On many-server queues in heavy traffic

The Annals of Applied Probability 2010, Vol. 20, No. 1, 129–195 DOI: 10.1214/09-AAP604 © Institute of Mathematical Statistics, 2010 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC B Y A NATOLII A. P UHALSKII AND J OSH E. R EED University of Colorado Denver and IITP, Moscow, and New York University We establish a heavy-traffic limit theorem on convergence in distribution for the number of customers in a many-server queue when the number of servers tends to infinity. No critical loading condition is assumed. Generally, the limit process does not have trajectories in the Skorohod space. We give conditions for the convergence to hold in the topology of compact convergence. Some new results for an infinite server are also provided. 1. Introduction. Heavy-traffic limits for many-server queues is hardly a new topic. In particular, there exists a substantial body of literature on the Halfin–Whitt regime which is singled out by the requirements of critical loading and certain initial conditions. In many instances, by assuming that the service time distribution lies within a specific class of probability distributions, it has been shown that in the Halfin–Whitt regime the suitably centered and normalized processes of the number of customers in the system converge in distribution for Skorohod’s J1 -topology to a process with continuous trajectories, which may be explicitly identified. This was first accomplished in the work of Halfin and Whitt [10] for the case of exponential service time distributions and has been continued by many additional authors as well for different classes of service time distributions, see, for instance, Jelenkovic, Mandelbaum, and Momcilovic [13], Mandelbaum and Momcilovic [18], Kaspi and Ramanan [14], Puhalskii and Reiman [21], Whitt [25] and the recent survey paper by Pang, Talreja and Whitt [19]. One exception to the above list is the work of Reed [22] in which no assumptions beyond a first moment are placed on the service time distribution. A related avenue of research concerns infinite servers in heavy traffic; see, for example, Krichagina and Puhalskii [15], Pang, Talreja and Whitt [19] and references therein. The purpose of this paper is to extend the aforementioned results to allow noncritical loading, generally distributed service times, and general initial conditions. We consider a sequence of G/GI /n queues. The number of servers, n, tends to infinity and the service time distribution is held fixed. It is assumed that the centered and normalized arrival processes converge in distribution. The main result asserts the weak convergence of finite-dimensional distributions of the suitably centered and normalized number-in-the-system processes to the finite-dimensional distributions of a unique strong solution of a stochastic differential equation of convolution Received June 2008; revised February 2009. AMS 2000 subject classifications. Primary 60K25; secondary 60F17, 60G15, 60G44. Key words and phrases. Many-server queues, heavy traffic, weak convergence, Skorohod space, martingales, Gaussian processes. 129 130 A. A. PUHALSKII AND J. E. REED type. In the general case, the trajectories of the limit process have discontinuities of the second kind. Under a certain condition on the fluid limit and the service time distribution, the limit stochastic process has trajectories in a Skorohod space and the convergence in distribution to this process holds for the J1 -topology. The results of Halfin and Whitt [10] and Reed [22] follow as particular cases. In fact, the convergence in distribution is stated for the stronger topology of uniform convergence on compact intervals, which we call the topology of compact convergence. This topology plays a prominent role throughout the paper so much so that the supporting results are established for this topology rather than for the J1 -topology. This emphasis is necessitated by the need to rely on certain continuity properties of equations describing the evolution of the system’s population, which we are able to establish only for the topology of compact convergence. Furthermore, the topology of compact convergence is almost too strong. Since it is nonseparable, the associated Borel σ -algebra is strictly larger than the Kolmogorov (i.e., cylindrical) σ -algebra, which leads to measurability problems; see, for example, Billingsley [1], Section 15. As a consequence, one needs to extend the notion of convergence in distribution. The relevant theory has been developed by Hoffman–Jørgensen and his followers and is expounded upon in van der Vaart and Wellner [23]. Its primary motivation was to provide tools to tackle convergence in distribution of empirical processes in strong topologies. Empirical processes also play an important part in the study of heavy traffic limits for many-server queues, see Louchard [17], Krichagina and Puhalskii [15]. This explains why we find methods developed in a different context useful for our setting. One of the challenges of working with convergence in distribution in nonseparable spaces has to do with Borel probability measures not being necessarily tight. Recall that on complete spaces the tightness and separability properties are equivalent. Many conclusions of the Hoffman–Jørgensen theory rely on the assumption that the limit probability measure is separable. It is therefore important to show that the stochastic processes that appear in the limit have separable ranges, so their distributions are separable probability measures. We accomplish this by establishing that certain limit processes are Gaussian semimartingales. Since Gaussian semimartingales jump at deterministic times, their ranges are separable sets. As a side remark, there are no known examples of nonseparable Borel probability measures and the axiom that such measures do not exist can be consistently added to the Zermelo–Fränkel set of axioms; see Dudley [8], Appendix C, and van der Vaart and Wellner [23], Section 1.3, for these and other observations. Our results, however, make no use of this fact. Another ingredient in the proof of the main theorem is a martingale argument which originates from Krichagina and Puhalskii [15]. It is instrumental in establishing tightness for the processes of interest. The key that makes an application of the methods of [15] possible is the insight of Reed [22] that the process of customers entering service in a many-server queue can be treated in analogy ON MANY-SERVER QUEUES IN HEAVY TRAFFIC 131 with the arrival process at an infinite server. There are, however, certain improvements on the derivation of the needed martingale properties as compared with the cited papers. In particular, our approach clarifies the connection with certain twoparameter processes being planar martingales, which was implicit in Krichagina and Puhalskii [15]. In addition, the construction of the limit process and a number of proofs in [15] relied on the continuity of the fluid limit. Here it is not necessarily the case, so a more subtle argument is called for. As a byproduct, in application to a G/GI /∞ system our methods enable us to remove the restriction of Krichagina and Puhalskii [15] that the fluid limit for the arrival process should be continuous. The paper is organized as follows. In Section 2, the main results are formulated and discussed and proofs are presented. We first recall basic facts about convergence in distribution for nonmeasurable mappings. We then state in Theorem 2.1 the fluid limit result of Reed [22], which is given a different proof. This proof is useful when dealing with a more general result of Theorem 2.3. Theorem 2.2 presents the stochastic approximation result. The proof relies heavily on the continuity property of convolution equations, on the one hand, and on a result of convergence in distribution of certain stochastic processes, on the other hand. The former result is proved in Appendix B and the latter result is proved in Section 4. The Gaussian limit for a G/GI /∞ system is stated in Section 2.3. This subsection also contains extended versions of Theorems 2.1 and 2.2. These results are proved by similar means, so we omit the actual proofs. Appendix C contains the martingale arguments. In Appendix D, we summarize the properties of convergence in distribution for nonmeasurable mappings which are, for the most part, borrowed from van der Vaart and Wellner [23]. Notation and conventions. The set of real numbers is denoted by R, the set of nonnegative reals is denoted by R+ , the set of nonnegative rational numbers is denoted by Q+ , the set of nonnegative integers is denoted by Z+ , and the set of natural numbers is denoted by N. For real numbers x and y, x ∧ y = min(x, y), x ∨ y = max(x, y), x + = x ∨ 0, and ⌊x⌋ denotes the integer part. Integrals of the   a  understood as except when a = 0: we interpret as form ab are 0 (a,b] [0,a] .   Similarly, R2 = [0,∞)×[0,∞) . For a function g(x) of a nonnegative real-valued + argument, g(x−) denotes the left-hand limit at x, g(x+) denotes the right-hand limit at x, and g(x) = g(x) − g(x−). We always assume that g(0−) = 0, so g(0) = g(0). If f (x, y) is a function with domain R2+ and x1 ≤ x2 and y1 ≤ y2 , we define f ((x1 , y1 ), (x2 , y2 )) = f (x2 , y2 ) − f (x2 , y1 ) − f (x1 , y2 ) + f (x1 , y1 ). 1A denotes the indicator function of set A. Products of topological spaces are assumed to be equipped with product topologies. All random entities are assumed to be defined on a common complete probability space (, F , P) with E denoting the associated expectation. Filtrations are defined as right-continuous flows of complete σ -algebras. 132 A. A. PUHALSKII AND J. E. REED 2. Main results. 2.1. Extended convergence in distribution. We recall the concept of convergence in distribution due to Hoffmann–Jørgensen as presented in van der Vaart and Wellner [23]. Let ξ be a real-valued function on , which does not have to be a random variable, i.e., to be appropriately measurable. The outer expectation E∗ ξ of ξ is defined as the infimum of Eζ over all random variables ζ on (, F , P) such that ζ ≥ ξ a.s. and Eζ is well defined. Let S be a metric space made into a measurable space by endowing it with the Borel σ -algebra B (S). Given a sequence Xn of maps from  to S and a measurable map X from (, F ) to (S, B (S)), we say that the Xn converge to X in distribution in S if lim E∗ f (Xn ) = Ef (X) n→∞ for all bounded continuous real-valued functions f on S. In this paper, S will have as its elements right-continuous with left-hand limits functions from an interval I on the real line to a complete metric space U . We denote this set by D(I, U ). If U is, in addition, separable, then it is customary to equip D(I, U ) with Skorohod’s J1 -topology and a complete separable metric; see, for example, Jacod and Shiryaev [12], Ethier and Kurtz [9]. The resulting Polish space will also be denoted by D(I, U ). We hope that no confusion will arise out of this ambiguity. For the most part, though, we will be concerned with the stronger topology of compact convergence. This topology is compatible with a complete metric which can be defined by d(x, y) = ∞  sup k=1 t∈[0,k]∩I  1 ∧ ρ(x(t), y(t)) 2−k ,  where x = (x(t), t ∈ I ) and y = (y(t), t ∈ I ) are elements of D(I, U ) and ρ is the metric on U . We denote this metric space by Dc (I, U ) and equip it with the Borel σ -algebra. One of the nice properties of the topology of compact convergence is that Dc (I, U1 ×U2 ) is homeomorphic to Dc (I, U1 )×Dc (I, U2 ) for arbitrary metric spaces U1 and U2 . [Recall that, by contrast, the topology of D(R+ , R2 ) is strictly finer than the topology of D(R+ , R)2 ]. The space Dc (I, U ), though complete, is not separable unless either I or U represent a one-point set. Therefore, the Kolmogorov σ -algebra on Dc (I, U ) is strictly smaller than the Borel σ -algebra, so a stochastic process X with trajectories in D(I, U ) need not be a random element of Dc (I, U ). However, if the range of the mapping from  to Dc (I, U ) defined by X is a separable set, then X is a random element of Dc (I, U ). We call such X a separable random element. In particular, if X has continuous trajectories a.s., or more generally, if its jumps occur at deterministic times a.s., then X is a separable random element of Dc (I, U ). We say that X is a tight random element if its distribution is a tight probability 133 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC measure on Dc (I, U ). As mentioned, X is a tight random element if and only if it is a separable random element. Suppose X is a stochastic process which is a random element of Dc (I, U ). We say that a sequence of stochastic processes Xn with trajectories in D(I, U ) converges in distribution in Dc (I, U ) (or, for the topology of compact convergence) to X if the associated maps Xn converge to X in distribution in Dc (I, U ). Convergence in distribution in D(I, U ) is defined in a standard fashion, provided U is Polish. If the latter is the case, convergence in distribution in Dc (I, U ) implies convergence in distribution in D(I, U ) and convergence in distribution in D(I, U ) to continuous-path processes implies convergence in distribution in Dc (I, U ). 2.2. Limit theorems for the many server queue. We assume as given a sequence of many server queues indexed by n, where n denotes the number of servers. Service is performed on a first-come–first-serve basis. We denote the number of customers present at time 0− by Qn0 . Out of those, Qn0 ∧ n customers are in service at time 0− and (Qn0 − n)+ customers are in the queue. We denote the remaining service times of the customers in service at time 0− by η̃1 , η̃2 , . . . . The service times of the customers in the queue at time 0− and the service times of customers exogenously arriving after time 0− are denoted by η1 , η2 , . . . and come from an i.i.d. sequence of nonnegative random variables with distribution F = (F (x), x ∈ R+ ). The ηi are not equal to zero a.s. Equivalently, we require that (2.1) F (0) < 1. We let E n (t) denote the number of exogenous arrivals by t. The entities Qn0 , {η̃1 , η̃2 , . . .}, {η1 , η2 , . . .}, and E n = (E n (t), t ∈ R+ ) are assumed to be independent. Let Qn (t) denote the number of customers either in the queue or in service at time t, let An (t) denote the number of customers that enter service after time 0− and by time t, and let Q̃n (t) denote the number of customers remaining in service at time t out of those that were in service at time 0−. The introduced stochastic processes are assumed to have trajectories from D(R+ , R). The following equations appear in Reed [22], cf. also Krichagina and Puhalskii [15]: for t ∈ R+ (2.2a) (2.2b) (2.2c) n Q (t) = (Qn0 + n n − n) + Q̃ (t) + E (t) −  t t 0 An (t) = (Qn0 − n)+ + E n (t) − Qn (t) − n n Q̃ (t) = Qn0 ∧n  i=1  1{η̃i >t} . 