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DECREUSEFOND AND P. MOYAL of normalized sequences ([3, 15]) are now classical results. Let us also men- tion the recent study on the existence/uniqueness of a stationary regime for the process counting the largest remaining processing time of a customer in the system ([21]). But when one aim is to have more accurate information on the state of the system, such as the total amount of work at current time (workload) or the number of customers having remaining processing time in a given range, no such simple state descriptor can be used. In order to address such ques- tions, one has to know at current time the exhaustive collection of residual processing times of all the customers present in the system. Consequently, we represent the queue by a point measure-valued process (µt)t≥0, putting Dirac measures at all residual processing times. The price to pay to have such a global information is therefore to work on a very big state space, in fact, of infinite dimension. In the past fifteen years an increasing interest has been dedicated to the study of such measure-valued Markov processes (for definition and main properties, see the reference survey of Dawson [5] on this subject). Such a framework is particularly adequate to describe particles or branching systems (see [5, 22, 24]), or queueing systems whose dynamics are too complex to be carried on with simple finite-dimensional processes: the processor sharing queue (see [12, 26]), queues with deadlines (see [6] for a queue under the earliest deadline first service discipline without reneging, [7, 8] for the same system with reneging and [13] for a processor sharing queue with reneging), or the Shortest Remaining Processing time queue ([1, 20]). In this paper we aim to identify the “mean behavior” of the measure valued process (µt)t≥0 describing the pure delay system, introduced above. In that purpose, we use the recent tools of normalization of processes to identify the fluid limit of the process, or formal law of large numbers. This fluid limit is the continuous and deterministic limit in law of a normalized sequence of these processes. We characterize as well the accuracy of this approximation by providing the corresponding functional central limit theorem, that is, the convergence in law of the normalized process of difference between the normalized process and its fluid limit to a diffusion. Formally, it is rather straightforward in our case to identify the infinitesi- mal generator of the Feller process (µt)t≥0. The natural but unusual term is that due to the continuous decreasing of the residual processing times at unit rate as time goes on. This term involves a “spatial derivative” of the measure µt, a notion which can only be rigorously defined within the framework of distributions. Because of this term, the fluid limit equation [see (4)] is the integrated version of a partial differential equation rather than an ordinary differential equation, as it is the rule in the previously studied queueing sys- tems. Thus, the classical Gronwall’s lemma is of no use here. Fortunately, A CENTRAL LIMIT THEOREM FOR THE M/GI/∞ QUEUE 3 we can circumvent this difficulty by solving the involved integrated equa- tion, known as transport equation, which is simple enough to have a closed form solution—see Theorem 1. Then, we can proceed using more classical techniques to show the convergence in law of the normalized sequence to the fluid limit (the authors have been informed during the review process of this paper that a similar result has been announced independently in [13]). As a second step, we show the weak convergence of the normalized sequence of deviation to the limit to a diffusion process. This paper is organized as follows. After some preliminaries in Section 2, we define properly the profile process (µt)t≥0 in Section 3, show, in partic- ular, that (µt)t≥0 is Feller–Dynkin, and give the corresponding martingale property. In Section 4, we give the fluid limit of (µt)t≥0. We deduce from this result fluid approximations of some performance processes in Section 5. We prove the functional central limit theorem for (µt)t≥0 in Section 6, and give the diffusion approximations of the performance processes in Section 6.3.
2. Preliminaries. We denote by b (resp. b, K ) the set of real-valued functions defined on R which are bounded,D right-continuousC C with left-limit (rcll for short) (resp. bounded continuous, continuous with compact sup- port). The space b is equipped with the Skorokhod topology and b with the topology of theD uniform convergence. The space of bounded (resp.C with 1 compact support) differentiable functions from R to itself is denoted by b (resp. 1 ) and for φ 1, C CK ∈Cb ′ φ ∞ := sup( φ(x) + φ (x) ). k k x∈R | | | | We denote by 1 the real function constantly equal to 1, I, the identity function I(x)= x, x R, and for all Borel set B B(R), 1B the indicator function of B. For all∈f and all x R, we denote∈ by τ f, the function ∈ Db ∈ x τxf( ) := f( x). The· Schwartz·− space, denoted by , is the space of infinitely differentiable functions, equipped with the topologyS defined by the semi-norms dγ β N N φ β,γ := sup x γ φ(x) , β , γ . | | x∈R dx ∈ ∈
Its topological dual, the space of tempered distributions, is denoted by ′, and the duality product is classically denoted µ, φ . The distributional derivativeS of some µ ′ is µ′ ′ such that µ′, φh = i µ, φ′ for all φ . For all µ ′ and x ∈R S, we denote∈ S by τ µ theh temperedi −h distributioni satis-∈ S ∈ S ∈ x fying τxµ, φ = µ,τxφ for all φ . h i h i ∈ S + The set of finite nonnegative measures on R is denoted by f (which is ′ M part of ) and p is the set of finite counting measures on R. The space + is equippedS M with the weak topology σ( +, ), for which + is Polish Mf Mf Cb Mf 4 L. DECREUSEFOND AND P. MOYAL
+ (we write µ,f = f dµ for µ f and f b). We say that a sequence n h i+ ∈ M ∈ D n w µ N∗ of converges weakly to µ, and denote µ µ, if for all f , { }n∈ Mf R ⇒ ∈Cb µn,f tends to µ,f . h Wei also denoteh for alli x R and all ν +, τ ν the measure satisfying for ∈ ∈ Mf x all Borel set B, τxν(B) := ν(B x). Remark that these last two definitions ′ − ′ + coincide with that in : for all φ and ν f , ν, φ S′ = ν, φ M+ , and S ∈ S ∈ M h i h i f for all x R, τxν, φ = ν,τxφ . For a∈ randomh variablei h (r.v.i for short) X defined on a fixed probability n space (Ω, ), we say that X n∈N∗ converges in distribution to X, and denote XnF X, if the sequence{ of} the distributions of the Xn’s tends weakly to that of X⇒. Let ( +, R) be the set of continuous functions from + to R. Let- C Mf Mf ting 0 < T , for E a Polish space, we denote ([0, T ], E), respectively ([0, T ], E),≤∞ the Polish space (for its usual strongC topology) of continu- ous,D respectively rcll, functions from [0, T ] to E. The mutual variation of two local martingales (M ) and (N ) in ([0, T ], R) is denoted by t t≥0 t t≥0 D (
1 when no ambiguity is possible, we often write for instance X for the process (Xt)t≥0. A CENTRAL LIMIT THEOREM FOR THE M/GI/∞ QUEUE 5
(resp. local martingale, martingale). According to [18], page 13, the tensor- quadratic process <
Theorem 1 ([8], Theorem 1). The only solution in ([0, T ], ′) of the integrated transport equation associated to (K, g) is givenD for all t S [0, T ] by ∈ (5) Xt = τtK + gt + N(g)t, where the mapping N is defined in (3).
