on some basic assumpions and he definiion of a cerain map, which are firs inroduced in Secions 5.1 and 5.2, respecively. Corollaries of he main resuls are discussed in Secion Basic assumpions. For Y = E, ν, X, K, le Y be he corresponding fluid limi as described in Theorem 1. For N N, he diffusion scaled quaniies Ŷ are defined as follows: (5.1 Ŷ. = N (Y Y. For simpliciy, we only consider arrival processes ha are eiher renewal processes or imeinhomogeneous Poisson processes. Assumpion 3. The sequence {E } N N of cumulaive arrival processes saisfies one of he following wo condiions:

20 2 H. KASPI AND K. RAMANAN (a here exis consans λ, σ 2 (, and β R such ha for every N N, E is a renewal process wih i.i.d. iner-renewal imes {ξ j } j N ha have mean 1/λ and variance (σ 2 /λ/(λ 2, where (5.2 λ. = λn β N. and he following Lindeberg condiion holds: for every ε >, ] lim N 2 E (ξ 1 2 I { N ξ } N>ε = ; (b here exis locally inegrable funcions λ and β on, such ha for every N N, E is an inhomogeneous Poisson process wih inensiy funcion (5.3 λ (. = λ(n β( N,,. Remark 5.1. Le λ( and β( be he locally inegrable funcions defined in Assumpion 3, and noe ha hey are in fac consan if Assumpion 3(a holds. Also, le σ( be he locally square inegrable funcion ha is equal o he consan σ 2 if Assumpion 3(a holds, and is equal o ( λ( 1/2 if Assumpion 3(b holds. Then, given a sandard Brownian moion B, he process Ê given by 1 (5.4 Ê(. = σ(s db(s β(s ds,,, is a well defined diffusion and herefore a semimaringale, wih σ(s db(s,, being he local maringale and β(s ds,, he finie variaion process in he decomposiion. If Assumpion 3 holds, hen i is easy o see ha E in Assumpion 1 is given by E( = λ(s ds,, and Ê Ê as N (a proof of he laer convergence can be found in Proposiion 8.4, which esablishes a more general resul. We now impose a echnical condiion on he service disribuion, which is used mainly o esablish he convergence of Ĥ (f o Ĥ(f in D R, for bounded and Hölder coninuous funcions f in Secion 8. Assumpion 4. The funcion y (1 G(x + y/(1 G(x is Hölder coninuous on,, uniformly wih respec o x, L, ha is, here exis C G <, γ G (, 1] and δ > such ha for every x, L and y, ỹ, L wih y ỹ < δ, (5.5 G(x + y G(x + ỹ 1 G(x C G y ỹ γ G. Remark 5.2. As shown below, Assumpion 4 is saisfied if eiher h is bounded, or if here exiss l < such ha sup x l, h(x < and G is uniformly Hölder coninuous on, L. In eiher case, i follows ha L = because he hazard rae funcion h is

21 SPDE LIMITS OF MANY-SERVER QUEUES 21 locally inegrable, bu no inegrable, on, L. Under he firs condiion above, for any x, y, ỹ,, ỹ < y, G(x + y G(x + ỹ ỹ 1 G(x = g(x + u ỹ y 1 G(x du h(x + u du h y ỹ, y and so Assumpion 4 is saisfied. On he oher hand, if here only exiss l < such ha sup x l, h(x <, bu G is uniformly Hölder coninuous on,, wih consan C < and exponen γ >, hen sraighforward calculaions show G(x + y G(x + ỹ 1 G(x ( max C 1 G(l, I l, (xh(x (y ỹ γ 1, and once again Assumpion 4 is saisfied. A relaively easily verifiable sufficien condiion for G o be uniformly Hölder coninuous is ha g L 1+α for some α > (recall ha since g is a densiy, we auomaically have g L 1 ; hus he laer condiion imposes jus a lile addiional regulariy on g. Indeed, in his case, Hölder s inequaliy implies ha y G(y G(ỹ = g(u du g L 1+α (y ỹα/(1+α, ỹ and so G is uniformly Hölder coninuous wih exponen γ = α/(1 + α < 1. Now, given s, recall ha ν s a ime s, and define J ν s (f =.,L f(x + represens he (scaled and cenered age disribuion 1 G(x + ν s (dx, f C b, L,. 1 G(x The process {J ν (f, } plays an imporan role in he analysis because i arises in he represenaion for f, ν given in Proposiion 6.4. In order o wrie J ν s more concisely, consider he following family of operaors: for,, define (5.6 (Φ f (x. = f(x + 1 G(x +, x, L. 1 G(x Since G is coninuous, each Φ maps he space of bounded and measurable (resp., coninuous funcions on, L ino iself and, moreover, (5.7 sup Φ f f., For fuure purposes, noe ha {Φ, } defines a semigroup, ha is, Φ f = f and (5.8 Φ (Φ s f = Φ +s f, s,. Also, recalling he definiion (4.19 of he family of operaors {Ψ, }, i is easily verified ha for every bounded and measurable funcion f on, L and s,, (5.9 (Ψ s Φ f(x, u = (Ψ s+ f(x, u, (x, u, L, s].