0 + , 1{s+x≤t} d n (s) A i=1 1{ηi ≤x} , 134 A. A. PUHALSKII AND J. E. REED We note that the integral on the right-hand side of (2.2a) represents the number of departures by time t of the customers that enter service after time 0− as seen from the representation  t t 0 0 1{s+x≤t} d n (s) A i=1 1{ηi ≤x} = n (t) A i=1 1{τin +ηi ≤t} , where τin = inf{t ∈ R+ : An (t) ≥ i}. It is shown in Lemma A.1 below that, given E n , Qn0 , and the sequences {ηi } and {η̃i }, equations (2.2a)–(2.2c) admit solutions An , Qn and Q̃n . The processes An and Qn are specified uniquely a.s. under the additional requirement that ηi > 0 a.s. We also provide an example of nonuniqueness when the ηi can equal zero. Since our results do not assume that the service times be positive a.s., we interpret processes An and Qn in what follows as some solutions to (2.2a) and (2.2b) [the process Q̃n is specified uniquely by (2.2c)]. We now introduce the fluid limit equations. Let F̃ = (F̃ (t), t ∈ R+ ) represent the distribution function of a nonnegative random variable. According to Lemma B.1, given q0 ∈ R+ and a nondecreasing function e = (e(t), t ∈ R+ ) ∈ D(R+ , R), there exists a unique function q = (q(t), t ∈ R+ ) ∈ D(R+ , R) which satisfies the equation (2.3) q(t) = q0 − (q0 − 1)+ F (t) − q0 ∧ 1F̃ (t) + e(t) −  t 0 e(t − s) dF (s) +  t  0 + q(t − s) − 1 dF (s). One can easily see that q is R+ -valued. We also define a function a = (a(t), t ∈ R+ ) ∈ D(R+ , R) by the relation (2.4) + a(t) = (q0 − 1)+ + e(t) − q(t) − 1  . The next theorem is, in essence, due to Reed [22]. Let F̃ n = (F̃ n (t), t ∈ R+ ) denote the empirical distribution function of η̃1 , . . . , η̃n , that is, F̃ n (t) = T HEOREM 2.1. n 1 1{η̃ ≤t} , n i=1 i Suppose that, for arbitrary T > 0 and ε > 0,   Qn lim P  0 − q0  > ε = 0, n→∞ n  lim P n→∞  sup |F̃ n (t) − F̃ (t)| > ε = 0 t∈[0,T ] ON MANY-SERVER QUEUES IN HEAVY TRAFFIC 135 and  lim P n→∞   n   E (t)  − e(t) > ε = 0. sup  n t∈[0,T ] Then, for arbitrary T > 0 and ε > 0,  lim P n→∞ and lim P n→∞   n   Q (t)   sup  − q(t) > ε = 0 n t∈[0,T ]  n   A (t)   − a(t) > ε = 0. sup  n t∈[0,T ] Since An (t) represents the number of customers that enter service by time t out of those that are either queued at time 0− or exogenously arrive after time 0−, the process An is R+ -valued and nondecreasing. Hence, under the hypotheses of Theorem 2.1, a = (a(t), t ∈ R+ ) is a nondecreasing R+ -valued function from D(R+ , R). According to the Glivenko–Cantelli theorem, the compact convergence in probability of the F̃ n to F̃ in the hypotheses holds if the η̃i are i.i.d. with distribution F̃ . An interpretation of equation (2.3) can be as follows. By (2.4), the function a = (a(t), t ∈ R+ ) represents “fluid customers entering service after time 0−.” If we write (2.3) in the form (2.5)  q(t) = q0 + e(t) − q0 ∧ 1F̃ (t) +  t 0 a(t − s) dF (s) , then the first two terms on the right have the meaning of “the number of fluid customers” seen by time t and the sum in parentheses represents “fluid departures” by time t. Condition (2.1) cannot be disposed of in general. If we suppose, for instance, that F (0) = 1, and F̃ (t0 ) = 0 for some t0 > 0, then we may obtain different fluid limits for the cases where Qn0 = n and Qn0 = n − 1. In the former case, Qn (t) = E n (t) + n for all t < t0 as all servers remain occupied until t0 , while in the latter case Qn (t) = n − 1 for t < t0 as the only available server will let all exogenously arriving customers out of the system immediately. We now state the stochastic approximation result. Define processes Xn = (X n (t), t ∈ R+ ), S n = (S n (t), t ∈ R+ ), and Y n = (Y n (t), t ∈ R+ ) by (2.6) (2.7) (2.8) √ Qn (t) − q(t) , n n √   S n (t) = n F̃ n (t) − F̃ (t) , X n (t) = Y n (t) =  √ E n (t) − e(t) . n n  136 A. A. PUHALSKII AND J. E. REED We note that Xn , S n and Y n are random elements of D(R+ , R). However, Xn and Y n might not be random elements of Dc (R+ , R). Let us also denote √ Qn0 n − q0 . n We introduce the following condition: X0n = (2.9) lim sup  t ε→0 t∈[0,T ] 0  1{0<|q(t−s)−1|<ε} dF (s) = 0 for all T > 0. T HEOREM 2.2. Suppose that, as n → ∞, the X0n converge in distribution in R to a random variable X0 , the S n converge in distribution in Dc (R+ , R) to a process S = (S(t), t ∈ R+ ), and the Y n converge in distribution in Dc (R+ , R) to a process Y = (Y (t), t ∈ R+ ), where S and Y are separable random elements of Dc (R+ , R). Then finite-dimensional distributions of the processes Xn weakly converge to finite-dimensional distributions of the process X = (X(t), t ∈ R+ ) that is a unique strong solution to the equation X(t) = X0 1{q0 >1} + X0+ 1{q0 =1}        × 1 − F (t) + X0 1{q0 <1} + X0 ∧ 01{q0 =1} 1 − F̃ (t) + q0 ∧ 1S(t) + Y (t) − +  t  0  t 0 Y (t − s) dF (s) + Z(t) X(t − s)1{q(t−s)>1} + X(t − s)+ 1{q(t−s)=1} dF (s),  where Z = (Z(t), t ∈ R+ ) is a zero-mean Gaussian semimartingale with trajectories from D(R+ , R) specified by the covariance EZ(s)Z(t) =  s∧t 0   F (s ∧ t − u) 1 − F (s ∨ t − u) da(u), the entities X0 , S, Z and Y being independent. The trajectories of X are Borel measurable and locally bounded a.s. If, in addition, condition (2.9) holds, then X is a separable random element of Dc (R+ , R) and the X n converge in distribution in Dc (R+ , R) to X. We now comment on this theorem. As with Theorem 2.1, the convergence of the S n in the hypotheses holds when the η̃i are i.i.d. with distribution F̃ . In such a case, S(t) = W 0 (F̃ (t)), where W 0 = (W 0 (t), t ∈ [0, 1]) is a Brownian bridge. Since the process Z is a Gaussian semimartingale, its jumps occur at deterministic times which are the jump times of its variance, see Jacod and Shiryaev [12], Chapter II, Section 4d, and Liptser and Shiryayev [16], Chapter 4, Section 9, so this process is a separable random element of Dc (R+ , R). If either F or a is a continuous function, then Z is a continuous path process. In particular, if Y , F and F̃ are continuous (with F (0) = 0), then X is continuous (see Lemma B.1). 137 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC Similarly to Krichagina and Puhalskii [15], Z admits a representation as a stochastic integral, which follows from the proof of Lemma 4.4 below. Specifically, we may assume that a.s. (2.10) Z(t) =  R2+ 1{s+x≤t} dŴ(s, x), where (Ŵ(s, x), s ∈ R+ , x ∈ R+ ) is a zero-mean Gaussian process with covariance    EŴ(s, x)Ŵ(t, y) = a(s) ∧ a(t) F (x) ∧ F (y) − F (x)F (y) . The integral on the right of (2.10) is understood as a mean-square limit of  R2+ Il,t (s, x) dŴ(s, x) = l   i=1 l l l Ŵ(sil , t − si−1 ) − Ŵ(si−1 , t − si−1 ) + Ŵ(0, t),  where 0 = s0l < s1l < s2l < · · · < sll = t, Il,t (s, x) = li=1 1{s∈(s l ,s l ]} i−1 i l ) → 0 as l → ∞. 1{0≤x≤t−s l } + 1{s=0} 1{0≤x≤t} , and maxi (sil − si−1 i−1 As mentioned, the trajectories of X may have discontinuities of the second kind. To illustrate, suppose that F (t) = 1{t≥1} , q is continuous, q(t) < 1 for t < 1, q(2) = 1 and, as t ↑ 2, q(t) infinitely many times changes from being less than 1 to being greater than 1 and back. Suppose also that e, F̃ , and Y are continuous. Then the function a is continuous, so Z is continuous. Consequently, all terms in the equation for X(t) except for the last one have limits as t ↑ 3. The last term equals X(t − 1) when q(t − 1) > 1, equals X(t − 1)+ when q(t − 1) = 1, and equals 0 when q(t − 1) < 1. Therefore, there are three subsequential limits as t ↑ 3: X(2), X(2)+ , and 0, which shows that X(3−) may be undefined. For example, we may take ⎧  1  ⎪ ⎪ −(t − 1)2 + πt + 1 , 0 ≤ t ≤ 1, ⎪ ⎪ ⎪ 1+π ⎪ ⎨ 1  e(t) = −(t − 2)2 + π(t − 1) ⎪ 1 + π ⎪ ⎪ ⎪ π ⎪ ⎪ ⎩ + 1, 1≤t <2 + 1 + (t − 2)2 sin 2−t and q0 = 0. Then q(t) = e(t) ≤ 1 for t ∈ [0, 1], so  + q(t) = e(t) − e(t − 1) + q(t − 1) − 1 =1+ 1 π (t − 2)2 sin 1+π 2−t for t ∈ [1, 2). As follows by Lemma B.1, if any interval [0, T ] can be partitioned into finitely many intervals such that on each of these intervals the function q either stays below 1, or stays at 1, or stays above 1, then the trajectories of X admit left and righthand limits. To show that even then X does not necessarily have trajectories from D(R+ , R), consider the example where Qn0 = 0, E n is a Poisson process of rate n, 138 A. A. PUHALSKII AND J. E. REED and F (t) = 1{t≥2} . Then e(t) = t, Y is a standard Wiener process, and Z(t) = 0. For the fluid limit, we have q(t) = t for t < 2, q(t) = 2 for 2 ≤ t < 3, q(t) = t − 1 for 3 ≤ t < 4, q(t) = 3 for 4 ≤ t < 5, q(t) = t − 2 for 5 ≤ t < 6, etc. The process X is of the form: X(t) = Y (t) for t < 2, X(t) = Y (t) − Y (t − 2) for 2 ≤ t < 3, X(3) = Y (3) − Y (1) ∧ 0, X(t) = Y (t) for 3 < t < 4, X(t) = Y (t) − Y (t − 4) for 4 ≤ t < 5, X(5) = Y (5) − Y (1) ∧ 0, X(t) = Y (t) for 5 < t < 6, etc. To sum up, there are two alternating patterns: X(t) = Y (t) which occurs when 0 ≤ t ≤ 2 or 2k + 1 < t ≤ 2k + 2 for k ∈ N and X(t) = Y (t) − Y (t − 2k) which occurs when 2k ≤ t < 2k + 1 for k ∈ N. We can also see that the trajectory X(t), t ∈ R+ , is right-continuous when Y (1) ≥ 0 and is left-continuous when Y (1) ≤ 0. To see that convergence in Skorohod’s J1 -topology to the right-continuous version of X does not hold for this example, note that [cf. (3.21) in the proof of Theorem 2.2] Xn (t) = Y n (t) − Y n (t − 2)1{t≥2} √  + √  +   − n q(t − 2) − 1 1{t≥2} . + X n (t − 2) + n q(t − 2) − 1 √ It follows that supt∈[0,4] |Xn (t)| ≤ 3 supt∈[0,4] |Y n (t)| = 3/ n. On the other hand, X(3+) − X(3−) = Y (1), so the jumps of X n “do not match” the jumps of X, which rules out convergence in distribution of the X n to the right-continuous version of X for the J1 -topology. One can talk of convergence in the M1 -topology, see Whitt [24] for the definition and basic properties. A sketch of the proof of the latter mode of convergence is as follows. It is obvious that the Xn converge in distribution to Y on [0, 2] for the topology of compact convergence. It is also easy to see that on [2, 3 − ε] and on [3 + ε, 4], for arbitrary ε ∈ (0, 1), there is convergence for the topology of compact convergence to Y (t) − Y (t − 2) and to Y (t), respectively. Suppose t ∈ [3 − ε, 3 + ε]. Then on assuming that the X n converge to Y in the topology of compact convergence on [0, 2] a.s. and noting that  √  + √  +  sup  X n (t − 2) + n q(t − 2) − 1 − n q(t − 2) − 1 t∈[3−ε,3+ε]  − Y (1) + √  + √  +  n q(t − 2) − 1 − n q(t − 2) − 1  ≤ sup |Xn (t) − Y (t)| + t∈[0,2] sup t∈[1−ε,1+ε] |Y (t) − Y (1)|, we have by continuity of Y that the latter left-hand side can be made arbitrarily small if n is large enough √ Now it is an easy matter to √ and ε is small enough. verify that the (Y (1) + n(q(t − 2) − 1))+ − n(q(t − 2) − 1)+ converge to Y (1)1{t∈[3,3+ε]} as n → ∞ in the M1 -topology on [3 − ε, 3 + ε], cf., Whitt [24], page 82. It follows that the X n converge to the right-continuous version of X for the M1 -topology on [0, 4]. Because of the periodic pattern observed above, we derive M1 -convergence on R+ . We conjecture that convergence in the M1 -topology holds more generally. ON MANY-SERVER QUEUES IN HEAVY TRAFFIC 139 The Halfin–Whitt regime is obtained when q0 = 1, e(t) = λt for some λ > 0,  and F̃ (t) = λ 0t (1 − F (s)) ds. By (2.3), q(t) = 1 for t ∈ R+ . The equation for X in the statement of Theorem 2.2 takes the form X(t) = X0+ 1 − F (t) + X0 ∧ 0 1 − F̃ (t) + S(t) + Y (t) −   t 0   Y (t − s) dF (s) + Z(t) +   t 0 X(t − s)+ dF (s). Usually, instead of there being a limit for the processes Y n as defined in (2.8) √ with n(1 − one assumes that 0∞ s dF (s) < ∞ and that there exist ρ n ∈ (0, 1) √ ρ n ) converging to a finite limit as n → ∞ such that the ( n(E n (t)/n − ∞ n in distribution. These hypotheses imply conρ t/ 0 s dF (s)), t ∈ R+ ) converge ∞ n vergence of the Y with e(t) = t/( 0 s dF (s)). We discuss condition (2.9). It obviously holds for the Halfin–Whitt regime. By Lemma B.3, it also holds when F is a continuous function with F (0) = 0. If the function F has atoms, then (2.9) holds if and only if q does not attain level 1 “in a continuous fashion,” loosely speaking. More precisely, if and only if limε→0 tε = ∞, where tε = inf{t : |q(t) − 1| ∈ (0, ε)}. To see that, note that if t0 represents a jump time of F and tε < ∞, then the left-hand side of (2.9) is greater than or equal to F (t0 ) for all T > t0 + tε . For instance, if F contains atoms, e is continuous, q0 < 1, and there exists some t ≥ 0 such that q(t) > 1, then (2.9) cannot hold. This follows by the fact that if e is continuous, then by (2.5) q cannot have upward jumps. Therefore, upcrossings of level 1 occur in a continuous fashion. On the other hand, q may have downward jumps. If it starts above 1, then it may get below this level without spending time in small neighborhoods of 1, so it is possible for (2.9) to hold. Consider the following example. Suppose that q0 = 2 and e(t) = 0 for t ≥ 0 so that (2.3) becomes q(t) = 2 − F (t) − F̃ (t) +  t  0 + q(t − s) − 1 dF (s). Furthermore, suppose that F (t) = F̃ (t) = 1{t≥1} . We then have that q(t) = 2 for 0 ≤ t < 1, q(t) = 1 for 1 ≤ t < 2, and q(t) = 0 for t ≥ 2. 2.3. Extensions. In this subsection, we state a number of results which admit similar proofs. We begin by extending Theorems 2.1 and 2.2 to the case where the customers in the queue at time 0− and the customers exogenously arriving after time 0− may have differently distributed service times and where there can be available servers at time 0− even when there are customers awaiting service. Thus, η1 , η2 , . . . represent service times of the exogenously arriving customers. Condition (2.1) is still assumed. We denote the number of customers in the queue at time 0− by Q̂n0 . Their 140 A. A. PUHALSKII AND J. E. REED service times are denoted η̂1 , η̂2 , . . . and come from an i.i.d. sequence of nonnegative random variables with distribution F̂ . As with the ηi , it is assumed that the η̂i are not equal to zero a.s., that is, F̂ (0) < 1. We retain the rest of the previous notation and assumptions. The entities Q̃n0 , Q̂n0 , {η̃1 , η̃2 , . . .}, {η̂1 , η̂2 , . . .}, {η1 , η2 , . . .}, and E n = (E n (t), t ∈ R+ ) are assumed to be independent. Let Ăn (t) denote the number of customers that enter service by time t out of those that have arrived after 0−, let Ân (t) denote the number of customers that enter service by time t out of those that were in the queue at time 0−, and let Q̂n (t) denote the number of customers out of those present at time 0− that have not left by time t. The introduced quantities satisfy the following equations for t ∈ R+ : n n n Q (t) = Q̂ (t) + E (t) −  t t 0 0 1{s+x≤t} d n (s) Ă i=1 1{ηi ≤x} ,  +  + Ă (t) = E (t) + Q̂n (t) − n − Qn (t) − n , Q̂ n n n (t) = Q̂n0 n + Q̃ (t) −  t t 0 Ân (t) = Q̂n0 − Q̂n (t) − n  Q̃n 0  n Q̃ (t) = i=1 1{s+x≤t} d 0 + n (s) Â i=1 1{η̂i ≤x} , , 1{η̃i >t} . The fluid limit equations are of the form q(t) = e(t) − +  t  t  0  0 e(t − s) dF (s) + q̂(t) − + q(t − s) − 1   t  0 + q̂(t − s) − 1 dF (s) dF (s),    t  + − q(t) − 1 q̂(t) = q̂0 1 − F̂ (t) + q̃0 1 − F̃ (t) + + q̂(t − s) − 1 0 d F̂ (s). Existence and uniqueness of solutions to these equations are addressed in Lemma A.1 and by Lemma B.1. We also define functions ă = (ă(t), t ∈ R+ ) ∈ D(R+ , R) and â = (â(t), t ∈ R+ ) ∈ D(R+ , R) by the equalities  ă(t) = e(t) + q̂(t) − 1  + â(t) = q̂0 − q̂(t) − 1 .  + , ON MANY-SERVER QUEUES IN HEAVY TRAFFIC T HEOREM 2.3. 141 Suppose that, for arbitrary T > 0 and ε > 0,   n  Q̂0   − q̂0  > ε = 0, lim P  n→∞ n   n  Q̃0   − q̃0  > ε = 0, lim P  n→∞ n sup |F̃ n (t) − F̃ (t)| > ε = 0 lim P n→∞ t∈[0,T ] and  lim P n→∞  E n (t)  − e(t) > ε = 0. sup  n t∈[0,T ]   Then, for arbitrary T > 0 and ε > 0, lim P n→∞ lim P n→∞   lim P n→∞ t∈[0,T ] lim P   n   Q̂ (t)  sup  − q̂(t) > ε = 0, n t∈[0,T ]  and n→∞  n   Q (t)   − q(t) > ε = 0, sup  n   Ăn (t) − ă(t) > ε = 0 sup  n t∈[0,T ]    Ân (t)  sup  − â(t) > ε = 0. n t∈[0,T ]    For a stochastic approximation result, we assume that the processes X n = (X n (t), t ∈ R+ ), S n = (S n (t), t ∈ R+ ), and Y n = (Y n (t), t ∈ R+ ) are defined by (2.6)–(2.8), respectively. We also denote X̂0n √ 1 = n Q̂n0 − q̂0 , n  X̃0n √ 1 = n Q̃n0 − q̃0 . n  T HEOREM 2.4. Assume that the X̂0n and X̃0n converge in distribution in R to random variables X̂0 and X̃0 , respectively, as n → ∞. If the S n converge in distribution in Dc (R+ , R) to a process S = (S(t), t ∈ R+ ) and the Y n converge in distribution in Dc (R+ , R) to a process Y = (Y (t), t ∈ R+ ) such that S and Y are separable random elements of Dc (R+ , R), then the processes Xn converge in the sense of finite-dimensional distributions to the process X = (X(t), t ∈ R+ ) that is 142 A. A. PUHALSKII AND J. E. REED a unique strong solution to the set of equations X(t) = Y (t) −  t  − 0  t  + 0 and   t 0 Y (t − s) dF (s) + Z̆(t) + X̂(t) X̂(t − s)1{q̂(t−s)>1} + X̂(t − s)+ 1{q̂(t−s)=1} dF (s)  X(t − s)1{q(t−s)>1} + X(t − s)+ 1{q(t−s)=1} dF (s)     X̂(t) = X̂0 1 − F̂ (t) + X̃0 1 − F̃ (t) + q̃0 S(t) + Ẑ(t) +  t  0 X̂(t − s)1{q̂(t−s)>1} + X̂(t − s)+ 1{q̂(t−s)=1} d F̂ (s),  where Z̆ = (Z̆(t), t ∈ R+ ) and Ẑ = (Ẑ(t), t ∈ R+ ) are zero-mean Gaussian semimartingales with trajectories in D(R+ , R) and with respective covariances EZ̆(s)Z̆(t) =  s∧t F (s ∧ t − u) 1 − F (s ∨ t − u) d ă(u) EẐ(s)Ẑ(t) =  s∧t F̂ (s ∧ t − u) 1 − F̂ (s ∨ t − u) d â(u), 0 and 0     the entities X̂0 , X̃0 , Z̆, Ẑ, S and Y being independent. The trajectories of X are Borel measurable and locally bounded a.s. If, in addition, for all T > 0, lim sup  t  ε→0 t∈[0,T ] 0 and  1{0<|q̂(t−s)−1|<ε} + 1{0<|q(t−s)−1|<ε} dF (s) = 0 lim sup  t ε→0 t∈[0,T ] 0 1{0<|q̂(t−s)−1|<ε} d F̂ (s) = 0, then X and X̂ are separable random elements of Dc (R+ , R) and the X n converge in distribution in Dc (R+ , R) to X. We conclude this subsection with the Gaussian approximation result for an infinite server. Consider a sequence of G/GI /∞ systems indexed by n. Adapting n the earlier notation, we denote by Q (t) the number of customers present at time t and we denote by E n (t) the number of exogenous arrivals by t. We also reuse the introduced earlier sequences {η̃i , i ∈ N} and {ηi , i ∈ N}. As above, they represent the remaining service times of customers in service at time 0−, whose number is 143 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC denoted by Q̃n0 , and the service times of exogenously arriving customers, respectively. The ηi are assumed to be i.i.d. with distribution F , however, by contrast with condition (2.1) we allow these random variables to equal zero a.s. The entities E n = (E n (t), t ∈ R+ ), Q̃n0 , {η̃i , i ∈ R+ } and {ηi , i ∈ R+ } are assumed to be n n independent. The evolution of the process Q = (Q (t), t ∈ R+ ) is governed by the equation Q̃n n Q (t) = 0  i=1 n 1{η̃i >t} + E (t) −  t t 0 0 1{s+x≤t} d n (s) E i=1 1{ηi ≤x} . n It specifies the process Q uniquely. In analogy with the earlier notation, given e = (e(t), t ∈ R+ ) ∈√D(R+ , R), we define q0 ∈ R+ and a nondecreasing function  n n q(t) = q0 (1 − F̃ (t)) + e(t) − 0t e(t − s) dF (s), X (t) = n(Q (t)/n − q(t)), √ X0n = n(Q̃n0 /n − q0 ), and we define S n (t) and Y n (t) by (2.7) and (2.8), respectively. T HEOREM 2.5. Suppose that, as n → ∞, the X0n converge in distribution in R to a random variable X0 , the processes S n = (S n (t), t ∈ R+ ) converge in distribution in Dc (R+ , R) to a process S = (S(t), t ∈ R+ ), and the processes Y n = (Y n (t), t ∈ R+ ) converge in distribution in Dc (R+ , R) to a process Y = (Y (t), t ∈ R+ ), where S and Y are separable random elements of Dc (R+ , R). n n Then the processes X = (X (t), t ∈ R+ ) converge in distribution in Dc (R+ , R) to the process X = (X(t), t ∈ R+ ) defined by   X(t) = 1 − F̃ (t) X0 + √ q0 S(t) + Y (t) −  t 0 Y (t − s) dF (s) + Z(t), where Z = (Z(t), t ∈ R+ ) is a zero-mean Gaussian semimartingale with trajectories in D(R+ , R) and with covariance EZ(s)Z(t) =  s∧t 0   F (s ∧ t − u) 1 − F (s ∨ t − u) de(u), the entities X0 , S, Z and Y being independent. For the case of a continuous Y , the latter limit theorem generalizes the earlier results by Borovkov [2] (which result can also be found in Borovkov [3], Chapter 2, Section 2) and Krichagina and Puhalskii [15]. We recall that Borovkov [2] imposed  a Hölder continuity condition on the function ( 0t (1 − F (t − s)) de(s), t ∈ R+ ) and Krichagina and Puhalskii [15] required e to be continuous. 3. Proofs of the main results. This section contains proofs of Theorems 2.1 and 2.2. 144 A. A. PUHALSKII AND J. E. REED 3.1. Preliminaries. We start by developing semimartingale representations for certain processes. On introducing n A (t)  1  V (t, x) = √ 1{ηi ≤x} − F (x) , n i=1 n (3.1) we have by (2.2a) (3.2) + 1 n 1 n 1 Q (t) = Q0 − 1 + Q̃n (t) n n n  t 1 1 1 + E n (t) − An (t − s) dF (s) + √ Z n (t), n n 0 n  where n Z (t) = − (3.3)  R2+ 1{s+x≤t} dV n (s, x). Let n (3.4)  ηi ∧x A (t) 1  dF (u) L (t, x) = √ . 1{ηi ≤x} − n i=1 1 − F (u−) 0 n Then by (3.1), n V (t, x) = − (3.5)  x n V (t, u−) 0 Hence, by (3.3), 1 − F (u−) dF (u) + Ln (t, x). Z n (t) = Gn (t) − M n (t), (3.6) where Gn (t) = (3.7)  t n V (t − x, x−) 1 − F (x−) 0 and M n (t) = (3.8)  R2+ dF (x) 1{s+x≤t} dLn (s, x). We note that by (3.1), (3.4) and (3.8) (3.9) n n M (t) = V (t, 0) +  R2+ 1{x>0} 1{s+x≤t} dLn (s, x). We also define, for k ∈ N and t ∈ R+ , (3.10) Mkn (t) =  R2+ 1{s+x≤t} dLnk (s, x), ON MANY-SERVER QUEUES IN HEAVY TRAFFIC 145 where n (3.11) Lnk (s, x) =  ηi ∧x A (s)∧k  1  dF (u) √ 1{0<ηi ≤x} − , 1{u>0} n i=1 1 − F (u−) 0 and Mkn (t) =  R2+ 1{s+x≤t} dLnk (s, x), where n Lnk (s, x) = A (s)∧k  ηi ∧x 1 − F (u) 1  dF (u). 1{u>0} n i=1 0 (1 − F (u−))2 For t ∈ R+ , let Ĝtn denote the complete σ -algebra generated by the random variables 1{τin ≤s} 1{ηi ≤x} , where x + s ≤ t and i ∈ N, and by the An (s) (or, equivalently, by the 1{τin ≤s} for i ∈ N), where s ≤ t. We introduce “the right-continuous modifi- n . Then Gn = (G n , t ∈ R ) is a filtration. cation” by Gtn = ε>0 Ĝt+ε + t The next lemma originates in Krichagina and Puhalskii [15], Lemma 3.5, see also Reed [22], Lemma 1, where larger filtrations are used. Our conditions are closer to minimal ones. The proof is also different from those in the cited papers and is more direct.  L EMMA 3.1. For each k ∈ N, the process Mkn = (Mkn (t), t ∈ R+ ) is a Gn square integrable martingale starting at 0 with predictable quadratic variation process Mkn  = (Mkn (t), t ∈ R+ ). P ROOF. According to Lemma C.3 it suffices to prove that, given i ∈ N, s ∈ R+ and x ∈ R+ , the random variable ηi is independent of the random variables ηj for j = i, of the τjn for j ≤ i, and of the τjn ∧ (s + x) for j > i, when conditioned on the event {τin ≥ s, ηi > x} and that ηi is independent of the τjn for j ≤ i. By (2.2a)–(2.2c), the τjn for j ≤ i are measurable with respect to the σ -algebra gen- erated by Q̃n0 , Q̂n0 , η̃j , j ∈ N, ηj , j < i, and E n (t), t ∈ R+ . By the independence assumptions in the hypotheses, this σ -algebra and ηi are independent, hence, τjn for j ≤ i and ηi are independent. Let An,i = (An,i (t), t ∈ R+ ) denote the process of exogenous arrivals entering service that would occur if the ith exogenously arriving customer had an infinite service time and let τjn,i = inf{t : An,i (t) ≥ j }. Then by (2.2a)–(2.2c) the τjn,i for j ∈ N are measurable with respect to the σ -algebra generated by Q̃n0 , Q̂n0 , η̃j , j ∈ N, ηj , j < i, and E n (t), t ∈ R+ . Hence, they are independent of ηi . Therefore, ηi , on the one hand, and ηj for j = i, τjn for j ≤ i, and τjn,i for j > i, on the other hand, are independent. In addition, as it follows by (2.2a)–(2.2c), on the event 146 A. A. PUHALSKII AND J. E. REED {τin ≥ s, ηi > x} the ith exogenously arriving customer does not depart until after time s + x, which means that she has the same effect on the epochs before time s + x when exogenously arriving customers enter service as if she never completed service. To put it precisely, τjn,i ∧ (s + x) = τjn ∧ (s + x) for j ∈ N on the event {τin ≥ s, ηi > x}. By the established independence property, we have for natural numbers r1 , r2 , . . . , rl none of which equals i and i < p1 < p2 < · · · < pm , and for bounded Borel functions f1 , . . . , fl , g1 , . . . , gi , h1 , . . . , hm , and f , on assuming that conditional probabilities given events of probability zero equal zero and adopting the convention that 0/0 = 0,  l  E fj (ηrj ) i  gj (τjn ) j =1 j =1 j =1  l  =E fj (ηrj ) m  j =1  l  =E fj (ηrj ) =E j =1 j =1  i  = fj (ηrj ) i  gj (τjn ) j =1 fj (ηrj ) m  j =1 j =1 l   hj τpn,i ∧ (s + x) 1{τin ≥s} j   P(τin ≥ s)P(ηi > x) i n m n,i j =1 gj (τj ) j =1 hj (τpj ∧ (s P(τin ≥ s, ηi > x) + x))1{τin ≥s} 1{ηi >x} ) E(f (ηi )1{ηi >x} 1{τin ≥s} ) × = gj (τjn )     n,i hj τpj ∧ (s + x) f (ηi )1{τin ≥s,ηi >x} P(τin ≥ s, ηi > x) × E f (ηi )1{ηi >x} E( gj (τjn ) j =1 m   l      n  hj τpj ∧ (s + x) f (ηi )1{τin ≥s,ηi >x} P(τin ≥ s, ηi > x) j =1 × i      hj τpnj ∧ (s + x) f (ηi )  τin ≥ s, ηi > x j =1 j =1 × m  P(τin ≥ s, ηi > x) l E( j =1 fj (ηrj ) i n m n j =1 gj (τj ) j =1 hj (τpj ∧ (s P(τin ≥ s, ηi > x) + x))1{τin ≥s} 1{ηi >x} ) 147 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC × =E E(f (ηi )1{ηi >x} 1{τin ≥s} ) P(τin ≥ s, ηi > x)  l  j =1 fj (ηrj ) i  gj (τjn ) m  j =1 j =1 × E f (ηi ) | τin ≥ s, ηi > x .      hj τpnj ∧ (s + x)  τin ≥ s, ηi > x  An application of Lemma C.3 completes the proof.  R EMARK 3.1. We stop short of claiming that M n is a Gn -locally square integrable martingale because there might not be a sequence of stopping times such that the “stopped” processes An are bounded. However, if, in addition, the jumps of An (including the “jump” at zero) are uniformly bounded, then M n is a Gn locally square integrable martingale. 3.2. Proof of Theorem 2.1. L EMMA 3.2. Under the hypotheses of Theorem 2.1, for T > 0, lim lim sup P b→∞ n→∞ P ROOF. (3.12) sup |Z n (t)| > b = 0. t∈[0,T ] In view of (3.6), it suffices to prove that lim lim sup P b→∞ n→∞ sup |Gn (t)| > b = 0 t∈[0,T ] and (3.13) lim lim sup P b→∞ n→∞ sup |M n (t)| > b = 0. t∈[0,T ] Let us firstly note that, for t > 0 and b large enough, (3.14) lim P n→∞  1 n A (t) > b = 0, n which follows by the bound An (t) ≤ (Qn0 − n)+ + E n (t) [see (2.2b)] and the convergence in probability of Qn0 /n to q0 and of E n (t)/n to e(t) in the hypotheses of Theorem 2.1. The proof of (3.12) is analogous to the proof of (3.23) in Krichagina and Puhalskii [15]. Let (3.15) ⌊nt⌋  1  1{ηi ≤x} − F (x) . V̆ n (t, x) = √ n i=1 148 A. A. PUHALSKII AND J. E. REED We have, for c > 0, on taking into account (3.1), (3.7) and (3.15), P n sup |G (t)| > b ≤ P t∈[0,T ]  1 n A (T ) > cT n +P  ∞ supt∈[0,cT ] |V̆ n (t, x−)| 1 − F (x−) 0 dF (x) > b . By (3.15), for fixed x, V̆ n (t, x) is a locally square-integrable martingale in t relative to the natural filtration with predictable quadratic variation process (⌊nt⌋/n)F (x)(1 − F (x)). Theorem 1.9.5 √ in Liptser and Shiryayev [16] yields the bound E supt∈[0,cT ] |V̆ n (t, x−)| ≤ 3 cT (1 − F (x−)). Hence,  ∞ E supt∈[0,cT ] |V̆ n (t, F (x−))| 1 − F (x−) 0 √ dF (x) ≤ 6 cT , so (3.12) follows by an application of Markov’s inequality and (3.14). Next, on noting that M n (t) = V̆ n (An (t)/n, 0) + Mkn (t) when An (t) ≤ k, we have by (3.9), Lemma 3.1, Kolmogorov’s inequality, and the Lenglart–Rebolledo inequality; see, for example, Liptser and Shiryayev [16], Theorem 1.9.3, for b > 0 and c > 0,    b P sup |M n (t)| > b ≤ P An (T ) > ⌊nbT ⌋ + P sup |V̆ n (t, 0)| > 2 t∈[0,T ] t∈[0,bT ]  n  b  sup M⌊nbT ⌋ (t) > 2 t∈[0,T ]  +P  n   4(T + c) + P M (T ) > c ⌊nbT ⌋ b2   4(T + c) ≤ P An (T ) > ⌊nbT ⌋ + b2 ≤ P An (T ) > ⌊nbT ⌋ +   ⌊nbT ⌋  1 − F (u) 1  ∞ dF (u) > c . 1{u≤ηi } +P n i=1 0 (1 − F (u−))2   By Markov’s inequality, ⌊nbT ⌋  1  ∞ 1 − F (u) bT , dF (u) > c ≤ 1{u≤ηi } P n i=1 0 (1 − F (u−))2 c   so, on picking c = b3/2 T 1/2 , P   n  4T T . sup |M (t)| > b ≤ P A (T ) > ⌊nbT ⌋ + 2 + 5 n t∈[0,T ] The convergence in (3.13) now follows by (3.14).  b b 149 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC P ROOF Qn (t)/n, T HEOREM 2.1. Denote en (t) = E n (t)/n, qn0 = Qn0 /n, qn (t) = and q̃n (t) = Q̃n (t)/n so that by (2.2b) and (3.2), OF qn (t) = (qn0 − 1)+ + q̃n (t) + en (t) − (3.16) +  t 0 en (t − s) dF (s)  t  n + 1 q (t − s) − 1 dF (s) + √ Z n (t). n 0 √ By Lemma 3.2, the supt∈[0,T ] |Z n (t)|/ n converge to zero in probability for every T > 0 as n → ∞. By hypotheses, the (qn0 − 1)+ and qn0 ∧ 1 converge in probability to (q0 − 1)+ and q0 ∧ 1, respectively. By (2.2c) and the hypotheses, the q̃n (t) converge in probability uniformly over compact intervals of t to q0 ∧ 1(1 − F̃ (t)). Also, the compact convergence in probability of the en to e implies that   sup en (t) − t∈[0,T ]  t 0  en (t − s) dF (s) − e(t) −  t 0   e(t − s) dF (s)  converges to zero in probability. Hence, on applying part 1 of Lemma B.2 to (3.17), we conclude by the fact that a sequence of random variables converges in probability if and only if its every subsequence contains a further subsequence that converges a.s. that the sequence supt∈[0,T ] |qn (t) − q(t)| converges in probability to zero as required. The convergence   n    A (t)  − a(t) > ε = 0 lim P sup  n→∞ n t∈[0,T ] follows by (2.2b) and the part of the theorem already proven.  3.3. Proof of Theorem 2.2. The key to the proof of Theorem 2.2 is the following result whose proof is deferred until the next section. Denote (3.17) H (t) = Y (t) − n n  t (3.18) H (t) = Y (t) − (3.19) (3.20) X̃n (t) = 0 Y (t − s) dF (s),  t 0 Y n (t − s) dF (s), √ 1 n   n Q̃ (t) − q0 ∧ 1 1 − F̃ (t) , n     X̃(t) = X0 1{q0 <1} + X0 ∧ 01{q0 =1} 1 − F̃ (t) + q0 ∧ 1S(t). Let H = (H (t), t ∈ R+ ), H n = (H n (t), t ∈ R+ ), X̃ = (X̃(t), t ∈ R+ ), X̃n = (X̃ n (t), t ∈ R+ ), and Z n = (Z n (t), t ∈ R+ ). We also denote  √ 1 n n X̂0 = n (Q0 − n)+ − (q0 − 1)+ , n X̂0 = X0 1{q0 >1} + X0+ 1{q0 =1} . 