Note that Propositions 3 and 4 show, in particular, that X defined by (5) is an element of ([0, T ], ′). D S 3. The profile process of the M/GI/∞ queue. Hereafter we consider a pure delay system M/GI/ : on a probability space (Ω, , P), consider a Poisson process ∞ F 1 Nt := {Ti≤t} ∗ iX∈N of positive intensity λ, representing the arrivals of the customers in a queue- ing system with an infinite reservoir of servers. Hence, any of them is im- mediately attended upon arrival. The time spent in the system by the ith arriving customer (denoted Ci) equals the service duration σi he requests. We assume that the sequence of marks σi i∈N∗ is i.i.d. with the random vari- able σ having the distribution α +{. We} denote at all time t 0, X the ∈ Mf ≥ t number of customers currently in the system, and St the number of already served customers at t, related to Nt and Xt by the relation Nt = Xt + St. The workload process (Wt)t≥0 equals at time t the total amount of service 6 L. DECREUSEFOND AND P. MOYAL requested by the customers in the system at t, in time units. The profile pro- cess of the queue is the point-measure valued process (µt)t≥0 whose units of mass represent the remaining processing times (i.e., time to the service completion) of all the customers who already entered the system at current time. In other words, for all t 0, ≥ Nt
µt = δ(Ti+σi−t). Xi=1 In this expression a nonpositive remaining processing time (T + σ t 0) i i − ≤ stands for a customer who already left the system, precisely t (T + σ ) − i i time units before t. As easily seen, for all t 0 and any x
Proposition 1. The process (µt)t≥0 is a Feller–Dynkin process of ([0, ), +). D ∞ Mf Proof. According to Lemma 3.5.1 and Corollary 3.5.2 of [5], the Markov process (µt)t≥0 enjoys the Feller–Dynkin property if: 1 E (i) For all f K , h> 0, the mapping µ [Ff (µh) µ0 = µ] is contin- uous. ∈C 7→ | (ii) For all h> 0, E[F1(µh) µ0 = µ] 0, as µ(R) + . (iii) For all µ + and f | 1 , E[F−→(µ ) µ = µ] → F∞(µ) as h 0. ∈ Mf ∈CK f h | 0 −→ f → Let us denote for all f 1 , F the mapping from + into R defined by ∈CK f Mf F (µ)= e−hµ,fi. In view of (7) and (8), we have for all µ + and f 1 f ∈ Mf ∈CK E[Ff (µh) µ0 = µ] (9) | = e−hτhµ,fi 1 λh + λh e−hτhδx,fi dα(x)+ ǫ (h) , − f Z where
ǫ (h)= P[ (h)] (1 λh) + P[ (h)] λh e−f(x−h) dα(x) f { A0 − − } { A1 − } Z + E[ e−f(σ1−(h−T1)) e−f(σ1−h) 1 ] { − } A1(h) i E −f(σj −(h−Tj )) 1 + e Ai(h) , "( ) # Xi≥2 jY=1 8 L. DECREUSEFOND AND P. MOYAL which is a o(h) from the same arguments as for (8). Note, moreover, that ǫf (h) does not depend on µ. Hence, the continuity in (i) is granted by the fact that, for all h> 0, the mapping µ τ µ is continuous from + into 7→ h Mf itself. For (ii), remark that the total mass on R of τhµ equals that of µ. Finally, the convergence in (iii) holds since µ,τhf h→0 µ,f by domi- nated convergence. The proposition is proved.h i −→ h i
As easily seen from (9), and since the set of linear combinations of Ff , f 1 , is dense in ( +, R) (see [26], Proposition 7.10), the infinitesimal ∈CK C Mf generator of (µ ) is given for all µ + by A t t≥0 ∈ Mf E[F (µ ) µ = µ] F (µ) F (µ) := lim h | 0 − A h→0 h
F (τhµ) F (µ) = lim − λF (µ)+ λ F (µ + δx) dα(x), h→0 h − Z for all F ( +, R) such that the latter limit exists (we say that F belongs ∈C Mf to the domain of ). A Proposition 2. For all φ 1, the process defined for all t 0 by ∈Cb ≥ t M (φ)= µ , φ µ , φ + µ , φ′ ds λt α, φ t h t i − h 0 i h s i − h i Z0 is an rcll square integrable t-martingale. For all φ, ψ , the mutual vari- ation of (M (φ)) with (MF (ψ)) is given for all t ∈ S0 by t t≥0 t t≥0 ≥ (10) < M.(φ), M.(ψ) > = λt α, φψ . t h i Proof. For all φ 1, the mapping Π from + into R defined by ∈Cb φ Mf Πφ(µ)= µ, φ clearly belongs to the domain of . As a consequence of Dynkin’sh lemmai ([9, 10]), the process defined for allA t 0 by ≥ t Mt(φ) = Πφ(µt) Πφ(µ0) Πφ(µs) ds − − 0 A (11) Z t = µ , φ µ , φ + µ , φ′ ds λt α, φ h t i − h 0 i h s i − h i Z0 is an rcll -local martingale. Let now ψ 1. The mapping Π from + Ft ∈Cb φ,ψ Mf into R defined by Πφ,ψ(µ) = Πφ(µ)Πψ(µ) also belongs to the domain of , implying that A t (12) M˜ (φ, ψ) = Π (µ )Π (µ ) Π (µ )Π (µ ) Π (µ ) ds t φ t ψ t − φ 0 ψ 0 − A φ,ψ s Z0 A CENTRAL LIMIT THEOREM FOR THE M/GI/∞ QUEUE 9 is as well an rcll -local martingale. But as easily checked, for all µ +, Ft ∈ Mf (13) Π (µ) = Π (µ) Π (µ) + Π (µ) Π (µ)+ λ α, φψ . A φ,ψ φ A ψ ψ A φ h i Itˆo’s formula yields together with (11) that, for all t 0, ≥ Πφ(µt)Πψ(µt) t t = Π (µ )Π (µ )+ Π (µ ) dM (ψ)+ Π (µ ) Π (µ ) ds φ 0 ψ 0 φ s s φ s A ψ s Z0 Z0 t t + Π (µ ) dM (φ)+ Π (µ ) Π (µ ) ds ψ s s ψ s A φ s Z0 Z0 + < Πφ(µ.), Πψ(µ.) >t, which, together with (12) and (13), implies t t Πφ(µs) dMs(ψ)+ Πψ(µs) dMs(φ)+ < Πφ(µ.), Πψ(µ.) >t Z0 Z0 = M˜ (φ, ψ)+ λt α, φψ . t h i By identifying the finite variation processes, we obtain that P-a.s. for all t 0, ≥ (14) < Π (µ.), Π (µ.) > = λt α, φψ , φ ψ t h i but in view of (11), this last quantity equals < M.(φ), M.(ψ) >t. In partic- ular, for all t 0, ≥ 2 E[< M.(φ) > ]= λt α, φ < , t h i ∞ hence, (Mt(φ))t≥0 is a square integrable martingale.