150 A. A. PUHALSKII AND J. E. REED T HEOREM 3.1. Under the hypotheses of Theorem 2.2, the processes X̃, H and Z are separable random elements of Dc (R+ , R). As n → ∞, the (X̂0n , X̃ n , H n , Z n ) converge in distribution in R × Dc (R+ , R)3 to (X̂0 , X̃, H, Z). Given Theorem 3.1 and Lemmas√B.1 and B.3, the proof of Theorem 2.2 is now routine. On recalling that X n (t) = n(Qn (t)/n − q(t)), we have by (2.2b), (2.3), (2.4), (3.2), (3.18) and (3.19), that X n (t) = X̂0n 1 − F (t) + X̃ n (t) + H n (t) + Z n (t) + (3.21)   t  0  X n (t − s) + √  + n q(t − s) − 1 − √  +  n q(t − s) − 1 dF (s). n n n n On writing (3.21) as Xn =  n (X̂√ part 2 of 0 , X̃ , H , Z ), we √ have by applying + + n Lemma B.2 with f (y, t) = (y + n(q(t) − 1)) − n(q(t) − 1) and f (y, t) = y1{q(t)>1} + y + 1{q(t)=1} that if xn → x in R × Dc (R+ , R)3 , then  n (xn )(t) → y(t) for all t, where y = (y(t), t ∈ R+ ) is determined by the following equations assuming that x = (x1 , x2 , x3 , x4 ) with x1 ∈ R and xi = (xi (t), t ∈ R+ ) for i = 2, 3, 4:   y(t) = x1 1 − F (t) + x2 (t) + x3 (t) + x4 (t) +  t  0 y(t − s)1{q(t−s)>1} + y(t − s)+ 1{q(t−s)=1} dF (s).  These equations specify y uniquely by Lemma B.1. The hypotheses and Theorem 3.1 imply that, as n → ∞, the (X̂0n , X̃ n , H n , Z n ) converge in distribution in R × Dc (R+ , R)3 to (X̂0 , X̃, H, Z), which is a separable random element. Therefore, by the continuous mapping principle (see Theorem D.2), the X n converge in the sense of weak convergence of finite-dimensional distributions to X = (X̂0 , X̃, H, Z), where (x1 , x2 , x3 , x4 ) = y. The trajectories of X are a.s. Borel measurable and locally bounded by Lemma B.1. If, in addition, condition (2.9) holds, then, by Lemmas B.2 and B.3, X has trajectories in D(R+ , R) and the convergence  n (xn ) → y holds in Dc (R+ , R). An application of the continuous mapping principle implies convergence in distribution in Dc (R+ , R) of the X n to X. By Lemma B.2,  is a continuous mapping from R × Dc (R+ , R)3 to Dc (R+ , R), so X is a tight random element. Hence, it is a separable random element. 4. Proof of Theorem 3.1. In Section 4.1, we derive certain properties of the processes that appear in the limit, emphasizing the property of being a separable random element. In Section 4.2, results on joint convergence in distribution are established. These developments culminate in the proof of Theorem 3.1 in Section 4.3. The hypotheses of Theorem 2.2 are assumed. ON MANY-SERVER QUEUES IN HEAVY TRAFFIC 151 4.1. Properties of processes associated with the Kiefer process. Let K = ((K(t, x), x ∈ [0, 1]), t ∈ R+ ) be a Kiefer process such that K, Y , X̂0 and X̃ are independent. Recall that this means that K is a zero-mean Gaussian process with EK(t, x)K(s, y) = (t ∧ s)(x ∧ y − xy). In particular, K(t, x) is a Brownian motion in t for fixed x. We choose a modification of K which is continuous in both variables a.s. Introduce for t ∈ R+ and x ∈ R+ , (4.1) V (t, x) = K(a(t), F (x)), (4.2) U (t, x) = K(t, F (x)), and define L(t, x) = V (t, x) + (4.3)  x V (t, u−) dF (u), 1 − F (u−)  x U (t, u−) ′ dF (u), L (t, x) = U (t, x) + 0 1 − F (u−)  t V (t − u, u−) dF (u). G(t) = 0 1 − F (u−) (4.4) (4.5) 0 The integrals on the right of (4.3)–(4.5) converge absolutely a.s. To see this, note that V (t, x) is a locally square integrable martingale in t for fixed x relative to the natural filtration with predictable quadratic variation process a(t)F (x)(1 − F (x)). Therefore, by Theorem √ 1.9.5 in Liptser and Shiryayev [16], for T > 0, E supt∈[0,T ] |V (t, x−)| ≤ 3 a(T )(1 − F (x−)). We thus obtain √  ∞  ∞ E supt∈[0,T ] |V (t, u−)| 3 a(t) √ dF (u) ≤ dF (u) 1 − F (u−) 1 − F (u−) 0 0 (4.6)  ≤ 6 a(t) so that  ∞ supt∈[0,T ] |V (t, u−)| (4.7) 0 1 − F (u−) dF (u) < ∞ a.s. for T > 0. A similar argument shows that for arbitrary T > 0 and δ > 0 (4.8) and (4.9)  x    V (t, u−)  lim P sup sup  1{F (u−)>1−ε} dF (u) > δ = 0 ε→0 t∈[0,T ] x∈R+ 0 1 − F (u−)   x    U (t, u−)  lim P sup sup  1{F (u−)>1−ε} dF (u) > δ = 0. ε→0 t∈[0,T ] x∈R+ 0 1 − F (u−)  Denote (4.10) F ′ (x) =  x 1 − F (u) 0 1 − F (u−) dF (u). 152 A. A. PUHALSKII AND J. E. REED Let U = ((U (t, x), x ∈ R+ ), t ∈ R+ ), V = ((V (t, x), x ∈ R+ ), t ∈ R+ ), L = ((L(t, x), x ∈ R+ ), t ∈ R+ ), L′ = ((L′ (t, x), x ∈ R+ ), t ∈ R+ ), and G = (G(t), t ∈ R+ ). L EMMA 4.1. The process L is a zero-mean Gaussian process with trajectories in D(R+ , Dc (R+ , R)). Its covariance is given by EL(t, x)L(s, y) = a(t ∧ s)F ′ (x ∧ y). Furthermore, the pair (L, V ) is Gaussian. P ROOF. By (4.1), (4.7) and Lebesgue’s dominated convergence theorem, the definition of L(t, x) in (4.3) implies that L(t, x) is right-continuous in x with lefthand limits for a given t. Similarly,  ∞ |V (t, u−) − V (s, u−)| dF (u) = 0,  ∞ |V (t, u−) − V (s−, u−)| dF (u) = 0. lim t↓s lim t↑s 1 − F (u−) 0 1 − F (u−) 0 It follows that L has trajectories in D(R+ , Dc (R+ , R)) a.s. Since the integrand in the integral on the right of (4.3) is left-continuous in u, Lebesgue’s dominated convergence theorem shows that this integral can be assumed to be a Stjeltjes integral, that is, to be a limit of Riemann sums. Since finitedimensional distributions of V are Gaussian, it follows that finite-dimensional distributions of (L, V ) are Gaussian too. The formula for the covariance of L is obtained by a direct calculation.  Lemma 4.1 implies that L defines an orthogonal random measure on R2+ in that EL((t1 , x1 ), (t2 , x2 ))L((s1 , y1 ), (s2 , y2 )) = 0, where t1 ≤ t2 , x1 ≤ x2 , s1 ≤ s2 , and y1 ≤ y2 , whenever the rectangles with the vertices (t1 , x1 ), (t1 , x2 ), (t2 , x1 ), (t2 , x2 ) and (s1 , y1 ), (s1 , y2 ), (s2 , y1 ), (s2 , y2 ) are disjoint. Similarly, EL((t1 , x1 ), (t2 , x2 ))(L(0, y2 ) − L(0, y1 )) = 0, EL((t1 , x1 ), (t2 , x2 ))(L(s2 , 0) − L(s1 , 0)) = 0, E(L(0, y2 ) − L(0, y1 ))(L(s2 , 0) − L(s1 , 0)) = 0, E(L(t2 , 0) − L(t1 , 0))(L(s2 , 0) − L(s1 , 0)) = 0 when (t1 , t2 ) and (s1 , s2 ) are disjoint, E(L(0, x2 ) − L(0, x1 )) (L(0, y2 ) − L(0, y1 )) = 0 when (x1 , x2 ) and (y1 , y2 ) are disjoint, EL((t1 , x1 ), (t2 , x2 ))L(0, 0) = 0, E(L(t2 , 0) − L(t1 , 0))L(0, 0) = 0, and E(L(0, x2 ) − L(0, x1 )) L(0, 0) = 0. These properties enable us to define in a standard fashion integrals with respect to L. To recapitulate, suppose h is a Borel function with (4.11)  R2+ h(s, x)2 da(s) dF ′ (x) < ∞. 153 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC If h(s, x) = (4.12) k  i=1 + ai 1{s∈(s1,i ,s2,i ]} 1{x∈(x1,i ,x2,i ]} + m  l  i=1 bi 1{s=0} 1{x∈(y1,i ,y2,i ]} ci 1{s∈(z1,i ,z2,i ]} 1{x=0} + d1{s=0} 1{x=0} , i=1 where the ai , bi , ci and d are real numbers, 0 ≤ x1,i < x2,i , 0 ≤ y1,i < y2,i , 0 ≤ z1,i < z2,i , and the sets (s1,i , s2,i ] × (x1,i , x2,i ], (y1,i , y2,i ], and (z1,i , z2,i ] are pairwise disjoint, we set  R2+ h(s, x) dL(s, x) = (4.13) k  ai L((s1,i , x1,i ), (s2,i , x2,i )) i=1 + l  bi L(0, y2,i ) − L(0, y1,i ) + m  ci L(z2,i , 0) − L(z1,i , 0) + dL(0, 0). i=1 i=1 Note that by Lemma 4.1 (4.14)  E R2+     2 h(s, x) dL(s, x) =  R2+ h(s, x)2 da(s) dF ′ (x). If h(s, x) is an arbitrary Borel function satisfying (4.11), then there exists a sequence hk of functions of the form (4.12) such that lim k→∞  R2+  2 h(s, x) − hk (s, x) da(s) dF ′ (x) = 0. This implies by (4.14) that the sequence R2 hk (s, x) dL(s, x) is Cauchy in +  L2 (, F , P), so it converges. We define R2 h(s, x) dL(s, x) as the limit. One + can see that the integral is a zero-mean Gaussian random variable and that  (4.14) still holds. By polarization, if g(s, x) is another function with R2 g(s,  + x)2 da(s) dF ′ (x) < ∞, then E (4.15)  R2+ h(s, x) dL(s, x) =  R2+  R2+ g(s, x) dL(s, x) h(s, x)g(s, x) da(s) dF ′ (x). The martingale property asserted in the next lemma is understood with respect to the natural filtration. Recall that Gaussian martingales are locally square integrable. 154 A. A. PUHALSKII AND J. E. REED L EMMA 4.2. x ∈ R+ , If g(s, x) is a Borel function on R2+ such that, for all t ∈ R+ and  R2+ 1{s≤t} 1{y≤x} g(s, y)2 da(s) dF ′ (y) < ∞, then the process N = (N(t), t ∈ R+ ) with N(t) =  R2+ g(s, x)1{s+x≤t} dL(s, x) is a zero-mean Gaussian martingale with predictable quadratic variation process N  = (N (t), t ∈ R+ ), where N (t) =  R2+ g(s, x)2 1{s+x≤t} da(s) dF ′ (x). The process N has a modification with trajectories in D(R+ , R). P ROOF. By construction, finite-dimensional distributions of N are limits of Gaussian distributions, so they are Gaussian too. The covariance of N is given, according to (4.15), by EN(s)N(t) =  R2+ g(s, x)2 1{u+x≤s∧t} da(u) dF ′ (x). It follows that N has uncorrelated, hence, independent increments. It is, thus, a zero-mean Gaussian martingale. By the independence-of-increments property, the natural filtration of N is right-continuous; see, for example, Doob [6], Part 2, Chapter VI, Section 8. Consequently, N admits a right-continuous with left-hand limits modification; see, for example, Doob [6], Part 2, Chapter IV, Section 1.  Let (4.16) M(t) =  R2+ 1{s+x≤t} dL(s, x). By Lemma 4.2, the process M = (M(t), t ∈ R+ ) admits a modification which is a Gaussian martingale with trajectories from D(R+ , R). We further consider such a modification throughout. The variance of M(t) is given by (4.17) C(t) =  t 0 a(t − s) dF ′ (s). We study measurability properties of the introduced processes. Recall that H was defined in (3.17). L EMMA 4.3. The processes H, G and M are separable random elements of Dc (R+ , R) and the processes L, L′ , U and V are separable random elements of Dc (R+ , Dc (R+ , R)). The pair (G, M) is Gaussian. The process G is a Gaussian semimartingale. ON MANY-SERVER QUEUES IN HEAVY TRAFFIC 155 P ROOF. By the hypotheses of Theorem 2.2, the process Y is a separable random element of Dc (R+ , R), so its distribution is a tight probability measure. By (3.17), H is obtained from Y by an application of a continuous operator on Dc (R+ , R). It follows that the distribution of H is a tight probability measure, hence, H is a separable random element of Dc (R+ , R). Since the process V is bounded on bounded domains [see (4.1)], by (4.5) the process G is of locally bounded variation a.s. By (4.7) and Lebesgue’s dominated convergence theorem, the trajectories of G are right-continuous and admit left-hand limits on a set of full probability. Since the integrand in (4.5) is a leftcontinuous function of y, the integral can be interpreted as a Stjeltjes integral, that is, as a limit of Riemann sums. As (L, V ) is a Gaussian pair by Lemma 4.1, the definition of M in (4.16) implies that (G, M) is a Gaussian pair. Hence, the processes G and M are Gaussian semimartingales with paths in D(R+ , R). By Liptser and Shiryaev [16], Theorem 4.9.1, their jump times are deterministic, so the ranges of these processes as elements of Dc (R+ , R) are separable. Since balls in Dc (R+ , R) belong to the Kolmogorov σ -algebra, the traces of the Kolmogorov and Borel σ -algebras on a separable set coincide, hence, G and M are separable random elements of Dc (R+ , R). By (4.2) and continuity of the Kiefer process in both variables, U (t, x), as a function of x, jumps only when F jumps, so its range is separable, hence, (U (t, x), x ∈ R+ ) is a separable random element of Dc (R+ , R) for each t. Next, the map t → (U (t, x), x ∈ R+ ) from R+ to Dc (R+ , R) is continuous, so U has a separable range in Dc (R+ , Dc (R+ , R)). It is, therefore, a separable random element of the latter space. The process V is a separable random element of Dc (R+ , Dc (R+ , R)) for a similar reason. The range of (L(t, x), x ∈ R+ ) for a given t is separable by the fact that the jumps of (L(t, x), x ∈ R+ ) occur at the times of jumps of F , hence, this process is a random element of Dc (R+ , R). Since the jumps of L as an element of Dc (R+ , Dc (R+ , R)) coincide with the jumps of a, it is a separable random element of Dc (R+ , Dc (R+ , R)). The assertion of the lemma for L′ is obtained analogously.  In what follows, we always assume the modifications as described in Lemmas 4.2 and 4.3. We now construct the process Z = (Z(t), t ∈ R+ ) in the statement of Theorem 2.2. We define (4.18) Z(t) = G(t) − M(t). By Lemma 4.3, Z is a Gaussian semimartingale, so it is a separable random element of Dc (R+ , R). In order to verify that its covariance function has the form stated in Theorem 2.2, we find it convenient to approximate this process with Gaussian processes of simpler structure. 156 A. A. PUHALSKII AND J. E. REED For l ∈ N, let 0 = s0l < s1l < s2l < · · · be a strictly increasing to infinity sequence of real numbers. Following the notation used in the Introduction, we set, for t ∈ R+ , Il,t (s, x) = (4.19) ∞  i=1 1{s∈(s l l i−1 ,si ]} 1{0≤x≤t−s l i−1 } + 1{s=0} 1{0≤x≤t} . We also define (4.20)  R2+ Il,t (s, x) dV (s, x) = ∞   i=1 l l l ) − V (si−1 , t − si−1 ) 1{s l V (sil , t − si−1  i−1 ≤t} + V (0, t) and introduce (4.21) Zl (t) = −  R2+ Il,t (s, x) dV (s, x). On recalling (4.1), we see that (Zl (t), t ∈ R+ ) is a zero-mean Gaussian process with covariance EZl (t)Zl (s) = ∞   i=1 l l a(sil ) − a(si−1 ) F (t ∧ s − si−1 )  l × 1 − F (t ∨ s − si−1 ) 1{s l  (4.22)   i−1 ≤t∧s}  + a(0)F (t ∧ s) 1 − F (t ∨ s) . l ) → 0 as l → ∞, then the Z (t) converge L EMMA 4.4. If supi (sil − si−1 l to Z(t) in the mean square sense for each t as l → ∞. The covariance of Z is given by EZ(t)Z(s) = P ROOF.  R2+ 0   F (t ∧ s − u) 1 − F (t ∨ s − u) da(u). By (4.3), (4.13), (4.19) and (4.20), Il,t (s, x) dV (s, x) = (4.23)  t∧s  R2+ Il,t (s, x) dL(s, x) − ∞  −  t V (0, y−) i=1 0 1{s l i−1 ≤t}  t−s l l l i−1 V (si , y−) − V (si−1 , y−) 1 − F (y−) 0 1 − F (y−) dF (y). dF (y) 157 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC Given ε > 0, for all l large enough, 1{s+x≤t} 1{s≥0} 1{x≥0} ≤ Il,t (s, x) ≤ 1{s+x≤t+ε} 1{s≥0} 1{x≥0} . Therefore,  R2+  2 Il,t (s, x) − 1{s+x≤t} da(s) dF ′ (x) ≤  R2+  2 1{s+x≤t+ε} − 1{s+x≤t} da(s) dF ′ (x). The right-hand side converges to zero as ε → 0, so by (4.16) and the definition of the integral M(t) = l.i.m. (4.24) l→∞  R2+ Il,t (s, x) dL(s, x), where l.i.m. stands for mean-square limit. Also ∞  i=1 1{s l i−1 ≤t} =  t−s l l l i−1 V (si , y−) − V (si−1 , y−) 1 − F (y−) 0  t V (s l , y−) i(y) 0 1 − F (y−) dF (y) +  t V (0, y−) 0 1 − F (y−) dF (y) dF (y), l l > t − y ≥ si(y)−1 . By right continuity of V (t, x) in t, (4.5), (4.8) and where si(y) Lebesgue’s dominated convergence theorem, G(t) = P- lim ∞  l→∞ (4.25) i=1 1{s l i−1 ≤t}  t−s l l l i−1 V (si , y−) − V (si−1 , y−) 1 − F (y−) 0 +  t V (0, y−) 0 1 − F (y−) dF (y)  dF (y) , where P- lim denotes limit in probability. By (4.18), (4.21), (4.23)–(4.25), Z(t) = P- liml→∞ Zl (t). Since the Zl are Gaussian processes, the latter limit holds in the mean square sense too; see, for example, Ibragimov and Rozanov [11], Lemma I.3.1. By (4.22), we can write EZl (t)Zl (s) = where  ∞ 0 F t ∧ s − r l (u) 1 − F t ∨ s − r l (u) 1{r l (u)≤t∧s} da(u),   l r (u) = ∞  i=1   l 1{u∈(s l si−1 l i−1 ,si ]} . l ) → 0 as l → ∞, r l (u) → u from the left. By right continuity Since maxi (sil − si−1 of F , F (t ∧ s − r l (u)) → F (t ∧ s − u) and F (t ∨ s − r l (u)) → F (t ∨ s − u) as 158 A. A. PUHALSKII AND J. E. REED l → ∞. Also, 1{r l (u)≤t∧s} → 1{u≤t∧s} . Therefore, by Lebesgue’s dominated convergence theorem, lim EZl (t)Zl (s) = l→∞  t∧s 0   F (t ∧ s − u) 1 − F (t ∨ s − u) da(u).  4.2. Convergence in distribution. Let (4.26) (4.27) ⌊nt⌋  1  U n (t, x) = √ 1{ηi ≤x} − F (x) , n i=1 ⌊nt⌋ 1  1{ηi ≤x} − L (t, x) = √ n i=1 ′n  ηi ∧x 0 dF (u) . 1 − F (u−) By (3.1) and (4.26), V n (t, x) = U n (4.28)  n A (t) n ,x . For what follows, we note that, given arbitrary T > 0 and δ > 0, (4.29)  x n    U (t, u−)  1{F (u−)>1−ε} dF (u) > δ lim lim sup P sup sup  ε→0 n→∞ t∈[0,T ] x∈R+ 0 1 − F (u−)  = 0. The limit in (4.29) is analogous to that in equation (3.23) in Krichagina and Puhalskii [15] [see also (3.24) in that paper], so a similar proof applies. We introduce the processes L′n = ((L′n (t, x), x ∈ R+ ), t ∈ R+ ), U n = ((U n (t, x), x ∈ R+ ), t ∈ R+ ), and L′ = ((L′ (t, x), x ∈ R+ ), t ∈ R+ ). L EMMA 4.5. As n → ∞, the (X̂0n , X̃ n , U n , L′n ) converge jointly in distribution in R × Dc (R+ , R) × Dc (R+ , Dc (R+ , R))2 to (X̂0 , X̃, U, L′ ). P ROOF. Recall that, by Lemma 4.3, L′ and U are separable random elements of the space Dc (R+ , Dc (R+ , R)). The process X̃ is a separable random element of Dc (R+ , R) by (3.20) and the assumption that S is a separable random element. The hypotheses of Theorem 2.2 and (2.2c) imply in a standard fashion that the (X̂0n , X̃ n ) converge in distribution in R × Dc (R+ , R) to (X̂0 , X̃). Since (X̂0n , X̃ n ) and (U n , L′n ) are independent and (X̂0 , X̃) and (U, L′ ) are also independent and are separable random elements of the associated metric spaces, by Theorem D.8 it suffices to establish convergence in distribution in Dc (R+ , Dc (R+ , R))2 for (U n , L′n ). By (4.26) and (4.27) (cf. (3.5)), (4.30) U n (t, x) = −  x n U (t, u−) 0 1 − F (u−) dF (u) + L′n (t, x). ON MANY-SERVER QUEUES IN HEAVY TRAFFIC 159 Let ⌊nt⌋  1  K (t, x) = √ 1{ζi ≤x} − x , n i=1 n (4.31) where the ζi are independent and uniform on [0, 1]. By Krichagina and Puhalskii [15], the K n = ((K n (t, x), x ∈ [0, 1]), t ∈ R+ ) converge to K in distribution in D(R+ , D([0, 1], R)). Since K is continuous in both variables, it follows that the convergence takes place in Dc (R+ , Dc ([0, 1], R)) too. (One can apply Corollary D.1.) Since by (4.26), we can assume that U n (t, x) = K n (t, F (x)), the U n converge in distribution in Dc (R+ , Dc (R+ , R)) to the process U . Let, for ε ∈ (0, 1), ′n,ε L n (t, x) = U (t, x) + ′ε L (t, x) = U (t, x) + L′n,ε ((L′n,ε (t, x), x  x n U (t, u−) 0  x 0 1{F (u−)≤1−ε} dF (u), 1 − F (u−) U (t, u−) 1{F (u−)≤1−ε} dF (u), 1 − F (u−) = ∈ R+ ), t ∈ R+ ), and L′ε = ((L′ε (t, x), x ∈ R+ ), t ∈ R+ ). An argument analogous to the one used in the proof of Lemma 4.3 shows that L′ε is a separable random element of Dc (R+ , Dc (R+ , R)). The continuous mapping principle (see Theorem D.1) yields the convergence in distribution in Dc (R+ , Dc (R+ , R))2 of the (U n , L′n,ε ) to (U, L′ε ). Thus, in view of (4.9), (4.30) and Theorem D.10, the result follows by (4.29).  L EMMA 4.6. As n → ∞, the (X̂0n , X̃ n , H n , Gn , Ln , V n ) converge in distribution in R × Dc (R+ , R)3 × Dc (R+ , Dc (R+ , R))2 to (X̂, X̃, H, G, L, V ). P ROOF. By Lemma 4.3, the processes H , G, L, V and U are separable random elements of the associated function spaces. Since the exogenous arrival process E n and (X̂0n , X̃ n , U n , L′n ) are independent, on the one hand, and Y and (X̂0 , X̃, U, L′ ) are independent, on the other hand, and are separable random elements, Lemma 4.5, the hypotheses of Theorem 2.2, and Theorem D.8 imply that the (X̂0n , X̃ n , Y n , U n , L′n ) converge in distribution in R × Dc (R+ , R)2 × Dc (R+ , Dc (R+ , R))2 to (X̂0 , X̃, Y, U, L′ ). Since the random element (X̂0 , X̃, Y, U, L′ ) is separable, by Theorem 2.1 and Slutsky’s lemma (Theorem D.9) the (X̂0n , X̃ n , Y n , U n , L′n , An /n) jointly converge in distribution in R × Dc (R+ , R)2 × Dc (R+ , Dc (R+ , R))2 × Dc (R+ , R) to (X̂0 , X̃, Y, U, L′ , a). On recalling that V n (t, x) = U n (An (t)/n, x), V (t, x) = U (a(t), x), Ln (t, x) = L′n (An (t)/n, x), and L(t, x) = L′ (a(t), x) [see (3.4), (4.1), (4.4), (4.27) and (4.28)], we conclude by the continuous mapping principle (Theorem D.1) that the (X̂0n , X̃ n , Y n , V n , Ln ) converge in distribution in R × Dc (R+ , R)2 × Dc (R+ , Dc (R+ , R))2 to (X̂0 , X̃, Y, V , L). 160 A. A. PUHALSKII AND J. E. REED Let, for ε ∈ (0, 1), Gn,ε (t) =  t n V (t − u, u−) 1{F (u−)≤1−ε} dF (u), 1 − F (u−)  t V (t − u, u−) 1{F (u−)≤1−ε} dF (u), Gε (t) = 0 1 − F (u−) 0 Gn,ε = (Gn,ε (t), t ∈ R+ ), and Gε = (Gε (t), t ∈ R+ ). We note that the Gε are separable random elements of Dc (R+ , R). On applying the continuous mapping principle and recalling (3.18), we have that the (X̂0n , X̃ n , H n , Gn,ε , Ln , V n ) converge in distribution in R × Dc (R+ , R)3 × Dc (R+ , Dc (R+ , R))2 to (X̂0 , X̃, H, Gε , L, V ). In view of Theorem D.10, (3.7), (4.5) and (4.8), the assertion of the lemma will follow if for arbitrary T > 0 and δ > 0, lim lim sup P ε→0 n→∞   x n  V (t, u−) sup sup  t∈[0,T ] x∈R+ 0 1 − F (u−)   1{F (u−)>1−ε} dF (u) > δ = 0. The latter limit is implied by (4.28) and (4.29).  Let, for ε > 0, (4.32)  1{a(s)F (x)>ε} 1{s+x≤t} dLn (s, x)  1{a(s)F (x)>ε} 1{s+x≤t} dL(s, x). M n,ε (t) = R2+ and (4.33) ε M (t) = R2+ We note that these two integrals are, in fact, finite sums: (4.34) M n,ε (t) =  Ln ((s−, x−), (s, x))  L((s−, x−), (s, x)). s,x:s+x≤t, a(s)F (x)>ε and (4.35) M ε (t) = s,x:s+x≤t, a(s)F (x)>ε In particular, M ε = (M ε (t), t ∈ R+ ) is a separable random element of Dc (R+ , R) and M n,ε = (M n,ε (t), t ∈ R+ ) is a continuous function of Ln for spaces Dc (R+ , R) and Dc (R+ , Dc (R+ , R)). L EMMA 4.7. Given t1 < · · · < tm ∈ R+ , as n → ∞, the (X̂0n , X̃ n , H n , Gn , M n,ε , M n (t1 ), . . . , M n (tm )) converge in distribution in R × Dc (R+ , R)4 × Rm to (X̂0 , X̃, H, G, M ε , M(t1 ), . . . , M(tm )). 161 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC P ROOF. Since Z n (t) = Gn (t) − M n (t) by (3.6) and Z(t) = G(t) − M(t) by (4.18), the continuous mapping principle implies that it suffices to prove that the (X̂0n , X̃ n , H n , Gn , M n,ε , Z n (t1 ), . . . , Z n (tm )) converge in distribution in R × Dc (R+ , R)4 × Rm to (X̂0 , X̃, H, G, M ε , Z(t1 ), . . . , Z(tm )). Let 0 = s0l < s1l < s2l < · · · be such that skl → ∞ as k → ∞ and supi=1,2,... (sil − l ) → 0 as l → ∞. Recall that the function I and the process Z = (Z (t), t ∈ si−1 l,t l l R+ ) are defined by (4.19) and (4.21), respectively. We also set Zln (t) = − (4.36)  R2+ Il,t (s, x) dV n (s, x), where the integral on the right is defined in analogy with (4.20), that is,  R2+ Il,t (s, x) dV n (s, x) = ∞  i=1 1{s l i−1 ≤t}  n l  l l l V (si , t − si−1 ) − V n (si−1 , t − si−1 ) + V n (0, t). Lemma 4.6, (4.34), (4.35) and the continuous mapping principle yield convergence in distribution in R×Dc (R+ , R)4 ×Rm of the (X̂0n , X̃ n , H n , Gn , M n,ε , Zln (t1 ), . . . , Zln (tm )) to (X̂0 , X̃, H, G, M ε , Zl (t1 ), . . . , Zl (tm )) as n → ∞. By Lemma 4.4, for arbitrary t ∈ R+ and γ > 0,   lim P |Z(t) − Zl (t)| > γ = 0. l→∞ It thus remains to prove that, for arbitrary t ∈ R+ and γ > 0, lim lim sup P |Z n (t) − Zln (t)| > γ = 0. (4.37) l→∞ n→∞   By (3.1), (3.3) and (4.36), for s ∈ R+ , Z n (s) − Zln (s) n A (s) ∞ 1   1 l 1 n l =√ l n i=1 j =1 {sj −1 ≤s} {τi ∈(sj −1 ,sj ]}  × 1{ηi ∈(s−τ n ,s−s l For k ∈ N, introduce j −1 ]} i − F (s − sjl −1 ) − F (s − τin ) .   n n Zl,k (s) = A (s)∧k ∞ 1   √ 1 l 1 n l l n i=1 j =1 {sj −1 ≤s} {τi ∈(sj −1 ,sj ]}  × 1{ηi ∈(s−τ n ,s−s l An (s)/n i j −1 ]} − F (s − sjl −1 ) − F (s − τin ) .   Since the converge to a(s) in probability as n → ∞ by Theorem 2.1, (4.37) would follow by (4.38) n lim lim sup P |Zl,k (t)| > γ = 0. l→∞ n→∞   162 A. A. PUHALSKII AND J. E. REED Let Fsn be complete σ -algebras generated by the random variables τjn ∧ τAnn (s)+1 and ηj ∧An (s) , where j ∈ N. By Brémaud [4], Appendix A3, Theorem 25, the flow Fn = (Fsn , s ∈ R+ ) is right-continuous, so it is a filtration. By part 4 of Lemma C.1 (see also Lemma 8 in Reed [22] and Lemma 5.2 in Krichagina and n (s), s ∈ R ) is an Fn -square integrable martinPuhalskii [15]), the process (Zl,k + n (s), s ∈ R ), where gale with predictable quadratic variation process (Zl,k + n n Zl,k (s) = A (s)∧k ∞   1   1{s l ≤s} 1{τ n ∈(s l ,s l ]} F (s − sjl −1 ) − F (s − τin ) i j −1 j n i=1 j =1 j −1 × 1 − F (s − sjl −1 ) − F (s − τin ) ∞ 1 ≤ 1 l n j =1 {sj −1 ≤s} = ∞ 1 1 l n j =1 {sj −1 ≤s}   n (s)  s A 0 i=1  1{τ n ∈(s l  s−s l j −1  s−sjl i l j −1 ,sj ]} 1{x∈(s−τ n ,s−s l j −1 ]} i dF (x) An (sjl ∧ s) − An (s − x) dF (x).  In view of the compact convergence in probability of the An /n to a as n → ∞ (Theorem 2.1), the latter sum converges in probability as n → ∞ to ∞  j =1 1{s l j −1 ≤s} = ∞  j =1 − = where ul (x) =  s−s l j −1  s−sjl 1{s l j −1 ≤s}  s 0  a(sjl ∧ s) − a(s − x) dF (x)  F (s − sjl −1 ) − F (s − sjl )+ a(sjl ∧ s)   a(s − x) dF (x)  s  s   a ul (s − x) dF (x) − a(s − x) dF (x), 0 ∞ l j =1 sj 0 ∧ s1{s l j −1 ≤s} 1{x∈[s l l j −1 ,sj )} for x ∈ [0, s]. Note that ul (x) ≥ x and ul (x) → x as l → ∞. Hence, ul (s − x) → s − x from the right, so a(ul (s − x)) → a(s − x). By Lebesgue’s dominated convergence theorem, 0s a(ul (s − x)) dF (x) → 0s a(s − x) dF (x) as l → ∞. We conclude that, for arbitrary δ > 0, n (t) > δ) = 0. lim lim sup P(Zl,k l→∞ n→∞ Limit (4.38) follows by an application of the Lenglart–Rebolledo inequality; see, for example, Liptser and Shiryayev [16], Theorem 1.9.3.  L EMMA 4.8. As n → ∞, the (X̂0n , X̃ n , H n , Gn , M n,ε , M n ) converge in distribution in R × Dc (R+ , R)4 × D(R+ , R) to (X̂0 , X̃, H, G, M ε , M). The latter random element has a tight distribution. ON MANY-SERVER QUEUES IN HEAVY TRAFFIC 163 P ROOF. Let us show that the sequence M n is tight in D(R+ , R) (for Skorohod’s J1 -topology). By Lemma 3.1, the process Mnn2 is a Gn -square integrable martingale with predictable quadratic variation process Mnn2 (t) = = An /n  R2+ 1{s+x≤t} dLnn2 (s, x) 2  t An (t−x)∧n  1 0 n i=1 1{0<x≤ηi } 1 − F (x) dF (x). (1 − F (x−))2 Since the tend to a in probability uniformly over compact sets, by the law of large numbers and (4.10), for T > 0 and δ > 0, lim P n→∞    sup Mnn2 (t) − t∈[0,T ]  t 0   1{x>0} a(t − x) dF ′ (x) > δ = 0. By Theorem VI.5.17 in Jacod and Shiryaev [12], the sequence Mnn2 is tight in D(R+ , R). Since the An /n converge in probability to a, the definition of V n (t, x) in (3.1), Donsker’s theorem, Slutsky’s lemma (Theorem D.9) and the continuous mapping principle imply that the√sequence Ṽ n = (V n (t, 0), t ∈ R+ ) converges in distribution in Dc (R+ , R) to ( F (0)(1 − F (0))W (a(t)), t ∈ R+ ), where (W (t), t ∈ R+ ) is a standard Brownian motion. Thus, this sequence is asymptotically tight in Dc (R+ , R) (Theorem D.3). By Theorem D.7, the sequence (Ṽ n , Mnn2 ) is asymptotically tight in Dc (R+ , R) × D(R+ , R). Since the map (x, y) → x + y is continuous as a map from Dc (R+ , R) × D(R+ , R) to D(R+ , R), it follows that the sequence Ṽ n + Mnn2 is asymptotically tight in D(R+ , R), see Theorem D.4. By Ulam’s theorem, Ṽ n + Mnn2 is a tight random element of D(R+ , R) for each n. Therefore, the sequence Ṽ n + Mnn2 is tight in D(R+ , R), van der Vaart and Wellner [23], Problem 1.3.9. Since, for T > 0, by (3.9)–(3.11), P     sup M n (t) − V̆ n An (t)/n, 0 − Mnn2 (t) > 0 ≤ P An (T ) > n2  t∈[0,T ]  and the latter probability tends to zero as n → ∞, it follows that the sequence M n is tight in D(R+ , R) (e.g., by Theorem VI.3.21 in Jacod and Shiryaev [12]). By Lemma 4.7, the (X̂0n , X̃ n , H n , Gn , M n,ε ) converge in distribution in R × Dc (R+ , R)4 to (X̂0 , X̃, H, G, M ε ), which is a separable, hence, tight, random element (recall the definition of X̃ in the statement of Theorem 2.2, Lemma 4.3, and (4.35)). Therefore, the sequence (X̂0n , X̃ n , H n , Gn , M n,ε ) is asymptotically tight in R × Dc (R+ , R)4 , see Theorem D.3. Since the sequence M n is tight in D(R+ , R), it follows that the sequence (X̂0n , X̃ n , H n , Gn , M n,ε , M n ) is asymptotically tight in R × Dc (R+ , R)4 × D(R+ , R), see Theorem D.7. Since the topology of R × D(R+ , R)5 is coarser than the topology of R × Dc (R+ , R)4 × D(R+ , R), the sequence (X̂0n , X̃ n , H n , Gn , M n,ε , M n ) is asymptotically tight in R × D(R+ , R)5 . It is thus tight because the (X̂0n , X̃ n , H n , Gn , M n,ε , 164 A. A. PUHALSKII AND J. E. REED M n ) are random elements of R × D(R+ , R)5 , see van der Vaart and Wellner [23], Problem 1.3.9. Lemma 4.7 and Prohorov’s theorem imply that the (X̂0n , X̃ n , H n , Gn , M n,ε , M n ) converge in distribution in R × D(R+ , R)5 to (X̂0 , X̃, H, G, M ε , M). On taking the set of Skorohod-continuous bounded functions as the separating subalgebra in Theorem D.6 we conclude that the sequence (X̂0n , X̃ n , H n , Gn , M n,ε , M n ) is asymptotically measurable in R × Dc (R+ , R)4 × D(R+ , R). The latter property, coupled with the asymptotic tightness of this sequence in R × Dc (R+ , R)4 × D(R+ , R), yields by Theorem D.5 the existence of a subsequence that converges in distribution in R × Dc (R+ , R)4 × D(R+ , R) to a random element with a tight probability law. The limit must be the same as in R × D(R+ , R)5 , that is, it is (X̂0 , X̃, H, G, M ε , M). It thus does not depend on a subsequence.  