4. Fluid limit. Consider a sequence of M/GI/ systems such that the n ∞ n nth system has an initial profile µ0 , is fed by a Poisson process (Nt )t≥0 of n n arrival times Ti i∈N∗ and of intensity λ , in which the customers request { } n n i.i.d. service durations σi i∈N∗ which have the nonatomic distribution α of a r.v. σn. We assume, furthermore,{ } that σn is integrable, that is, αn,I < . n h i ∞ The process (µt )t≥0, defined for all t by n Nt n µ = δ n n , t Ti +σi −t Xi=1 n is the profile process of this nth system. Denote by ( t )t≥0 the associated filtration. Let us also define as previously the performanceF processes of the nth system: Xn, Sn and W n. We normalize the process µn in time, space and weight by defining for all t 0 ≥ n Nnt n 1 µ¯ = δ(T n+σn−nt)/n. t n i i Xi=1 10 L. DECREUSEFOND AND P. MOYAL
Thus, for all Borel set B and all t, µn (nB) µ¯n(B)= nt , t n n where nB := nx,x B . In words,µ ¯t is 1/n times the point measure having { ∈ } n n atoms at levels n times smaller than that of µnt. We also define ( t )t≥0 := n G ( nt)t≥0, the associated filtration, and normalize the arrival process as well asF the performance processes of the nth system the corresponding way, that is, for all t 0, ≥ n n n Nnt n Xnt n N¯ := , X¯ := = µ¯ , 1R∗ , t n t n h t + i n n n Snt n n Wnt n S¯ := = µ¯ , 1R− , W¯ := = µ¯ ,I1R∗ . t n h t i t n2 h t + i ∗ n n n We denote for all i N ,σ ¯i := σi /n. Thus, the σ¯i i∈N∗ are i.i.d. with the nonatomic distribution∈ α ¯n of the r.v. σn/n. The distribution{ } α ¯n is such that α¯n,I < , and satisfies for all Borel set B h i ∞ α¯n(B)= αn(nB). We assume hereafter that the following hypothesis holds.
Hypothesis 1. There exists λ> 0 such that (15) λn λ, n−→→∞ For all ε> 0, there exists M > 0 such that, for all n N∗, ε ∈ (16) P[ µn, 1 > nM ] ε. h 0 i ε ≤ For some measureµ ¯∗ of + such that 0 Mf µ¯∗,I < , h 0 i ∞ for all f , ∈ Db (17) µ¯n,f µ¯∗,f in probability. h 0 in −→→∞h 0 i There exists a nonatomic probability distributionα ¯∗ such that α¯n w α¯∗, ⇒ α¯n,I α¯∗,I < . h in −→→∞ h i ∞
1 n Let φ b and ψ ( )= φ( /n)/n. According to Proposition 2, the process defined for∈C all t by · · t (18) M¯ n(φ) := M n (ψn)= µ¯n, φ µ¯n, φ + µ¯n, φ′ ds λnt α¯n, φ t nt h t i − h 0 i h s i − h i Z0 A CENTRAL LIMIT THEOREM FOR THE M/GI/∞ QUEUE 11 is a square integrable n-martingale of ([0, ), R), such that Gt D ∞ n n λ n 2 (19) < M¯ . (φ) > = t α¯ , φ . t n h i In other words, the process M¯ n defined for all t by t M¯ n =µ ¯n µ¯n (¯µn)′ ds λntα¯n t t − 0 − s − Z0 ′ n is a -valued t -martingale of tensor-quadratic process given for any φ and ψ inS by G S λn << M¯ n >> (φ, ψ)= t α¯n, φψ . t n h i
Theorem 2. Assume that Hypothesis 1 holds. Then µ¯n = µ¯∗ in ([0, ), +), ⇒ D ∞ Mf ∗ + where µ¯ is the deterministic element of ([0, ), f ) defined for all t 0 and all φ by C ∞ M ≥ ∈ Db t (20) µ¯∗, φ = µ¯∗,τ φ + λ α¯∗,τ φ ds. h t i h 0 t i h s i Z0 n + Proof. First we prove that µ¯ n∈N∗ is tight in ([0, ), f ). To this end, it suffices to show conditions{ } C.1 and C.2 of TheoremD ∞ A.2M in the Appendix. To show C.1, fix φ 1 and T > 0. Remarking that, for all s 0, ∈Cb ≥ µ¯n, 1 N¯ n + µ¯n, 1 , h s i≤ s h 0 i equation (18) yields P-a.s. for all u < v T , ≤ µ¯n, φ µ¯n, φ |h v i − h u i| v n ′ n n ¯ n ¯ n µ¯s , φ ds + λ α¯ , φ v u + Mφ (v) Mφ (u) ≤ u |h i| |h i|| − | | − | (21) Z v u φ′ (N¯ n + µ¯n, 1 )+ φ λn ≤ | − |{k k∞ T h 0 i k k∞ } + M¯ n(v) M¯ n(u) . | φ − φ | Let ξ> 0. From Doob’s inequality, n n 4 n 4 λ 2 P sup M¯ (φ) ξ E[< M¯ . (φ) >T ] φ ∞T 0, t 2 2 n→∞ t≤T | |≥ ≤ ξ ≤ ξ n k k −→ in view of (19). Hence, it follows from the standard convergence criterion n on ([0, T ], R) that (M¯ (φ)) N∗ converges in distribution to the null D { t t≥0}n∈ 12 L. DECREUSEFOND AND P. MOYAL process. This sequence is, in particular, tight, and it is routine in view of (21) and Hypothesis 1 to check the standard tightness criterion in ([0, T ], R): for all ε> 0 and η> 0, there exists δ> 0 and N N such that, forD all n N, ∈ ≥ P n n (22) sup µ¯v , φ µ¯u, φ η ε. u,v≤T,|v−u|≤δ |h i − h i| ≥ ≤ Letting β := M φ , assumption (16) implies that, for all n N∗, ε εk k∞ ∈ (23) P[ µ¯n, φ > β ] P[ φ µ¯n, 1 > M φ ] ε. |h 0 i| ε ≤ k k∞h 0 i εk k∞ ≤ n Hence, (22) and (23) show that ( µ¯t , φ )t≥0 n∈N∗ is tight in ([0, T ], R) for all T > 0 (see [25], Theorem D.9).{ Thish sequencei } is thus tight inD ([0, ), R). D ∞ We now prove condition C.2 (compact containment). Fix T > 0. Let us first apply [12], Lemma A.2.: under Hypothesis 1, we have
¯ n nNt 1 n ∗ (24) φ(¯σi ) = (λt α¯ , φ )t≥0 in ([0, T ], R) ( n ! ) ∗ ⇒ h i D Xi=1 t≥0 n∈N for any measurable φ, such that φ is continuous on the supports ofα ¯∗ andα ¯n, n N∗, and such that φ , α¯∗ < and φ , α¯n < , n N∗. In particular, this∈ yields for any 0
Since ζ,I1R M implies that for all y> 0, ζ([y, )) M /y, we have h +i≤ T ∞ ≤ T lim sup ζ([y, )) = 0, lim sup ζ(( ,y]) = 0, y→∞ y→−∞ ζ∈KT,η ∞ ζ∈KT,η −∞ + which implies in turns that T η f is relatively compact (see [16]). Now, since up to time T no alreadyK served⊂ M customer can have a remaining pro- cessing time less than T , we have − n 1 sup ν¯t , (−∞,−T ] = 0. t≤T h i This, together with (26), implies that n lim inf P[¯µ T,η, for all t [0, T ]] 1 η. n→∞ t ∈ K ∈ ≥ − K K + T,η being the closure of T,η, we found a compact subset T,η f such that K ⊂ M n lim inf P[¯µ KT,η, for all t [0, T ]] 1 η, n→∞ t ∈ ∈ ≥ − which completes the proof of tightness. n Let now (χt)t≥0 be a subsequential limit of µ¯ n∈N∗ . We have for all t 0 { } ≥ t ∗ ′ ∗ χt =µ ¯0 + (χs) ds + λtα¯ , Z0 ∗ which is nothing but the integrated transport equation associated to (¯µ0, g), where g := λtα¯∗, t 0. From Theorem 1, we have for all t 0 and φ t ≥ ≥ ∈ S χ , φ = τ µ¯∗, φ + g , φ + N(g ), φ h t i h t 0 i h t i h t i t d = µ¯∗,τ φ + λt α¯∗, φ λ s α¯∗,τ φ ds = µ¯∗, φ , h 0 t i h i− dsh t−s i h t i Z0 integrating by parts. The subsequential limit is therefore unique in the space ([0, ), +), and equal toµ ¯∗, since is a separating class of +. D ∞ Mf S Mf 5. Fluid limits of some performance processes. Let us provide some ap- plications of Theorem 2 to the asymptotic estimation of some performance processes describing the queueing system. Assume that Hypothesis 1 hold ∗ and, in addition, that the limiting initial profileµ ¯0 in Hypothesis 1 is such that ∗ (27) µ¯0 has no atom. ∗ Assumption (27) implies in view of (20) thatµ ¯t has no atom for any t 0. Thus, Theorem 2 and the Continuous Mapping Theorem yield that,≥ for n any x ∗ 1 real deterministic function ( µ¯t , (x,y) )t≥0 (the boundaries may as well be included or infinite). Thus, ah fluid limiti approximation of the process count- ing the customers having residual service times in the range (x,y) is given for all t 0 by ≥ t y+s µ¯∗, 1 =µ ¯∗((x + t,y + t)) + λ dα¯∗(x) ds. h t (x,y)i 0 Z0 Zx+s In particular, a fluid approximation of the normalized congestion process X¯ n is given by the process X¯ ∗ defined for all t by t +∞ ∗ ∗ ∗ ∗ X¯ = µ¯ , 1R∗ =µ ¯ ((t, )) + λ dα¯ (x) ds, t h t + i 0 ∞ Z0 Zs whereas the normalized service process S¯n can be approximated by S¯∗, where t s ∗ ∗ ∗ ∗ S¯ = µ¯ , 1R =µ ¯ (( ,t]) + λ dα¯ (x) ds. t h t −i 0 −∞ Z0 Z0 Remark now that for all t T , all x 0 and all n N∗, ≤ ≥ ∈ n Nnt n n n 1 n W¯ µ¯ ,I1 + µ¯ ,I1 + σ¯ 1σ¯n≥x. t ≤ h t (0,x]i h 0 (t+x,∞)i n i i Xi=1 ∗ 1 In view of (24), the third process on the right tends to (λt α¯ ,I (x,∞) )t≥0. h ∗ 1i Hence, it is bounded over compact sets and uniformly in n since α¯ ,I (x,∞) is finite. The same argument applies to the second process onh the righti n 1 since µ¯0 ,I (x,∞) is finite as well, and the first one tends in distribution h∗ 1 i ¯ n to ( µ¯t ,I (0,x] )t≥0. Hence, W n∈N∗ is tight, and any subsequential limit h ∗i { } n thus reads ( µ¯ ,I1R∗ )t≥0. In other words, the fluid limit of W¯ is given by h t + i W¯ ∗, where ∗ ∗ W¯ = µ¯ ,I1R∗ t h t + i +∞ t +∞ = (x t) dµ¯∗(x)+ λ (x s) dα¯∗(x) ds. − 0 − Zt Z0 Zs 6. Functional central limit theorem. 6.1. Preliminary results. In this section we prove two technical results (Propositions 3 and 4), which will be useful in the sequel. We refer again the reader to the definitions and notation introduced in Section 2. Throughout this whole section, fix T > 0. Lemma 1. For all φ , the collection τrφ, r [0, T ] is a pre-compact set of . ∈ S { ∈ } S A CENTRAL LIMIT THEOREM FOR THE M/GI/∞ QUEUE 15 Proof. For all semi-norm associated to the integers β, γ, for all r [0, T ], ∈ β β β (γ) j β−j j β−j τrφ β,γ = sup x φ (x r) Cβr φ j,γ CβT φ j,γ. | | x | − |≤ | | ≤ | | jX=1 jX=1 The set τrφ, r [0, T ] is thus bounded in the nuclear space : it is a pre-compact{ set∈ in view} of Proposition 4.4.7, page 81 of [23]. S Lemma 2. For all φ , the mapping r τrφ is continuous from R into . ∈ S 7−→ S Proof. Let r′ < r. Let β and γ N. For some s ]r′, r[, ∈ ∈ β (γ) ′ (γ) τr′ φ τrφ β,γ = sup x (φ (x r ) φ (x r)) | − | x | − − − | β r r′ Cj sβ φ ≤ | − | β | |j,γ+1 jX=1 β r r′ Cj (r + 1)β φ =: M . ≤ | − | β | |j,γ+1 r,β,γ,φ jX=1 −1 Thus, for all ε> 0, fixing η := ε(Mr,β,γ,φ) , we have τr′ φ τrφ β,γ < ε whenever