4.3. Completion of the proof of Theorem 3.1. Let M(t) = M1 (t) + M2 (t) (4.39) be the decomposition of the Gaussian martingale M into the sum of a continuous Gaussian martingale M1 = (M1 (t), t ∈ R+ ) and a pure-jump Gaussian martingale M2 = (M2 (t), t ∈ R+ ) with jumps occurring at the jump times of C(t); see Jacod and Shiryaev [12], Chapter II, Section 4d, Liptser and Shiryayev [16], Chapter 4, Section 9. Formally, M1 (t) = (4.40) M2 (t) =  t 0  t 0 1{C(s)=0} dM(s), 1{C(s)>0} dM(s). Let also M ′ε (t) = M(t) − M ε (t). (4.41) We show that, for T > 0 and δ > 0, (4.42a) (4.42b) sup |M ε (t) − M2 (t)| > δ = 0, lim P ε→0 t∈[0,T ] sup |M ′ε (t) − M1 (t)| > δ = 0. lim P ε→0 t∈[0,T ] Since by (4.17) C(t) = M2 (t) =  s+x=t R2+ a(s)F ′ (x), by (4.16), 1{a(s)F (x)>0} 1{s+x≤t} dL(s, x). Hence, by (4.33) and (4.40), ε M2 (t) − M (t) =  R2+ 1{0<a(s)F (x)≤ε} 1{s+x≤t} dL(s, x). ON MANY-SERVER QUEUES IN HEAVY TRAFFIC 165 Lemma 4.2 implies that (M2 (t) − M ε (t), t ∈ R+ ) is a Gaussian martingale relative to the natural filtration with predictable quadratic variation process  ( R2 1{0<a(s)F (x)≤ε} 1{s+x≤t} da(s) dF ′ (x), t ∈ R+ ). The latter function tends + to zero as ε → 0 for every t ∈ R+ , so the Lenglart–Rebolledo inequality yields (4.42a). The limit in (4.42b) follows by (4.41), (4.42a) and the identity in (4.39). We introduce processes M ′n,ε = (M ′n,ε (t), t ∈ R+ ) by letting M ′n,ε (t) = n M (t) − M n,ε (t). Let also M ′ε = (M ′ε (t), t ∈ R+ ). By Lemma 4.8 and the continuous mapping principle, as n → ∞, the (X̂0n , X̃ n , H n , Gn , M n,ε , M ′n,ε ) converge in distribution in R × Dc (R+ , R)4 × D(R+ , R) to (X̂0 , X̃, H, G, M ε , M ′ε ). The latter random element has a separable range. Therefore, by Theorem D.11, lim d ∗ ((X̂0n , X̃ n , H n , Gn , M n,ε , M ′n,ε ), (X̂0 , X̃, H, G, M ε , M ′ε )) = 0, n→∞ BL1 ∗ is the distance on the space of mappings from  to R × D (R , R)4 × where dBL c + 1 D(R+ , R) as defined in Appendix D. The limits in (4.42a) and (4.42b) imply that ∗ ((X̂ , X̃, H, G, M ε , M ′ε ), (X̂ , X̃, H, G, M , M ) = 0, so limε→0 dBL 0 0 2 1 1 ∗ ((X̂0n , X̃ n , H n , Gn , M n,ε , M ′n,ε ), (X̂0 , X̃, H, G, M2 , M1 )) = 0. lim lim sup dBL 1 ε→0 n→∞ Therefore, there exists a sequence εn → 0 such that lim d ∗ ((X̂0n , X̃ n , H n , Gn , M n,εn , M ′n,εn ), (X̂0 , X̃, H, G, M2 , M1 )) = 0, n→∞ BL1 which implies by Theorem D.11 that the (X̂0n , X̃ n , H n , Gn , M n,εn , M ′n,εn ) converge in distribution in R × Dc (R+ , R)4 × D(R+ , R) to (X̂0 , X̃, H, G, M2 , M1 ). Since the process M1 has continuous paths and convergence in Skorohod’s J1 topology to continuous functions is equivalent to compact convergence, by Corollary D.1 the (X̂0n , X̃ n , H n , Gn , M n,εn , M ′n,εn ) converge in distribution in R × Dc (R+ , R)5 to (X̂0 , X̃, H, G, M2 , M1 ). By the continuous mapping principle, the (X̂0n , X̃ n , H n , Gn − M n,εn − M ′n,εn ) converge in distribution in R × Dc (R+ , R)3 to (X̂0 , X̃, H, G − M1 − M2 ). Recalling that M n = M n,ε + M ′n,ε , Z n = Gn − M n , M = M1 + M2 , and Z = G − M completes the proof of Theorem 3.1. APPENDIX A: EXISTENCE AND UNIQUENESS FOR THE QUEUEING EQUATIONS We prove existence and uniqueness of solutions to (2.2a)–(2.2c). L EMMA A.1. Given a nondecreasing nonnegative integer-valued process E n with trajectories in D(R+ , R), a nonnegative integer-valued random variable Qn0 , and sequences of nonnegative random variables {ηi } and {η̃i }, there exist nonnegative integer-valued processes Qn and Q̃n and nonnegative integer-valued nondecreasing process An with trajectories in D(R+ , R) such that equations (2.2a)– (2.2c) are satisfied. The process Q̃n is specified uniquely. If ηi > 0 for all i, then the processes An and Qn are specified uniquely. 166 A. A. PUHALSKII AND J. E. REED P ROOF. We fix ω ∈  throughout. Obviously, Q̃n is specified uniquely by (2.2c). For Qn and An , we start with the case where Qn0 < n. First, the existence issue is addressed. Introduce the process Qn,1 = (Qn,1 (t), t ∈ R+ ) as the process of the number of customers in the infinite server with the same arrival and service times and the initial number of customers, that is, (A.1) Q n,1 n n (t) = Q̃ (t) + E (t) −  t t 0 0 1{s+x≤t} d n (s) E i=1 1{ηi ≤x} . Let τ n,1 = inf{t : Qn,1 (t) ≥ n} ≤ ∞. Suppose τ n,1 > 0. We define Qn (t) = Qn,1 (t) and An (t) = E n (t) for t < τ n,1 . Obviously, equations (2.2a) and (2.2b) are satisfied for t < τ n,1 . If τ n,1 = ∞, the proof of existence is over. Suppose the contrary. Clearly, τ n,1 is a jump time of E n . We choose An (τ n,1 ) such that (A.2) An (τ n,1 ) i=E n (τ n,1 −)+1  1{ηi >0} n = n − Q (τ n,1 n −) + Q̃ (τ n,1 )− n,1 E n (τ  −) i=1  1{τ n +ηi =τ n,1 } . i It exists and is not greater than E n (τ n,1 ) due to the facts that Qn (τ n,1 −) < n and that Qn,1 (τ n,1 ) ≥ n, so by (A.1), n Q (τ n,1 (A.3) − n −) + Q̃ (τ E n (τ n,1 ) i=E n (τ n,1 −)+1 n,1 n ) + E (τ n,1 )− n,1 E n (τ  −) i=1 1{τ n +ηi =τ n,1 } i 1{ηi =0} ≥ n. We also let (A.4) Qn (τ n,1 ) = n + E n (τ n,1 ) − An (τ n,1 ). Hence, Qn (τ n,1 ) ≥ n, so (2.2b) holds for t = τ n,1 . By (A.2) and (A.4), (A.5) Qn (τ n,1 ) = Qn (τ n,1 −) + E n (τ n,1 ) + Q̃n (τ n,1 ) − An (τ n,1 ) i=1 1{τ n +ηi =τ n,1 } . i We obtain that equations (2.2a) and (2.2b) are satisfied on [0, τ n,1 ]. We next extend the construction past τ n,1 . Let us define a nondecreasing nonnegative integervalued process An,2 = (An,2 (t), t ∈ R+ ) with right-continuous trajectories admitting left-hand limits and with An,2 (0) = 0 by specifying its jumps An,2 (t) for 167 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC t > 0 as follows: n,2 A  (t) = inf k ∈ N : (A.6) An,2 (t−)+k i=An,2 (t−)+1 ≥ An,2 (t−)  i=1 1{ηn,2 >0} i n,2 1{τ n,2 +ηn,2 =t} − Q̃ i i  (t) , where τin,2 = inf{t : An,2 (t) ≥ i}, ηin,2 = ηi+An (τ n,1 ) , and n,2 (A.7) Q̃ n (t) = Q̃ (t + τ n,1 )+ An (τ n,1 ) i=1 1{τ n +ηi >t+τ n,1 } . i It follows by (A.1) and (A.2) on recalling that Qn (τ n,1 −) = Qn,1 (τ n,1 −) and An (τ n,1 −) = E n (τ n,1 −) that Q̃n,2 (0) = n and Q̃n,2 (t) ≤ n for t ≥ 0 as the righthand side of (A.7) is decreasing in t. In words, An,2 (t) is the number of customers that enter service by time t for an n-server queue that always has a nonzero queue length and for which residual service times of customers in service at time 0 are those of customers in the nserver queue under consideration at time τ n,1 and the service times of customers entering service after time zero are equal to the service times of customers entering service after time τ n,1 in the queue in question. We note that this process can run away to infinity on finite time. (For instance, if Q̃n,2 (t) has its first jump at s > 0 and ηin,2 = 0 for all i, then An,2 (t) = ∞ for t ≥ s.) We therefore introduce τ̃ n,1 = sup{t : An,2 (t) < ∞} and note that τ̃ n,1 > 0 by the right continuity of Q̃n,2 . By (A.6), we have for t < τ̃ n,1 that  n,2 (t) A i=1 1{τ n,2 +ηn,2 >t} = −Q̃n,2 (t), i i so, since Q̃n,2 (0) = n, n,2 (t) A (A.8) i=1 1{τ n,2 +ηn,2 >t} + Q̃n,2 (t) = n. i i Introduce (A.9) σ n,1 = inf t : An,2 (t) > Qn (τ n,1 ) − n + E n (t + τ n,1 ) − E n (τ n,1 ) .    By right continuity of An,2 , σ n,1 > 0. Also, σ n,1 ≤ τ̃ n,1 . Intuitively, σ n,1 is the time when An,2 (t) and An (t +τ n,1 ) − An (τ n,1 ) diverge. To substantiate this, define 168 A. A. PUHALSKII AND J. E. REED for t ∈ [τ n,1 , τ n,1 + σ n,1 ), (A.10) Qn (t) = Qn (τ n,1 ) − n + Q̃n,2 (t − τ n,1 ) + E n (t) − E n (τ n,1 ) − n,1 An,2 (t−τ  ) i=1   1{τ n,2 +ηn,2 ≤t−τ n,1 } . i i By (A.8), we can also write Qn (t) = Qn (τ n,1 ) + E n (t) − E n (τ n,1 ) − An,2 (t − τ n,1 ).  (A.11)  Therefore, Qn (t) ≥ n by (A.9), and by (A.4), Qn (t) = n + E n (t) − An (τ n,1 ) − An,2 (t − τ n,1 ). (A.12) On letting An (t) = An (τ n,1 ) + An,2 (t − τ n,1 ), (A.13) we can see that (2.2b) holds on [τ n,1 , τ n,1 + σ n,1 ). To obtain (2.2a), we substitute (A.4) in (A.10) and recall the definitions of τin,2 , ηin,2 and Q̃n,2 (t). If σ n,1 = ∞, the proof of existence is over. If σ n,1 < ∞, we let (A.14) An (τ n,1 + σ n,1 ) = Qn (τ n,1 + σ n,1 )− − n + E n (τ n,1 + σ n,1 )  and  Qn (τ n,1 + σ n,1 ) = E n (τ n,1 + σ n,1 ) − (A.15) n n,1 n,1 An,2 (σ n,1 −)+A  (τ +σ ) i=1 1{τ n,2 +ηn,2 =σ n,1 } i i + Q̃n,2 (σ n,1 ). We note that by (A.11) and (A.14), (A.16) An (τ n,1 + σ n,1 ) = Qn (τ n,1 ) − n + E n (τ n,1 + σ n,1 ) − E n (τ n,1 ) − An,2 (σ n,1 −),  so by (A.9) (A.17)  0 ≤ An (τ n,1 + σ n,1 ) < An,2 (σ n,1 ). Let us show that Qn (τ n,1 + σ n,1 ) < n. By (A.6) and the right-hand inequality in (A.17), n n,1 n,1 An,2 (σ n,1 −)+A  (τ +σ ) i=An,2 (σ n,1 −)+1 1{ηn,2 >0} < i An,2 (σ n,1 −) i=1 1{τ n,2 +ηn,2 =σ n,1 } − Q̃n,2 (σ n,1 ), i i 169 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC so n A (τ n,1 +σ n,1 )< n n,1 n,1 An,2 (σ n,1 −)+A  (τ +σ ) i=1 (A.18) 1{τ n,2 +ηn,2 =σ n,1 } i i − Q̃n,2 (σ n,1 ), and by (A.15) and (A.14), (A.19) Qn (τ n,1 + σ n,1 ) < E n (τ n,1 + σ n,1 ) − An (τ n,1 + σ n,1 ) = n − Qn (τ n,1 + σ n,1 )− .   Hence, Qn (τ n,1 + σ n,1 ) < n. Also, Qn (τ n,1 + σ n,1 ) ≥ 0 which is shown as follows. By (A.15) and (A.16), Qn (τ n,1 + σ n,1 ) ≥ Qn (τ n,1 + σ n,1 )− + E n (τ n,1 + σ n,1 )− An (τ n,1 + σ n,1 )  − An,2 (σ n,1 −) =n− i=1  1{τ n,2 +ηn,2 =σ n,1 } + Q̃n,2 (σ n,1 ) An,2 (σ n,1 −) i=1 i i 1{τ n,2 +ηn,2 =σ n,1 } + Q̃n,2 (σ n,1 ). i i Since −Q̃n,2 (σ n,1 ) ≤ Q̃n,2 (σ n,1 −), we have by (A.8) that An,2 (σ n,1 −) i=1 ≤ 1{τ n,2 +ηn,2 =σ n,1 } − Q̃n,2 (σ n,1 ) i An,2 (σ n,1 −) i=1 i 1{τ n,2 +ηn,2 ≤σ n,1 } + Q̃n,2 (σ n,1 −) = n. i i The required inequality has been proven. By (A.14), (2.2b) for t < τ n,1 + σ n,1 , and the inequality Qn ((τ n,1 + σ n,1 )−) ≥ n, An (τ n,1 +σ n,1 ) = E n (τ n,1 +σ n,1 )−An ((τ n,1 +σ n,1 )−). We thus have (2.2b) for t = τ n,1 + σ n,1 . Equation (2.2a) for t = τ n,1 + σ n,1 follows by (A.15) and (A.7). Existence has been proven on [0, τ n,1 + σ n,1 ]. At time τ n,1 + σ n,1 , we are in a similar situation to the one we faced at t = 0. We can thus define τ n,2 and σ n,2 and proceed until we get to the desired time t. This is bound to happen after a finite number of steps the reason being that in any interval n,j + σ n,j ), i−1 (τ n,j + σ n,j ) + τ n,i ], where σ n,1 = 0 by definition, we [ ji−1 =1 (τ j =1 have at least one upward jump of E n . Since E n is finite-valued, there are only finitely many such intervals on any interval [0, t]. We now prove uniqueness. Thus, we suppose that Qn and An satisfy (2.2a) and (2.2b) and that ηi > 0 for all i. Assume for the moment that τ n,1 > 0. If Qn 170 A. A. PUHALSKII AND J. E. REED and An differed from those defined above on [0, τ n,1 ), then there would exist t ′ < τ n,1 with Qn (t ′ ) ≥ n. We note that t ′ > 0 which is checked as follows. By (2.2a) and (2.2b) for t = 0, n A (0)  + 1{ηi =0} Q (0) = Q̃ (0) + Qn (0) − n + An (0) − n n i=1 n E (0)  + ≤ Q̃ (0) + Qn (0) − n + E n (0) − 1{ηi =0} n i=1  +  + = Qn (0) − n + Qn,1 (0) < Qn (0) − n + n, so Qn (0) < n. For τ̃ n,1 = inf{t < τ n,1 : Qn (t) ≥ n} ∈ (0, ∞), we would have that Qn (t) = n,1 Q (t) and An (t) = E n (t) when t < τ̃ n,1 . Also, An (τ̃ n,1 ) ≤ E n (τ̃ n,1 ). Since by (2.2a) and (A.1), n Q (τ̃ n,1 n,1 )=Q (τ̃ n,1 )+ E n (τ̃ n,1 ) i=An (τ̃ n,1 )+1 1{ηi =0} < n + E n (τ̃ n,1 ) − An (τ̃ n,1 ), we arrive at a contradiction with (2.2b). We have thus proved that a solution to (2.2a) and (2.2b) is specified uniquely on [0, τ n,1 ). Next, (2.2a) and (2.2b) imply (A.5), so by (A.3) and the inequality E n (τ n,1 ) ≥ n,1 A (τ n,1 ) we have that Qn (τ n,1 ) ≥ n, and (2.2b) implies (A.4). Also, (A.4) and (A.5) imply (A.2). Since ηi > 0, the latter condition specifies An (τ n,1 ) uniquely. Uniqueness on [0, τ n,1 ] follows. We now show that Qn (t) ≥ n for t ∈ [τ n,1 , τ n,1 + σ n,1 ). Let σ̃ n,1 = inf{t > n,1 τ : Qn (t) < n} and suppose that σ̃ n,1 < τ n,1 + σ n,1 . By right continuity of Qn , we have that σ̃ n,1 > τ n,1 . By (2.2b), for t ∈ [τ n,1 , σ̃ n,1 ), (A.20) An (t) − An (τ n,1 ) = E n (t) − E n (τ n,1 ) + Qn (τ n,1 ) − Qn (t). Since by (2.2a) and (A.7) for t ≥ τ n,1 , (A.21) Qn (t) = Qn (τ n,1 ) − n + Q̃n,2 (t − τ n,1 ) + E n (t) − E n (τ n,1 ) −  n (t) A i=An (τ n,1 )+1 1{τin +ηi ≤t} , we have that, for t ∈ [τ n,1 , σ̃ n,1 ), (A.22) n (t) A i=An (τ n,1 )+1 1{τin +ηi >t} = n − Q̃n,2 (t − τ n,1 ),  171 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC so that n (t) A  i=An (τ n,1 )+1 1{τin +ηi >t} = −Q̃n,2 (t − τ n,1 ). Consequently, if t is a jump time of An , then n (t) A i=An (t−)+1 1{ηi >0} = n (t−) A i=An (τ n,1 )+1 1{τin +ηi =t} − Q̃n,2 (t − τ n,1 ). It follows by (A.6) and the assumption that ηi > 0 that An (t) − An (τ n,1 ) = An,2 (t − τ n,1 ), which implies that Qn (t) is given by the right-hand side of (A.10) (or the right-hand side of (A.11)) for t ∈ [τ n,1 , σ̃ n,1 ). Besides, σ̃ n,1 − τ n,1 ≤ τ̃ n,1 . We also have by (2.2b) since Qn (σ̃ n,1 ) < n, that An (σ̃ n,1 ) − An (τ n,1 ) = n E (σ̃ n,1 ) − E n (τ n,1 ) + Qn (τ n,1 ) − n, so by (A.21) with t = σ̃ n,1 (σ̃ n,1 ) An i=An (τ n,1 )+1 1{τ n +ηi >σ̃ n,1 } = Qn (σ̃ n,1 ) − Q̃n,2 (σ̃ n,1 − τ n,1 ) i < n − Q̃n,2 (σ̃ n,1 − τ n,1 ). It follows by (A.22) on noting that σ̃ n,1 is a jump time of An that An (σ̃ n,1 ) An (σ̃ n,1 −)+1 1{ηi >0} < An ( σ̃ n,1 −) i=An (τ n,1 )+1 1{τ n +ηi =σ̃ n,1 } − Q̃n,2 (σ̃ n,1 − τ n,1 ), i so by the fact that An (t) − An (τ n,1 ) = An,2 (t − τ n,1 ) for t < σ̃ n,1 and (A.6), An (σ̃ n,1 ) < An,2 (σ̃ n,1 − τ n,1 ). Therefore, An (σ̃ n,1 ) i=An (τ n,1 )+1 1{τ n +ηi ≤σ̃ n,1 } ≤ i n,1 −τ n,1 ) An,2 (σ̃ i=1 1{τ n,2 +ηn,2 ≤σ̃ n,1 −τ n,1 } . i i Since the right-hand side of (A.10) is not less than n at σ̃ n,1 , so is the right-hand side of (A.21), which contradicts the definition of σ̃ n,1 . The obtained contradiction proves that Qn (t) ≥ n on [τ n,1 , σ n,1 ). Consequently, (A.20) holds for t from this interval. By (A.21), (A.22) holds for those t, hence, as we have seen, (A.13) holds. By (A.13) and (A.20), Qn (t) is given by the right-hand side of (A.11). Thus, Qn and An are specified uniquely on [0, τ n,1 + σ n,1 ). For t = τ n,1 + σ n,1 , we have that (2.2a) implies (A.15) (with possibly different An (τ n,1 + σ n,1 )). By (2.2b) on [0, τ n,1 + σ n,1 ], (A.23) + An (τ n,1 + σ n,1 ) = E n (τ n,1 + σ n,1 ) − Qn (τ n,1 + σ n,1 ) − n  + Qn (τ n,1 + σ n,1 )− − n .     172 A. A. PUHALSKII AND J. E. REED In particular, An (τ n,1 + σ n,1 ) is not greater than the right-hand side of (A.14), so by (A.11) it is not greater than the right-hand side of (A.16), and we obtain (A.17) by applying (A.9). A similar argument to the one we used above yields (A.18). Then, in analogy with (A.19) on taking into account (A.23), Qn (τ n,1 + σ n,1 ) < E n (τ n,1 + σ n,1 ) − An (τ n,1 + σ n,1 ) = n ∨ Qn (τ n,1 + σ n,1 ) − Qn (τ n,1 + σ n,1 )− ,   so that Qn (τ n,1 + σ n,1 ) < n ∨ Qn (τ n,1 + σ n,1 ). Hence, Qn (τ n,1 + σ n,1 ) < n. By (A.23), we have (A.14). Having established uniqueness on [0, τ n,1 + σ n,1 ], we can use this argument repeatedly until a given t is reached. We have thus established existence and uniqueness for the case where Qn0 < n. If Qn0 ≥ n, then we can use an analogous argument to the one employed when τ n,1 = 0.  