r r′ < η. | − | | − | We therefore have the following results. Proposition 3. The mapping G defined by (1) is continuous from ([0, T ], ′) into itself. C S Proof. We first prove that G takes its values in ([0, T ], ′). Let φ , X ([0, T ], ′), and let GX be the mapping fromCR2 intoSR defined∈ byS ∈C S φ GX (t, r)= X ,τ φ . Let t, r T . For all t′, r′, φ h t r i ≤ X ′ ′ X (28) G (t , r ) G (t, r) X ′ X ,τ φ + X ,τ ′ φ τ φ . | φ − φ | ≤ |h t − t r i| |h t r − r i| On the one hand, ( X , φ ) ([0, T ], R), and is thus bounded on [0, T ]: h s i s≥0 ∈ D sup Xs, φ = sup Xs, φ < . s≤T |h i| s∈[0,T ]∩Q |h i| ∞ Therefore, := Xs,s [0, T ] Q is a weakly bounded subset of the set of continuousX mappings{ from∈ into∩ }R. The space is Fr´echet, and hence is a tonnel in view of corollary 0,S page III.25 of [4]. FromS the Banach–Steinhaus 16 L. DECREUSEFOND AND P. MOYAL theorem (Theorem 1, page III.25 of [4]), is equicontinuous. Hence, for all ε> 0, X −1 ε ε ε Vε := Xs , = φ, sup Xs, φ < −2 2 s∈[0,T ]∩Q |h i| 2 s∈[0\,T ]∩Q is a neighborhood of 0, the zero of . The mapping r τ φ being continuous S 7→ r in view of Lemma 2, for some ηr,φ > 0 ′ ε r r < η = (τ ′ φ τ φ) V = X ,τ ′ φ τ φ < . | − | r,φ ⇒ r − r ∈ ε ⇒ |h t r − r i| 2 On the other hand, ( Xs,τrφ )s≥0 ([0, T ], R), hence, for some δr,t,φ and all t′, h i ∈C ′ ε t t < δ = X ′ X ,τ φ < . | − | r,t,φ ⇒ |h t − t r i| 2 For all t, r and all ε> 0, the two previous relations in (28) imply that, for some ηr,φ and δr,t,φ, t′ t < δ , r′ r < η = GX (t′, r′) GX (t, r) < ε. | − | r,t,φ | − | r,φ ⇒ | φ − φ | X The mapping Gφ is thus continuous, and hence uniformly continuous on the 2 compact set ([0, T ]) : for some δφ, ηφ > 0, X ′ ′ X sup sup Gφ (t , r ) Gφ (t, r) < ε. ′ ′ ′ ′ r,r ,|r−r |<ηφ t,t |t−t |<δφ | − | Finally, letting ξ := η δ , φ φ ∧ φ sup G(X)t′ , φ G(X)t, φ ′ ′ t,t ,|t−t |<ξφ |h i − h i| X ′ ′ X sup sup Gφ (t , r ) Gφ (t, r) < ε. ′ ′ ′ ′ ≤ r,r ,|r−r |<ηφ t,t ,|t−t |<δφ | − | ′ Thus, G(X) ([0, T ], ) since the map t G(X)t, φ is continuous for all φ . ∈C S 7→ h i ∈ S n Let us now show the continuity of G. Let X n∈N∗ be a sequence of ′ { } n ([0, T ], ) tending to X. In particular, for all φ , ( Xt , φ )t≥0 n∈N∗ D S ∈ S { h ni } tends in ([0, T ], R) to ( Xt, φ )t≥0, and thus supn∈N∗ supt≤T Xt , φ < . Hence, inD view of the Banach–Steinhaush i theorem, |h i| ∞ := Xn,n N,t Q [0, T ] H { t ∈ ∈ ∩ } is equicontinuous. Therefore, for all φ , ∈ S n sup Xt Xt, φ 0, t≤T |h − i|n −→→∞ A CENTRAL LIMIT THEOREM FOR THE M/GI/∞ QUEUE 17 and consequently, for all k, all collection φ , i = 1,...,k of and all i k, { i } S ≤ n sup max Xt Xt, φi 0. t≤T i=1,...,k |h − i|n −→→∞ The latter means that for all semi-norm pw of the weak topology, n sup pw(Xt Xt) 0. t≤T − n−→→∞ Thus, in view of Pietsch’s theorem ([23], Proposition 0.6.7, page 9), for all pre-compact set K of , S n sup sup Xt Xt, φ 0. t≤T φ∈K |h − i|n −→→∞ This result can be applied for a fixed φ to the collection τrφ, r [0, T ] (which is a pre-compact set of from Lemma∈ S 1) to obtain { ∈ } S n sup sup Xt Xt,τrφ 0. t≤T r≤T |h − i|n −→→∞ It follows that G is continuous, since n n sup G(X )t G(X)t, φ = sup Xt Xt,τtφ t≤T |h − i| t≤T |h − i| n sup sup Xt Xt,τrφ 0. ≤ t≤T r≤T |h − i|n −→→∞ Proposition 4. The mapping N defined by (3) is continuous from ([0, T ], ′) into ([0, T ], ′). D S C S Proof. That N takes values in ([0, T ], ′) can be shown similarly to C S Proposition 3. For the continuity of N, remark that the mapping H defined by (2) is continuous from ([0, T ], ′) into ([0, T ], ′) since, as well known, D t S C S the mapping (Xt)t≥0 ( 0 Xs ds)t≥0 is continuous from ([0, T ], R) into ([0, T ], R) for the Skorokhod7−→ topology. The proof then proceedsD as that of R CProposition 3. n n 6.2. Central limit theorem. Fix ξ, λ n∈N∗ , λ, and σ n∈N∗ satisfying n { } { } Hypothesis 1. µ¯ n∈N∗ is the corresponding sequence of normalized pro- { } n ∗ file processes. From Theorem 2, µ¯ n∈N∗ tends in distribution toµ ¯ in + { } ([0, ), f ), defined by (20). Throughout this section, we make the fol- Dlowing∞ additionalM assumptions. Hypothesis 2. (29) √n(λn λ) 0. − n−→→∞ 18 L. DECREUSEFOND AND P. MOYAL There exists ′, such that for all φ , Y0 ∈ S ∈ S (30) √n µ¯n µ¯∗, φ , φ in probability, h 0 − 0 i −→hY0 i (31) √n α¯n,I α¯∗,I 0. |h i − h i|n −→→∞ Let n N∗. Define the following processes for all t by ∈ n = √n(¯µn µ¯∗), Yt t − t n = √nM¯ n, Mt t n(φ, ψ)= λnt α¯n, φψ , φ, ψ . At h i ∈ S n ′ n The process is an -valued t -martingale, hence, for all φ , the nM S G ∈ S process ( t , φ )t≥0 is a real square integrable martingale of increasing process ( hMn(φ, φi)) [by (19)]. At t≥0 n Lemma 3. Under Hypotheses 1 and 2, for all T > 0, the sequence n∈N∗ converges in distribution to the martingale of ([0, T ], ′), whose{M tensor-} quadratic process satisfies for all t [0, T ] andM allDφ, ψ S ∈ ∈ S (32) << >> (φ, ψ)= λt α¯∗, φψ . M t h i Proof. We use the convergence criterion, Theorem A.3 in the Appendix. Define for all t 0 and φ, ψ ≥ ∈ S γ (φ, ψ) := λt α¯∗, φψ . t h i n 2 n n For all φ , γ0(φ, φ) = 0 and for all n, (( t , φ ) t (φ, φ))t≥0 is a t - local martingale,∈ S thus conditions (37) andhM (38) arei −A verified. On the otherG hand, for all T > 0 and ψ , we have P-a.s. for all t T ∈ S ≤ n(φ, ψ) γ (φ, ψ) = (λn λ)t α¯n, φψ + λt α¯n α¯∗, φψ , At − t − h i h − i hence, n 2 sup t (φ, ψ) γt(φ, ψ) t≤T {A − } 2T 2 (λn λ)2 φψ 2 + λ2 (φψ)′ 2 α¯n α¯∗,I 0, ≤ { − k k∞ k k∞h − i}n −→→∞ which proves (41) taking ψ := φ. Finally, for all T > 0, condition (40) is met since ( n(φ, φ)) ([0, T ], R) and At t≥0 ∈C E n n 2 E n n 2 sup( t , φ t−, φ ) sup( µ¯t , φ µ¯t−, φ ) t≤T hM i − hM i t≤T h i − h i φ k k∞ 0, ≤ n n−→→∞ A CENTRAL LIMIT THEOREM FOR THE M/GI/∞ QUEUE 19 n kφk∞ the jumps of ( µ¯t , φ )t≥0 being a.s. of size less than n : we have (39) as well. Therefore,h we cani apply Theorem A.3 to state that n (33) ( , φ ) N∗ = (P ) in ([0, ), R), { hMt i t≥0}n∈ ⇒ t t≥0 D ∞ where P is a continuous martingale of increasing process (γt(φ, φ))t≥0. In n particular, for all T > 0 and all φ , ( t , φ )t≥0 n∈N∗ is tight in ([0, T ], R), ∈ S { hM in } D ′ and hence (Theorem A.1 in the Appendix), N∗ is tight in ([0, T ], ). {M }n∈ D S All subsequential limit thus satisfies t, φ = Pt for all t [0, T ] and M hMn i ∈ all φ , which means with (33) that n∈N∗ tends in distribution to the martingale∈ S of ([0, T ], ′). Finally{M and} according to the previous arguments, M D S n n n (< . , φ , . , ψ > ) N∗ = ( (φ, ψ)) N∗ { hM i hM i t t≥0}n∈ { At t≥0}n∈ tends to (<< >>t(φ, ψ))t≥0 as well as to (γt(φ, ψ))t≥0, which completes the proof. M Consider now the Hilbert space L2(dα¯∗). Sinceα ¯∗ is a probability mea- sure, we have L2(dα¯∗), and hence, (L2(dα¯∗))′ ′. Moreover, L2(dα¯∗) S⊂ ⊂ S 2 ∗ admits a countable basis, denoted hi i∈N. For all i 0 and all ψ L (dα¯ ), denote { } ≥ ∈ ∗ ci(ψ)= ψ(x)hi(x) dα¯ (x), Z the ith coordinate of ψ in that basis. We have the following result. Theorem 3. Under Hypotheses 1 and 2, for all T > 0, the sequence n ′ ∗ n∈N∗ tends in distribution in ([0, T ], ) to the process defined for {Yall t} 0 and all φ by D S Y ≥ ∈ S t ∗ √ i (34) t , φ = 0,τtφ + λ ci(τt−sφ) dBs, hY i hY i 0 Xi≥0 Z where Bi is a sequence of independent real standard Brownian motions. { }i≥0 Proof. For all n N and all t [0, T ], P-a.s., ∈ ∈ t n = n + ( n)′ ds + √n λntα¯n λtα¯∗ + n. Yt Y0 Ys { − } Mt Z0 n Hence, ( t )t≥0 solves a transport equation, and in view of Theorem 1, the above equationY amounts P-a.s. for all t T to ≤ n = G( n) + √n λntα¯n λtα¯∗ + n Yt Y0 t { − } Mt + √nλnN( α¯n) √nλN( α¯∗)+ N( n) · − · M t (35) = G( n) + √n(λn λ)tα¯n λt√n(¯α∗ α¯n)+ n Y0 t − − − Mt + √n(λn λ)N( α¯n) λ N(√n α¯∗) N(√n α¯n) − · t − { · t − · t} + N( n) , M t 20 L. DECREUSEFOND AND P. MOYAL n ′ n n where ( 0 )t≥0 denotes the -valued process constantly equal to 0 and α¯ Y∗ n S∗ ′ Yn · (resp. α¯ , √n α¯ , √n α¯ ) denotes the -valued process (tα¯ )t≥0 [resp. (tα¯∗) · , (√ntα¯· n) , (√· ntα¯∗) ]. First, fromS (30), for all φ , ξ> 0, t≥0 t≥0 t≥0 ∈ S n P[ 0, φ ξ] 0. |hY0 −Y i| ≥ n−→→∞ n Hence, in view of Corollary A.1 in the Appendix, we have that ( 0 )t≥0 ′ Y ⇒ ( 0)t≥0 in ([0, T ], ). Thus, from Proposition 3 and the Continuous Map- pingY TheoremD (see [S2]), (G( n) ) (G( ) ) in ([0, T ], ′). Y0 t t≥0 ⇒ Y0 t t≥0 D S On the other hand, we have n n n sup √n λ λ t α¯ , φ √n λ λ T φ ∞ 0, 0≤t≤T | − | |h i| ≤ | − | k k n−→→∞ n n ′ n 2 ′ sup √n λ λ N( α¯ )t, φ √n λ λ T φ ∞ 0, 0≤t≤T | − ||h · i| ≤ | − | k k n−→→∞ ∗ n ′ ∗ n sup λt√n α¯ , φ α¯ , φ λT √n φ ∞ α¯ ,I α¯ ,I 0. 0≤t≤T |h i − h i| ≤ k k |h i − h i|n −→→∞ n n n n Hence, the sequences (λ λ)√n α¯ n∈N∗ , (λ λ)N(√n α¯ ) n∈N∗ and ∗ n { − · } { − · } ′ λ(√n α¯ √n α¯ ) n∈N∗ converge to the null process (0)t≥0 of ([0, T ], ). {The convergence· − · of this} last sequence implies in view of PropositionC 4 thatS ∗ n ′ λ N(√n α¯ ) N(√n α¯ ) (0)t≥0 in ([0, T ], ). { · − · }n −→→∞ C S Now, from Lemma 3, n = in ([0, T ], ′), thus, in view of Proposi- tion 4 and the ContinuousM Mapping⇒ M Theorem,D NS( n)= N( ) in ([0, T ], ′). n M ⇒ M D ∗ S Consequently, n∈N∗ is tight and from (35), its limit in distribution satisfies P-a.s.