We now provide an example of nonuniqueness in the case where there are zero service times. Consider a single server queue (so n = 1) with no customers initially. There are two arrivals: the first arrival occurs at t = 1 and the second at t = 2. The service times are η1 = 2 and η2 = 0, respectively. Thus, E n (t) = 1{t≥1} + 1{t≥2} . The equations for Qn (t) and An (t) are as follows Qn (t) = 1{t≥1} + 1{t≥2} − 1{An (t)≥1} 1{1+2≤t} − 1{An (t)≥2} 1{2+0≤t} , + An (t) = 1{t≥1} + 1{t≥2} − Qn (t) − 1  . They admit two sets of solutions: Qn (t) = 1{1≤t<3} , An (t) = 1{1≤t<2} + 21{t≥2} and Qn (t) = 1{1≤t<2} + 21{2≤t<3} , An (t) = 1{1≤t<3} + 21{t≥3} . APPENDIX B: CONTINUITY PROPERTIES OF CONVOLUTION EQUATIONS Let B = (B(x), x ∈ R+ ) be a distribution function on R+ with B(0) < 1. Given T > 0, let L∞ ([0, T ], R) denote the Banach space of R-valued bounded Borel measurable functions on [0, T ] which is equipped with the uniform norm x∞ = supt∈[0,T ] |x(t)|, where x = (x(t), t ∈ [0, T ]). Consider the equation (B.1) y(t) = x(t) +  t 0   f y(t − s), t − s dB(s), where x = (x(t), t ∈ [0, T ]) ∈ L∞ ([0, T ], R) and the function f : R × R+ → R is Borel measurable. L EMMA B.1. If |f (y, t)| ≤ |y| and |f (y1 , t) − f (y2 , t)| ≤ |y1 − y2 | for all y, y1 , y2 , and t from the domain, then for every x ∈ L∞ ([0, T ], R) there exists a unique y = (y(t), t ∈ [0, T ]) ∈ L∞ ([0, T ], R) which satisfies equation (B.1). This solution can be obtained by the method of successive approximations and ON MANY-SERVER QUEUES IN HEAVY TRAFFIC 173 there exists a function ρ(t) which only depends on the function B such that supt∈[0,T ] |y(t)| ≤ ρ(T ) supt∈[0,T ] |x(t)|. The latter bound holds more generally if |y(t)| ≤ |x(t)| +  t 0 |y(t − s)| dB(s). If, in addition, x is right-continuous with left-hand limits (respectively, admits limits on the right and limits on the left) and f (y, t) is right-continuous with lefthand limits (respectively, admits limits on the right and limits on the left) in the second argument, then y is right-continuous with left-hand limits (respectively, admits limits on the right and limits on the left). If x is continuous and either f (y, t) is continuous in the second argument, f (0, 0) = 0, and x(0) = 0, or f (y, t) admits limits on the right and limits on the left in the second argument, and B is continuous on [0, T ] with B(0) = 0, then y is continuous. P ROOF. (B.2) Let t0 ∈ (0, T ] be such that B(t0 ) < 1. Define the map φx by φx (z)(t) = x(t) +  t 0   f z(t − s), t − s dB(s). R). The boundedness condition on f implies that φx is an operator  t on L∞ ([0, t0 ], ′ ′ We show it is a contraction. Since |φx (z)(t) − φx (z )(t)| ≤ 0 |z(t − s) − z (t − s)| dB(s), we have that supt∈[0,t0 ] |φx (z)(t) − φx (z′ )(t)| ≤ B(t0 ) supt∈[0,t0 ] |z(t) − z′ (t)|, which proves the claim because B(t0 ) < 1. Since L∞ ([0, t0 ], R) is a Banach space, the operator φx has a unique fixed point so that the equation y(t) = x(t) + t 0 f (y(t − s), t − s) dB(s) has a unique solution for t ∈ [0, t0 ], which is obtained by the method of successive approximations.  Next, on introducing yt0 (t) = y(t + t0 ), xt0 (t) = x(t + t0 ) + tt+t0 f (y(t + t0 − s), t +t0 −s) dB(s), and ft0 (u, v) = f (u, t0 +v), we can write (B.1) for t ∈ [0, T − t0 ] in the form (B.3) yt0 (t) = xt0 (t) +  t 0   ft0 yt0 (t − s), t − s dB(s). The function xt0 = (xt0 (t), t ∈ [0, T − t0 ]) is uniquely specified by x and the values of y(t) for t ≤ t0 . The preceding argument applied to (B.3) shows that given y(t) for t ∈ [0, t0 ], there exists a unique extension of y(t) to the interval [t0 , 2t0 ∧ T ] that satisfies the equation. The method of successive approximations converges to this solution. By applying this argument repeatedly, we deduce existence and uniqueness of a solution in L∞ ([0, T ], R). This solution is obtained by the method of successive approximations. If x is right-continuous with left-hand limits (respectively, admits limits on the right and limits on the left) and f (y, t) is right-continuous with left-hand limits (respectively, admits limits on the right and limits on the left) in t, then the successive approximations starting from the zero function are right-continuous with 174 A. A. PUHALSKII AND J. E. REED left-hand limits (respectively, admit limits on the right and limits on the left), so y is right-continuous with left-hand limits (respectively, admits limits on the right and limits on the left). A similar argument applies in the case where x is continuous. Suppose that |y(t)| ≤ |x(t)| + Then  t 0 |y(t − s)| dB(s) for t ∈ [0, T ]. sup |y(t)| ≤ sup |x(t)| + sup |y(t)|B(t0 ), t∈[0,t0 ] so t∈[0,t0 ] sup |y(t)| ≤ t∈[0,t0 ] Next, sup t∈[t0 ,(2t0 )∧T ] |y(t)| ≤ t∈[0,t0 ] 1 sup |x(t)|. 1 − B(t0 ) t∈[0,t0 ] sup t∈[t0 ,(2t0 )∧T ] + |x(t)| + sup |y(t)| t∈[0,t0 ] sup t∈[t0 ,(2t0 )∧T ] sup t∈[0,(2t0 )∧T ] |y(t)| ≤ sup t∈[0,(2t0 ))∧T ] |x(t)|  |y(t)| 1 − B(t0 ) . Thus,   1 1 + . 1 − B(t0 ) (1 − B(t0 ))2 It follows that sup |y(t)| ≤ sup |x(t)| t∈[0,T ] t∈[0,T ] ⌊T /t 0 ⌋+1  i=1 1 . (1 − B(t0 ))i  The next lemma is concerned with convergence of solutions to (B.1). Let yn = (yn (t), t ∈ [0, T ]) ∈ L∞ ([0, T ], R) solve equations n n y (t) = x (t) + (B.4)  t 0 f n yn (t − s), t − s dB(s),   where xn = (xn (t), t ∈ [0, T ]) ∈ L∞ ([0, T ], R) and the f n satisfy the hypotheses on f in Lemma B.1 (in particular, are Lipshitz continuous in the first argument). L EMMA B.2. 1. If lim sup  t  n    f y(t − s), t − s − f y(t − s), t − s  dB(s) = 0 n→∞ t∈[0,T ] 0 and the xn converge to x in L∞ ([0, T ], R), then the yn converge to y in L∞ ([0, T ], R). ON MANY-SERVER QUEUES IN HEAVY TRAFFIC 175 2. Suppose that f n (y, t) → f (y, t) as n → ∞ for every y and t and that f n (y, t) is monotonically decreasing in n and monotonically increasing in y, that is, f m (y, t) ≤ f n (y, t) for m ≥ n and f n (y1 , t) ≤ f n (y2 , t) for y1 ≤ y2 . If xn (t) → x(t) for all t ∈ [0, T ] and supn supt∈[0,T ] |xn (t)| < ∞, then yn (t) → y(t) for all t ∈ [0, T ]. P ROOF. In order to prove part 1, introduce  t     n f y(t − s), t − s − f y(t − s), t − s dB(s). x̃ (t) = x (t) − x(t) + n n 0 We have,  t      |y (t) − y(t)| ≤ |x̃ (t)| + f n yn (t − s), t − s − f n y(t − s), t − s  dB(s) n n 0 ≤ |x̃n (t)| +  t 0 |yn (t − s) − y(t − s)| dB(s), so by Lemma B.1 supt∈[0,T ] |yn (t) − y(t)| ≤ ρ(T ) supt∈[0,T ] |x̃n (t)|. The hypotheses imply that limn→∞ supt∈[0,T ] |x̃n (t)| = 0. Thus, limn→∞ supt∈[0,T ] |yn (t) − y(t)| = 0. We now prove part 2. Given m ∈ N, we define xm (t) = sup xn (t), (B.5) xm (t) = inf xn (t). n≥m n≥m Let yn,m and yn,m be defined by the respective equations (B.6a) (B.6b) n,m y m (t) = x (t) + yn,m (t) = xm (t) +  t 0 f n yn,m (t − s), t − s dB(s),      t f n yn,m (t − s), t − s dB(s).  t f ym (t − s), t − s dB(s), 0 Since f n (y, t) is monotonically increasing in y and is monotonically decreasing in n, an application of the method of successive approximations to (B.6a) and (B.6b) with the initial approximations being zero functions shows that the yn,m (t) and yn,m (t) monotonically decrease in n for every t and m. Besides, the sequences {supt∈[0,T ] |yn,m (t)|, n ∈ N} and {supt∈[0,T ] |yn,m (t)|, n ∈ N} are bounded, as it follows by Lemma B.1. Thus, for t ∈ [0, T ], there exist finite limits ym (t) = limn→∞ yn,m (t) and ym (t) = limn→∞ yn,m (t). On letting n → ∞ in (B.6a) and (B.6b) and applying Lebesgue’s bounded convergence theorem, we obtain that (B.7a) (B.7b) m m y (t) = x (t) + ym (t) = xm (t) + 0  t 0     f ym (t − s), t − s dB(s). 176 A. A. PUHALSKII AND J. E. REED As m → ∞, the xm (t) monotonically decrease to x(t) and the xm (t) monotonically increase to x(t). An application of the method of successive approximations to (B.7a) and (B.7b) shows that the ym (t) monotonically decrease and the ym (t) monotonically increase. The limits satisfy (B.1), so ym (t) → y(t) and ym (t) → y(t), as m → ∞. Also, since xm (t) ≤ xn (t) ≤ xm (t) for n ≥ m by (B.5), we have by (B.1), (B.7a) and (B.7b) that yn,m (t) ≤ yn (t) ≤ yn,m (t) for n ≥ m. It follows that yn (t) → y(t) as n → ∞.  We now study regularity properties of solutions to (B.1) for equations arising in Theorem 2.2. Given z = (z(t), t ∈ [0, T ]) ∈ L∞ ([0, T ], R), consider the condition lim sup (B.8)  t ε→0 t∈[0,T ] 0 1{0<|z(t−s)|<ε} dB(s) = 0. Note that (B.8) implies the condition of part 1 of Lemma B.2 for f n (y, t) = (y + nz(t))+ − nz(t)+ and f (y, t) = y1{z(t)>0} + y + 1{z(t)=0} . L EMMA B.3. If z admits limits on the right and limits on the left and B(t) is continuous on [0, T ] with B(0) = 0, then condition (B.8) holds. Assume that f (y, t) = y1{z(t)>0} + y + 1{z(t)=0} in (B.1). If both z and x are right-continuous with left-hand limits and condition (B.8) holds, then y is rightcontinuous with left-hand limits. If z is right-continuous with left-hand limits, x is continuous, and B(t) is continuous on [0, T ] with B(0) = 0, then y is continuous. P ROOF. If z admits limits on the right and limits on the left, then the function 1{0<|z(s)|<ε} has at most countably many points of discontinuity, which implies  that the function 0t 1{0<|z(t−s)|<ε} dB(s) is continuous in t when B(t) is a continuous function with B(0) = 0. By Dini’s theorem, the monotonic convergence of t 0 1{0<|z(t−s)|<ε} dB(s) to zero as ε → 0 is uniform in t ∈ [0, T ], as required. Suppose that x and z are right-continuous with left-hand limits and condition (B.8) holds. Let ŷn = (ŷn (t), t ∈ [0, T ]) solve the equation ŷn (t) = x(t) +  t   n + ŷ (t − s) + nz(t − s) − nz(t − s)+ dB(s). 0 By Lemma B.1, the functions ŷn are right-continuous with left-hand limits. By part 1 of Lemma B.2, the ŷn converge to y for the topology of compact convergence. Therefore, the latter function is also right-continuous with left-hand limits. The argument for the case where z is right-continuous with left-hand limits, x is continuous, and B(t) is continuous on [0, T ] with B(0) = 0 is similar.  ON MANY-SERVER QUEUES IN HEAVY TRAFFIC 177 APPENDIX C: MARTINGALE LEMMAS This section contains results on certain processes being martingales which are instrumental in the proofs above. Let A = (A(t), t ∈ R+ ) be an integer–valued nonnegative process with nondecreasing trajectories from D(R+ , R). We denote τi = inf{t : A(t) ≥ i}, i ∈ N, so the jumps of A occur at the τi . Obviously, A(t) = ∞ i=1 1{τi ≤t} . Let ξi , i ∈ N, be nonnegative random variables. The next lemma is an extension of Lemma 5.2 in Krichagina and Puhalskii [15] (see also Lemma 8 in Reed [22]). L EMMA C.1. Let A = (At , t ∈ R+ ) be a filtration such that the random variables τj ∧(A(t)+1) and ξj ∧A(t) , where j ∈ N, are At -measurable. Let Ãi represent the complete σ -algebras generated by the events  ∩ {τi > t}, where t ∈ R+ and  ∈ At . Suppose that (βi (x, y), x ∈ R+ , y ∈ R+ ), i ∈ N, are real-valued Borel functions such that Eβi (τi , ξi )2 < ∞. The following assertions hold. 1. The τi are A-predictable stopping times. 2. Suppose, in addition, that there exists a nondecreasing sequence of σ -algebras Âi , i ∈ N, such that Ãi ⊂ Âi and such that the random variables τj , where j = 1, 2, . . . , i, and ξj , where j = 1, 2, . . . , i − 1, are Âi -measurable. Let, for k ∈ N, Rk (t) = A(t)∧k  βi (τi , ξi ), Rk (t) = A(t)∧k  E(βi (τi , ξi )2 |Âi ). i=1 i=1 If E(βi (τi , ξi )|Âi ) = 0, then the processes Rk = (Rk (t), t ∈ R+ ) and (Rk (t)2 − Rk (t), t ∈ R+ ) are A-square-integrable martingales. 3. The inclusions Ãi ⊂ Âi hold provided the At are the complete σ -algebras generated by the random variables τj ∧(A(t)+1) and ξj ∧A(t) , where j ∈ N, and the Âi are the complete σ -algebras generated by the random variables τj , where j = 1, 2, . . . , i, and ξj , where j = 1, 2, . . . , i − 1. 4. If the ξi , i ∈ N, are independent, and if, for each i ∈ N, ξi is independent of the τj for j ≤ i, and Eβi (x, ξi ) = 0 for every x, then, for A and Âi defined as in part 3 and for Rk and Rk  defined as in part 2, the processes Rk are A-squareintegrable martingales with the predictable quadratic variation processes Rk . The τi are A-stopping times because, for t ∈ R+ and i ∈ N, {τi ≤ t} = {τi∧(A(t)+1) ≤ t} which follows on noting that τA(t)+1 > t. The latter set belongs to At . The proof of the A-predictability of the τi follows the argument of Krichagina and Puhalskii [15]. The time τ1 is A-predictable since the times (1 − 1/ l)τ1 , l = P ROOF. 178 A. A. PUHALSKII AND J. E. REED 1, 2, . . . , are stopping times predicting τ1 (note that τ1 is A0 -measurable). For i > 1, introduce σi,l = τi − (τi − τν(i) )/ l, l ∈ N, where τν(i) = max{τp : τp < τi } and τν(i) = 0 if no such p exists. Obviously, σi,l ↑ τi as l → ∞ and σi,l < τi on the set {τi > 0}. We show that the σi,l are A-stopping time. For t ∈ R+ , in view of the fact that {τi ≤ t} ⊂ {σi,l ≤ t} ⊂ {τν(i) ≤ t},      {σi,l ≤ t} = {τi ≤ t} ∪ {σi,l ≤ t} ∩ τν(i) ≤ t ∩ τi∧(A(t)+1) > t . The first set on the right belongs to At , as has been proved. Let τ̂ν(i) = max{τp∧(A(t)+1) : τp∧(A(t)+1) < τi∧(A(t)+1) } and σ̂i,l = τi∧(A(t)+1) − (τi∧(A(t)+1) − τ̂ν(i) )/ l. The random variables τ̂ν(i) and σ̂i,l are At -measurable. If τν(i) ≤ t and τi∧(A(t)+1) > t, then τν(i) = τ̂ν(i) . On the other hand, if τ̂ν(i) ≤ t and τi∧(A(t)+1) > t, then τν(i) = τ̂ν(i) . We conclude that    {σi,l ≤ t} ∩ τν(i) ≤ t ∩ τi∧(A(t)+1) > t      = {σ̂i,l ≤ t} ∩ τ̂ν(i) ≤ t ∩ τi∧(A(t)+1) > t . The set on the right is in At . Thus, the σi,l , l ∈ N, are A-stopping times which predict τi . Part 1 has been proved. We prove part 2. As {A(t) ≥ i} = {τi∧(A(t)+1) ≤ t}, the random variables At are At -measurable, so the Rk (t) are At -measurable. Since supt∈R+ ERk (t)2 < ∞, to prove that Rk is an A-square-integrable martingale, it is enough to prove that (C.1) E A(s∧σ )∧k i=1 βi (τi , ξi ) = 0 for any A-stopping time σ (see, e.g., Jacod and Shiryaev [12], I.1.44). We have E A(s∧σ )∧k i=1 βi (τi , ξi ) = k  i=1 E1{τi ≤s∧σ } βi (τi , ξi ). Since {τi > s ∧ σ } = ({τi > u} ∩ {u ≥ s ∧ σ }), where the union is over all positive rational u, we have, by the fact that s ∧ σ is an A-stopping time and hence {u ≥ s ∧ σ } ∈ Au , and the definition of Ãi , that {τi > s ∧ σ } ∈ Ãi . By the inclusion Ãi ⊂ Âi , E1{τi ≤s∧σ } βi (τi , ξi ) = E(1{τi ≤s∧σ } E(βi (τi , ξi )|Âi )). The latter conditional expectation equals zero by hypotheses. Equality (C.1) has been proved. In order to prove that (Rk (t)2 − Rk (t), t ∈ R+ ) is an A-square-integrable martingale, we show that ERk (s ∧ σ )2 = ERk (s ∧ σ ) for any A-stopping time σ . The definition implies that 2 Rk (s ∧ σ ) = k  i=1 1{τi ≤s∧σ } βi (τi , ξi )2 +2 k  k  i=1 j =i+1 1{τi ≤s∧σ } 1{τj ≤s∧σ } βi (τi , ξi )βj (τj , ξj ). 179 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC Next, for i < j , in analogy with an earlier argument, using the facts that τi and ξi are Âj -measurable and that τj is Âj -measurable, E1{τi ≤s∧σ } 1{τj ≤s∧σ } βi (τi , ξi )βj (τj , ξj )   = E 1{τi ≤s∧σ } 1{τj ≤s∧σ } βi (τi , ξi )E(βj (τj , ξj )|Âj ) = 0. Therefore, 2 ERk (s ∧ σ ) = E k  i=1 1{τi ≤s∧σ } E(βi (τi , ξi )2 |Âi ) = ERk (s ∧ σ ). Part 2 has been proved. By Brémaud [4], Appendix A3, Theorem 25, the flow At , t ∈ R+ , is rightcontinuous, so we prove part 3 by showing that each generator of Ãi is in Âi . Since {τi > t} = {A(t) + 1 ≤ i}, we have that, if j > i, then {τj ∧(A(t)+1) ≤ u} ∩ {τi > t} = {τi∧(A(t)+1) ≤ u} ∩ {τi > t} which event is seen to be in Âi , and {τj ∧(A(t)+1) ≤ u} ∩ {τi > t} ∈ Âi when j ≤ i. Similarly, if j ≥ i, then {ξj ∧A(t) ≤ u} ∩ {τi > t} = {ξ(i−1)∧A(t) ≤ u} ∩ {τi > t} ∈ Âi , and if j < i, then {ξj ∧A(t) ≤ u} ∩ {τi > t} ∈ Âi . Part 3 has been proved. We prove part 4. Under the hypotheses, E(βi (τi , ξi )|Âi ) = E(βi (x, ξi )| Âi )|x=τi = 0. By parts 2 and 3, it remains to prove that Rk  is A-predictable. As Rk (t) = ki=1 1{τi ≤t} Eβi (x, ξi )2 |x=τi , the τi are A-predictable, and the Eβi (x, ξi )2 |x=τi are Aτi − -measurable, it follows that Rk  is A-predictable, Dellacherie [5], Subsection 5 of Section 1 of Chapter 4.  