{Y for all} t [0, T ]: Y ∈ t (36) ∗ = G( ) + + N( ) = τ + + τ ( )′ ds. Yt Y0 t Mt M t tY0 Mt t−s Ms Z0 In view of (32), the process reads = √λB(¯α∗), where B(¯α∗) is the cylindrical Brownian motion onM (L2(dα¯∗M))′ (see [18]). Thus, from (36), we have for all φ and for all t ∈ S t ∗, φ = ,τ φ + √λ B(¯α∗) , φ B(¯α∗) ,τ φ′ ds hYt i hY0 t i h t i− h s t−s i Z0 t √ i ′ i = 0,τtφ + λ ci(φ)Bt ci(τt−sφ )Bs ds , hY i − 0 Xi≥0 Z i where B i≥0 is a sequence of independent real standard Brownian motions (see again{ } [18]). Note that the latter series converges in L2(Ω) since φ L2(dα¯∗). Remark now that, for all i 0, ∈ ≥ d(c (τ φ)) i t−s = c (τ φ′). ds i t−s A CENTRAL LIMIT THEOREM FOR THE M/GI/∞ QUEUE 21 Using Itˆo’s integration by parts formula, this yields to t t c (φ)Bi c (τ φ′)Bi ds = c (τ φ) dBi , i t − i t−s s i t−s s Z0 Z0 which completes the proof. 6.3. Diffusion approximations of the performance processes. In this sec- tion we show that Theorem 3 can be used to provide diffusion approxima- tions for the congestion, service and workload processes. The stochastic in- tegrals on the right-hand side of (34) make sense for any φ L2(dα¯∗), hence, ∗ L2 ∗ ′ ∈ the process ( t )t≥0 takes values in ( (dα¯ )) , provided that the limiting initial state Y belongs to (L2(dα¯∗))′. Showing that in that case the con- Y0 vergence announced in Theorem 3 holds in ([0, T ], (L2(dα¯∗))′) is an open problem at this point, that is beyond the scopeD of this paper. However, it is easily seen that under Hypothesis 1 the limiting service time distributionα ¯∗ 2 ∗ is such that the functions 1 ∗ and 1 belong to L (dα¯ ). Hence, diffusion R+ R− approximations for the congestion and service processes can be provided by the real processes obtained when fixing φ = 1 ∗ and φ = 1 in (34), that R+ R− is, for all t 0, ≥ t +∞ ∗ ∗ i 1 ∗ √ t , R = 0((t, + )) + λ hi(x) dα¯ (x) dBs hY + i Y ∞ 0 t−s Xi≥0 Z Z and t t−s ∗ 1 √ ∗ i t , R− = 0(( ,t]) + λ hi(x) dα¯ (x) dBs. hY i Y −∞ 0 0 Xi≥0 Z Z +∞ 2 ∗ Moreover, provided that 0 x dα¯ (x) < (i.e., the limiting service time has a finite second moment), a diffusion∞ approximation for the workload R process is given for all t 0 by ≥ +∞ ∗ ,I1R∗ = (x t) d 0(x) hYt + i − Y Zt t +∞ √ ∗ i + λ (x + s t)hi(x) dα¯ (x) dBs. 0 t−s − Xi≥0 Z Z Assume, for example, thatα ¯∗ is the exponential distribution ε(1) [which is the limiting distribution obtained when taking σn ε(1/n) for all n N∗]. ∼ ∈ In that case, it is well known that a possible hi i∈N is given by the sequence of Laguerre’s polynomials, defined for all i { 0} by ≥ ex di h (x)= (e−xxi). i i! dxi 22 L. DECREUSEFOND AND P. MOYAL Then a diffusion approximation of the congestion process (and accordingly, of the service and workload processes) is given for all t 0 by ≥ t +∞ i ∗ 1 d −x i i , 1 ∗ = ((t, + )) + √λ (e x ) dx dB . t R+ 0 i s hY i Y ∞ 0 t−s i! dx Xi≥0 Z Z APPENDIX: TIGHTNESS AND WEAK CONVERGENCE ON FUNCTIONAL SPACES Let us recall some results about tightness and weak convergence on metric spaces. Theorem A.1 (cf. [19], Theorem 4.1). Let F be a nuclear Fr´echet space, ′ n ′ F be its topological dual, and X n∈N∗ be a sequence of ([0, T ], F )-valued n { } D n r.v. Then X N∗ is tight if for all φ F , the sequence ( X , φ ) N∗ n∈ t i t≥0}n∈ is tight in{ ([0}, T ], R). ∈ { h D This shows, in particular, since is a nuclear Fr´echet space, that the tight- n ′ S n ness of X n∈N∗ in ([0, T ], ) is granted by that of ( X , φ )t≥0 n∈N∗ for all φ { . Moreover,} D since isS naturally a separating{ classh of itsi topological} dual ∈′ S, we have the following:S S n ′ n Corollary A.1. For all sequence X n∈N∗ of ([0, T ], ), X = X if, for all φ , ( Xn, φ ) = ( X ,{ φ ) } in ([0D, T ], ′).S ⇒ ∈ S h t i t≥0 ⇒ h t i t≥0 D S The space + has no such good topological properties as that of ′. Mf S Nevertheless, a tightness criterion is established for sequences of +-valued Mf processes. Theorem A.2 (Jakubowski’s criterion, cf. [5], Theorem 3.6.4). The se- n + quence X n∈N∗ of ([0, T ], f ) is tight if the following two conditions hold: { } D M 1 n C.1. For all φ , the sequence ( X , φ ) N∗ is tight in ([0, ), R), ∈Cb { h t i t≥0}n∈ D ∞ C.2. For all T > 0 and 0 <η< 1, there exists a compact subset KT,η of + such that Mf n lim inf P[X KT,η t [0, T ]] 1 η. n→∞ t ∈ ∀ ∈ ≥ − Let us finally give the following result, which establishes a criterion for the convergence in distribution of a sequence of real processes to a diffusion process. A CENTRAL LIMIT THEOREM FOR THE M/GI/∞ QUEUE 23 Theorem A.3 ([10], Theorem 1.4). Let Rn and An be two se- quences of ([0, ), R), and an increasing function{ } γ {([0}, ), R) such that D ∞ ∈ D ∞ (37) Rn(0) = 0 and γ(0) = 0, (38) Rn and (Rn)2 An are two local martingales, − Ft − and for all T > 0, (39) E sup Rn(t) Rn(t ) 2 0, n→∞ t REFERENCES [1] Baccelli, F. and Bremaud,´ P. (2002). Elements of Queueing Theory, 2nd ed. Springer, Berlin. [2] Billingsley, P. (1968). Convergence of Probability Measures. John Wiley & Sons Inc., New York. MR0233396 [3] Borovkov, A. A. (1967). Limit laws for queueing processes in multichannel systems. Sibirsk. Mat. Z.ˇ 8 983–1004. MR0222973 [4] Bourbaki, N. (1973). Topological Vector Spaces. Springer, Berlin. [5] Dawson, D. A. (1993). Measure-valued Markov processes. In Ecole´ d’Et´ede´ Prob- abilit´es de Saint-Flour XXI—1991. Lecture Notes in Mathematics 1541 1–260. Springer, Berlin. MR1242575 [6] Doytchinov, B., Lehoczky, J. and Shreve, S. (2001). Real-time queues in heavy traffic with earliest-deadline-first queue discipline. Ann. Appl. Probab. 11 332– 378. MR1843049 [7] Decreusefond, L. and Moyal, P. (2005). Fluid limit of the M/M/1/1+GI-EDF queue. In IEEE Proceedings Symposium on Applications and the Internet Work- shops 05 248–251. [8] Decreusefond, L. and Moyal, P. (2008). The fluid limit of a heavily loaded EDF queue with impatient customers. Markov Process. Related Fields 14 131–158. [9] Dynkin, E. B. (1965). Markov Processes, I–II. Springer, Berlin. [10] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York. MR838085 [11] Erlang, A. K. (1917). Solutions of some problems in the theory of probabilities of significance in automatic telephone exchange. Traduction Anglaise: P.O. Elec. Eng. J. 10 189–197. [12] Gromoll, H. C., Puha, A. L. and Williams, R. J. (2002). The fluid limit of a heav- ily loaded processor sharing queue. Ann. Appl. Probab. 12 797–859. MR1925442 24 L. DECREUSEFOND AND P. MOYAL [13] Gromoll, C., Robert, P., Zwart, B. and Bakker, R. (2006). The impact of reneging in processor sharing queues. Sigmetrics Performance Evaluation Review 34 87–96. [14] Guillemin, F. and Simonian, A. (1995). Transient characteristics of an M/M/∞ system. Adv. in Appl. Probab. 27 862–888. MR1341889 [15] Iglehart, D. L. (1973). Weak convergence of compound stochastic process. I. Stochastic Processes Appl. 1 11–31; corrigendum, ibid. 1 (1973), 185–186. MR0410830 [16] Kallenberg, O. (1983). Random Measures, 3rd ed. Akademie-Verlag, Berlin. MR818219 [17] Metivier,´ M. (1982). Semimartingales: A Course on Stochastic Processes. de Gruyter Studies in Mathematics 2. de Gruyter, Berlin. MR688144 [18] Metivier, M. (1986). Stochastic Differential Equations in Infinite Dimensional Spaces. de Gruyter, Berlin. [19] Mitoma, I. (1983). Tightness of probabilities on C([0, 1]; S′) and D([0, 1]; S′). Ann. Probab. 11 989–999. MR714961 [20] Moyal, P. (2007). Stationarity of measure-valued stochastic recursions: Applications to the pure delay system and the SRPT queue. C. R. Math. Acad. Sci. Paris 345 233–237. MR2352926 [21] Moyal, P. (2007). Stationarity of pure delay systems and queues with impatient customers via stochastic recursions. Preprint. arxiv:PR/0603709. [22] Mel´ eard,´ S. and Roelly, S. (1993). Sur les convergences ´etroite ou vague de proces- sus `avaleurs mesures. C. R. Acad. Sci. Paris, S´erie I 317 785–788. MR1244431 [23] Pietsch, A. (1966). Nuclear Locally Convex Spaces. Grenzgebiete Band. [24] Roelly-Coppoletta, S. (1986). A criterion of convergence of measure-valued processes: application to measure branching processes. Stochastics 17 43–65. MR878553 [25] Robert, P. (2000). R´eseaux et Files D’attente: M´ethodes Probabilistes. Math´ematiques & Applications (Berlin) [Mathematics & Applications] 35. Springer, Berlin. MR2117955 [26] Robert, P. (2003). Stochastic Networks and Queues, French ed. Applications of Mathematics (New York) 52. Springer, Berlin. MR1996883 [27] Ustunel, S. (1982). Stochastic integration on nuclear spaces and its applications. Ann. Inst. H. Poincar´eProbab. Statist. 18 165–200. MR662449 GET/Tel´ ecom´ Paris—CNRS UMR 5141 Ceremade—CNRS UMR 5745 46 rue Barrault Universite´ Paris-Dauphine 75634 Paris Place du Marechal´ de Lattre de Tassigny France 75016 Paris E-mail: [email protected] France and Laboratoire de Mathematiques´ Appliquees´ Universite´ de Technologie de Compiegne` Centre de Recherches de Royallieu BP 20 529 60 205 Compiegne` Cedex France E-mail: [email protected]