For k ∈ N, introduce the two-parameter processes Lk = (Lk (t, x), t ∈ R+ , x ∈ R+ ) with (C.2) Lk (t, x) = A(t)∧k   i=1 1{0<ξi ≤x} −  ξi ∧x 0 1{u>0} dFi (u) , 1 − Fi (u−) where Fi denotes the distribution function of ξi . Define also (C.3) Lk (t, x) = A(t)∧k   ξi ∧x i=1 0 1{u>0} 1 − Fi (u) dFi (u). (1 − Fi (u−))2 Let, for t ∈ R+ and x ∈ R+ , complete σ -algebras F̂t,x be generated by the random variables τi∧(A(t)+1) , ξi∧A(t) , and 1{τi ≤s} , where i ∈ N, s ∈ R+ , and s ≤ t + x, and by 1{τi ≤s} 1{ξi ≤y} , where i ∈ N, s ∈ R+ , y ∈ R+ , and s + y ≤ t + x. We de note Ft,x = ε>0,δ>0 F̂t+ε,x+δ (see Figure 1) and note that both Lk (t, x) and Lk (t, x) are Ft,x -measurable. (As a matter of fact, these random variables are Ft,0 -measurable.) In the next lemma, we define conditional probabilities as being equal to zero when the conditioning events are of probability zero. 180 A. A. PUHALSKII AND J. E. REED F IG . 1. The σ -algebra Ft,x . L EMMA C.2. Suppose that, given i ∈ N, s ∈ R+ , and x ∈ R+ , the random variable ξi is independent of the random variables ξj for j = i, of the τj for j ≤ i, and of the τj ∧ (s + x) for j > i, conditioned on the event {τi ≥ s, ξi > x}. Suppose also that each ξi , for i ∈ N, is independent of the τj for j ≤ i. Then for s ≤ t and x ≤ y, E(Lk ((s, x), (t, y))|Fs,x ) = 0 and E((Lk ((s, x), (t, y)))2 |Fs,x ) = Lk ((s, x), (t, y)). P ROOF. Assuming x and y are held fixed, we apply Lemma C.1 with At = Ft,x and Âi being the complete σ -algebra generated by ξj for j < i, by τj for j ≤ i, by 1{τj ≤s} 1{τi >t} for j > i, t ∈ R+ , and s ≤ t + x, and by 1{τj ≤s} 1{τi >t} 1{0<ξj ≤u} for j ≥ i, s ∈ R+ , t ∈ R+ , u ∈ R+ , and s + u ≤ t + x. Let us check that Ãi ⊂ Âi . Pick j ∈ N. The inclusions {τj ∧(A(t)+1) ≤ u} ∩ {τi > t} ∈ Âi and {ξj ∧A(t) ≤ u} ∩ {τi > t} ∈ Âi follow by part 3 of Lemma C.1. For t ∈ R+ and s ≤ t + x, {τj ≤ s} ∩ {τi > t} ∈ Âi by definition. Similarly, {τj ≤ s} ∩ {0 < ξj ≤ u} ∩ {τi > t} ∈ Âi , where i ∈ N, s ∈ R+ , u ∈ R+ , and s + u ≤ t + x. Thus, if  ∈ At , then  ∩ {τi > t + ε} ∈ Âi for all ε > 0, hence,  ∩ {τi > t} ∈ Âi . Introduce  v∧y dFi (u) (C.4) , βi (v) = 1{x<v≤y} − 1{u>0} 1 − Fi (u−) v∧x so that (C.5) Lk (t, y) − Lk (t, x) = A(t)∧k  i=1 βi (ξi ). 181 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC We show that E(βi (ξi )|Âi ) = 0. (C.6) We have for natural numbers i < k1 < k2 < · · · < kp , i < n1 < n2 < · · · < ′ , nm , nonnegative real numbers ũ, t˜, s̃, t1 , t2 , . . . , tp , s1 , s2 , . . . , sp , t1′ , t2′ , . . . , tm ′ ′ ′ ′ s1 , s2 , . . . , sm , and u1 , . . . , um such that sj ≤ tj + x, s̃ + ũ ≤ t˜ + x, and sj + uj ≤ (1) (1) tj′ + x, and for Borel bounded functions f1 , . . . , fl , g1 , . . . , gi , h1 , . . . , hp , (2) (2) h1 , . . . , hm , and f , on noting that by (C.4) βi (v) = 1{v>x} βi (v) and that the condition s̃ + ũ ≤ t˜ + x implies the identity 1{ξi >x} 1{t˜<τi ≤s̃} 1{0<ξi ≤ũ} = 0,  i−1  E fj (ξj ) j =1 × i  m  j =1 gj (τj ) j =1 p  (1)  hj j =1 1{τkj ≤sj } 1{τi >tj } hj(2) 1{τn  ′ 1 ′ 1{0<ξ ≤u } nj j j ≤sj } {τi >tj }      × f 1{t˜<τi ≤s̃} 1{0<ξi ≤ũ} βi (ξi )  i−1  =E fj (ξj ) (C.7) gj (τj ) m   (2)  hj j =1 p   (1)  hj j =1 j =1 j =1 × i  1{τkj ≤sj } 1{τi >tj } + h(1) j (0)1{τi ≤tj } 1{τn ′ 1{0<ξ ≤u } nj j j ≤sj }   1{τi >tj′ } + h(2) j (0)1{τi ≤tj′ }    × f (0)1{ξi >x} βi (ξi ) = f (0)  J ⊂{1,2,...,p}, J ′ ⊂{1,2,...,m}  i−1  E fj (ξj ) j =1 × × i  gj (τj ) hj (0)1{τi ≤tj } j ∈{1,2,...,p}\J j ∈J ′ ×  1{τkj ≤sj } 1{τi >tj } (1)  (2)  hj hj j ∈J j =1   (1)  1{τn  ′ 1{0<ξ ≤u } nj j j ≤sj } j ∈{1,2,...,m}\J ′  1{τi >tj′ }  h(2) j (0)1{τi ≤tj′ } 1{ξi >x} βi (ξi ) . We show that a generic term in the sum on the rightmost side of (C.7) equals zero. For given J and J ′ , let tˆ = maxj ∈J tj ∨ maxj ∈J ′ tj′ and t˘ = minj ∈{1,2,...,p}\J tj ∧ 182 A. A. PUHALSKII AND J. E. REED minj ∈{1,2,...,m}\J ′ tj′ . Since sj ≤ tˆ + x and sj′ ≤ tˆ + x, we have  i−1  E fj (ξj ) gj (τj )  j ∈J ′ (2)  1{τn hj  (1)  hj j ∈J j =1 j =1 × i  ′ j ≤sj }  1{τkj ≤sj } 1{τi >tj }  h(1) j (0)  ×E j ∈{1,2,...,m}\J ′  hj(2) (0)1{τi ≤tj′ } 1{ξi >x} βi (ξi ) h(2) j (0) j ∈{1,2,...,m}\J ′ fj (ξj ) j =1 i  gj (τj ) hj j ∈J ′  (1)  hj j ∈J j =1 (2)   ×   j ∈{1,2,...,p}\J  i−1  j ∈{1,2,...,p}\J hj (0)1{τi ≤tj } 1{0<ξnj ≤uj } 1{τi >tj′ } × = (1)  1{τk j ∧(tˆ+x)≤sj } 1{τn ′ 1{0<ξn ≤uj } ˆ j j ∧(t +x)≤sj }      ×1{tˆ<τi ≤t˘} βi (ξi )τi ≥ tˆ, ξi > x P(τi ≥ tˆ, ξi > x). The independence hypotheses imply that  i−1  E j =1 × i  fj (ξj ) gj (τj ) j ∈J ′ h(2) j 1{τn   i−1  =E hj j ∈J j =1   (1)  fj (ξj )  j ∈J ′  ∧(tˆ+x)≤sj } j 1{0<ξnj ≤uj }     × 1{tˆ<τi ≤t˘} βi (ξi )τi ≥ tˆ, ξi > x j =1 × ′ ˆ j ∧(t +x)≤sj } 1{τk i  gj (τj )  (1)  hj j ∈J j =1 1{τk  ∧(tˆ+x)≤sj } j      h(2) j 1{τnj ∧(tˆ+x)≤sj′ } 1{0<ξnj ≤uj } 1{tˆ<τi ≤t˘} τi  ≥ tˆ, ξi > x  × E βi (ξi )|ξi > x . By (C.4), E(βi (ξi )|ξi > x) = 0. We conclude that the leftmost side of (C.7) equals zero, which establishes (C.6). ON MANY-SERVER QUEUES IN HEAVY TRAFFIC F IG . 2. 183 The σ -algebra Mt . We now show that (C.8) E(βi (ξi )2 |Âi ) =  ξi ∧y ξi ∧x 1{u>0} 1 − Fi (u) dFi (u). (1 − Fi (u−))2 Denoting 2 β i (ξi ) = βi (ξi ) −  ξi ∧y ξi ∧x 1{u>0} 1 − Fi (u) dFi (u) (1 − Fi (u−))2 and replicating the argument that established (C.6), we can see that (C.8) follows provided E(β i (ξi )|ξi > x) = 0, which is a consequence of (C.4). The required properties now follow by part 2 of Lemma C.1.  Let M̂t denote the complete σ -algebra generated by the events {τi ≤ s} ∩ {ξi ≤ x} and {τi ≤ s} where i ∈ N, s ∈ R+ , x ∈ R+ , and s + x ≤ t, and let Mt =  ε>0 M̂t+ε (see Figure 2). The flow M = (Mt , t ∈ R+ ) is a filtration. Note that Lk (t, x) and Lk (t, x) are Mt+x -measurable and Ft,x ⊃ Mt+x . Define, for k ∈ N and t ∈ R+ , (C.9) Mk (t) =  1{s+x≤t} dLk (s, x) Mk (t) =  1{s+x≤t} dLk (s, x). R2+ and (C.10) R2+ L EMMA C.3. Under the independence hypotheses of Lemma C.2, the processes Mk = (Mk (t), t ∈ R+ ) are M-square integrable martingales with predictable quadratic variation processes Mk  = (Mk (t), t ∈ R+ ). 184 A. A. PUHALSKII AND J. E. REED P ROOF. Since Mk and Mk  are right-continuous, it suffices to prove that Mk is a square integrable martingale relative to the flow M̂ = (M̂t , t ∈ R+ ) with predictable quadratic variation process Mk . Fix s < t. We first show that    E Lk (0, t) − Lk (0, s) |M̂s = 0, (C.11) 2     (C.12) E Lk (0, t) − Lk (0, s) |M̂s = E Lk (0, t) − Lk (0, s)|M̂s . The argument is similar to the one used in the proof of Lemma C.2. Denote  ξi ∧t dFi (u) ζi = 1{s<ξi ≤t} − (C.13) , 1{u>0} 1 − Fi (u−) ξi ∧s ζ i = ζi2 − (C.14) j  ξi ∧t ξi ∧s 1{u>0} 1 − Fi (u) dFi (u). (1 − Fi (u−))2 Let M̂s denote the σ -algebra generated by the ξk for k ≤ j and by M̂s when j j ∈ N and let M̂0s = M̂s . Obviously, ζj is M̂s -measurable. We prove that (C.15) E(ζi |M̂si−1 ) = 0, (C.16) E(ζ i |M̂si−1 ) = 0. We have for distinct natural numbers k1 , k2 , . . . , kp and n1 , n2 , . . . , nm , none of which equals i, nonnegative real numbers t˜, s̃, x̃, s1 , s2 , . . . , sp , x1 , x2 , . . . , xp , and t1 , t2 , . . . , tm , such that s̃ + x̃ ≤ s, t˜ ≤ s, sj + xj ≤ s, and tj ≤ s, and for Borel (1) (2) (2) bounded functions f1 , f2 , g1 , . . . , gk , h(1) 1 , . . . , hp , and h1 , . . . , hm ,     E f1 1{τi ≤s̃} 1{ξi ≤x̃} f2 1{τi ≤t˜} × i−1  j =1 p  (1)  hj gj (ξj ) j =1   1{τkj ≤sj } 1{ξkj ≤xj }  = f1 (0)E f2 1{τi ≤t˜}   i−1 gj (ξj )  = f1 (0)E f2 1{τi ≤t˜}   i−1 hj j =1 p  (1)  hj j =1 j =1 ×  m  (2)  gj (ξj ) j =1 hj p  (1)  hj j =1 × × E(ζi |ξi > s)P(ξi > s). hj    1{τnj ∧s≤tj } 1{ξi >s} ζi 1{τkj ∧s≤sj } 1{ξkj ≤xj } m  (2)  j =1 1{τnj ≤tj } ζi  1{τkj ∧s≤sj } 1{ξkj ≤xj } m  (2)  j =1    1{τnj ∧s≤tj } ξi > s  ON MANY-SERVER QUEUES IN HEAVY TRAFFIC 185 Since E(ζi |ξi > s) = 0 by (C.13), (C.15) follows. Equality (C.16) is obtained analogously. By (C.2) and (C.13), Lk (0, t) − Lk (0, s) = ki=1 1{τi =0} ζi . Since {τi = 0} ∈ M̂s , by (C.15)    E Lk (0, t) − Lk (0, s) |M̂s = k  i=1 1{τi =0} E(ζi |M̂s ) = 0. By (C.15), if j < i, then E(ζj ζi |M̂s ) = E(ζj E(ζi |M̂si−1 )) = 0. We thus obtain on recalling (C.3), (C.14) and (C.16),  2 E Lk (0, t) − Lk (0, s) |M̂s = k  i=1  1{τi =0} E(ζi2 |M̂s ) + 2   j <i 1{τj =0} 1{τi =0} E(ζj ζi |M̂s )  = E Lk (0, t) − Lk (0, s)|M̂s . This completes the proof of (C.11) and (C.12). By (C.2) and (C.9), (C.17) Mk (t) = A(t)∧k   i=1 1{0<ξi ≤t−τi } −  ξi ∧(t−τi ) 0 1{u>0} dFi (u) , 1 − Fi (u−) which implies that Mk (t) is M̂t -measurable. We prove that Mk is a martingale. l ) → 0 as l → ∞ Let 0 = s0l < s1l < s2l < · · · < sll = t be such that maxi (sil − si−1 l for some m. Define (see Figure 3) and s = sm (C.18a) (C.18b) Jl,s (u, x) = m  1{u∈(s l 1{x≤s−s l + 1{u=0} 1{x≤s} , Jl,t (u, x) = l  1{u∈(s l 1{x≤t−s l + 1{u=0} 1{x≤t} , i=1 i=1 (C.18c) Mk,l (s) =  (C.18d) Mk,l (t) =  R2+ Note that (C.19a) (C.19b) l i−1 ,si ]} R2+ l i−1 ,si ]} i−1 } i−1 } Jl,s (u, x) dLk (u, x), Jl,t (u, x) dLk (u, x). Mk,l (s) = m  l l ) + Lk (0, s), , 0), (sil , s − si−1 Lk (si−1 Mk,l (t) = l  l l , 0), (sil , t − si−1 ) + Lk (0, t). Lk (si−1 i=1 i=1     186 A. A. PUHALSKII AND J. E. REED F IG . 3. Approximating martingales. Since Jl,s (u, x) → 1{u+x≤s} and Jl,t (u, x) → 1{u+x≤t} as l → ∞, it follows by (C.9), (C.18c) and (C.18d) that Mk,l (s) → Mk (s) and Mk,l (t) → Mk (t) as l → ∞. By Lemma C.2 and (C.19a), 2 EMk,l (s) = m  i=1 l l ) ≤ ELk (s, s) < ∞. , 0), (sil , s − si−1 ELk  (si−1   Similarly, EMk,l (t) ≤ ELk (t, t) < ∞. Hence, lim E(Mk,l (s)|M̂s ) = Mk (s), (C.20a) l→∞ lim E(Mk,l (t)|M̂s ) = E(Mk (t)|M̂s ). (C.20b) Lemma l→∞ C.2 l , 0), (s l , s E(Lk ((si−1 i l E(Lk ((si−1 , 0), (sil , t ( l , 0), (s l , t − s l ))|F E(Lk ((si−1 i i sl l ) = i−1 ,s−si−1 l − si ))|Fs l ,s−s l ) for i = 1, 2, . . . , m and that i−1 i−1 − sil ))|Fs l ,0 ) = 0 for i = m + 1, . . . , l. Since M̂s ⊂ i−1 implies that   l ,s−s l ) ∩ ( i=1,2,...,m Fsi−1 i−1 l ,0 ), i=m+1,...,l Fsi−1 by (C.11), (C.19a) and (C.19b), E(Mk,l (t)|M̂s ) = E(Mk,l (s)|M̂s ). The martingale property of Mk follows by (C.20a) and (C.20b). We now compute the predictable quadratic variation process. By (C.17), recalling (C.19b), Mk (t) − Mk,l (t) (C.21) =− A(t)∧k l   i=1 j =1 1{τi ∈(s l l j −1 ,sj ]}  1{ξi ∈(t−τi ,t−s l − j −1 ]}  ξi ∧(t−s l ) j −1 ξi ∧(t−τi ) 1{u>0} dFi (u) . 1 − Fi (u−) 187 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC l j =1 1{x∈(sjl −1 ,sjl ]} We apply part 4 of Lemma C.1 with βi (x, y) = − (1{y∈(t−x,t−s l ]} − j −1 lemma,   y∧(t−sjl −1 ) y∧(t−x) E Mk (t) − Mk,l (t) =E A(t)∧k l   k  = k   E l  j =1 i=1 i=1 2 i=1 j =1 ≤ 1{u>0} dFi (u)/(1 − Fi (u−))). According to the 1{τi ∈(s l l j −1 ,sj ]} 1{τi ∈(s l l j −1 ,sj ]}  t−s l j −1 t−τi 1{u>0} 1 − Fi (u) dFi (u) 1 − Fi (u−) Fi (t − sjl −1 ) − Fi (t − τi )   E Fi t − sjl (τi )−1 − Fi (t − τi ) ,    where j (τi ) is defined by the requirement that τi ∈ (sjl (τi )−1 , sjl (τi ) ]. Since t − sjl (τi )−1 converges from the right to t − τi as l → ∞, F (t − sjl (τi )−1 ) converges to F (t − τi ). By bounded convergence,  2  2 lim E Mk (t) − Mk,l (t) = 0. (C.22) l→∞ Similarly, lim E Mk (s) − Mk,l (s) = 0. (C.23) l→∞ By (C.19a) and (C.19b), Mk,l (t) − Mk,l (s) = (C.24) By Lemma C.2, for i < j , l  i=1 l l l ) ∧ s), (sil , t − si−1 , s − si−1 Lk (si−1     + Lk (0, t) − Lk (0, s) . l l l ) ∧ s), (sil , t − si−1 , s − si−1 E Lk (si−1    × Lk (sjl −1 , s − sjl −1 ∧ s), (sjl , t − sjl −1 ) |Fs l   l j −1 ,s−sj −1 ∧s l l l = Lk (si−1 , s − si−1 ∧ s), (sil , t − si−1 )    × E Lk (sjl −1 , s − sjl −1 ∧ s), (sjl , t − sjl −1 ) |Fs l   and  2 l l l ) ∧ s), (sil , t − si−1 , s − si−1 E Lk (si−1   l j −1 ,s−sj −1 ∧s |Fs l l j −1 ,s−sj −1 ∧s l l l ), ∧ s), (sil , t − si−1 , s − si−1 = Lk  (si−1     =0 188 A. A. PUHALSKII AND J. E. REED so, due to the fact that M̂s ⊂ Fs l l j −1 ,s−sj −1 ∧s , l l l ) ∧ s), (sil , t − si−1 , s − si−1 E Lk (si−1  (C.25) and   ×Lk (sjl −1 , s − sjl −1 ∧ s), (sjl , t − sjl −1 ) |M̂s = 0  2 l l l ) ∧ s), (sil , t − si−1 , s − si−1 E Lk (si−1  (C.26)    |M̂s  l l l , s − si−1 ∧ s), (sil , t − si−1 ) |M̂s . = E Lk  (si−1     Since Lk (0, t) and Lk (0, s) are F0,0 -measurable, l l l ) Lk (0, t) − Lk (0, s) |Fs l ∧ s), (sil , t − si−1 , s − si−1 E Lk (si−1      = Lk (0, t) − Lk (0, s)  l i−1 ,s−si−1 ∧s l l l ) |Fs l ∧ s), (sil , t − si−1 , s − si−1 × E Lk (si−1 so    l i−1 ,s−si−1 ∧s  = 0, l l l , s − si−1 ∧ s), (sil , t − si−1 ) E Lk (si−1  (C.27)      × Lk (0, t) − Lk (0, s) |M̂s = 0. Putting together (C.12), (C.18a), (C.18b), (C.24)–(C.27) yields  2 E Mk,l (t) − Mk,l (s) |M̂s  l  =E (C.28) i=1  =E  l l l ) ∧ s), (sil , t − si−1 , s − si−1 Lk  (si−1    + Lk (0, t) − Lk (0, s) M̂s  R2+      Jl,t (u, x) − Jl,s (u, x) dLk (u, x)M̂s . Since Jl,s (u, x) → 1{u+x≤s} and Jl,t (u, x) → 1{u+x≤t} as l → ∞, (C.29) lim  Jl,s (u, x) dLk (u, x) =  1{u+x≤s} dLk (u, x), lim  Jl,t (u, x) dLk (u, x) =  1{u+x≤t} dLk (u, x), l→∞ R2+ l→∞ R2+ so by bounded convergence, a.s.,  lim E l→∞ (C.30) =E R2+   R2+ R2+    Jl,t (u, x) − Jl,s (u, x) dLk (u, x)M̂s R2+   1{s<u+x≤t} dLk (u, x)M̂s .  189 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC Putting together (C.22), (C.23), (C.28) and (C.30) and recalling the definition of Mk  in (C.10) obtains the equality 2     E Mk (t) − Mk (s) |M̂s = E Mk (t) − Mk (s) |M̂s  a.s. It remains to verify that the process Mk  is M̂-predictable. By (C.3), Lk (t, x) = k  L̂i (t, x), i=1 where (C.31) L̂i (t, x) = 1{τi ≤t}  ξi ∧x 0 1{u>0} Denote M̂i (t) = (C.32)  R2+ 1 − Fi (u) dFi (u). (1 − Fi (u−))2 1{u+x≤t} d L̂i (u, x). Since Mk  = ki=1 M̂i , it suffices to prove that each process M̂i = (M̂i (t), t ∈ R+ ) is M̂-predictable. It is M̂-adapted which follows by the representation M̂i (t) = 1{τi ≤t}  ξi ∧(t−τi ) 1{u>0} 1 − Fi (u) dFi (u). (1 − Fi (u−))2  ξi ∧(t−τi ) 1{x>0} 1 − Fi (x) dFi (x). (1 − Fi (x−))2 0 By (C.31) and (C.32), (C.33) M̂i (t) = 1{τi ≤t} 0 Consider the decomposition M̂i = M̂ic + M̂id , where M̂ic is a continuous-path adapted process and M̂id is a pure-jump adapted process. By being continuous and adapted, M̂ic is M̂-predictable. The process M̂id is of the form (C.34) M̂id (t) =  u>0 1{τi +u≤t} 1{u≤ξi } 1 − Fi (u) Fi (u), (1 − Fi (u−))2 where the sum is taken over all positive times of jumps of Fi . Note that τi is an M̂-stopping time by the definition of M̂t . Since u > 0 in (C.34), τi + u is an M̂predictable stopping time. We show that 1{u≤ξi } is M̂(τi +u)− -measurable. Note that {ξi < u} = t∈Q+ ({τi + ξi < t} ∩ {τi + u > t}). By the representation {τi + ξi < t} = s,x∈Q+ :s+x<t ({τi ≤ s} ∩ {ξi ≤ x}), we have that {τi + ξi < t} ∈ M̂t , hence, {ξi < u} ∈ M̂(τi +u)− . By Dellacherie [5], IV.1.5, each summand on the right of (C.34) is an M̂-predictable process. It follows that M̂id is too.  190 A. A. PUHALSKII AND J. E. REED APPENDIX D: HOFFMANN–JØRGENSEN’S CONVERGENCE IN DISTRIBUTION In this section, we state the properties of Hoffmann–Jørgensen’s convergence in distribution envoked in the proofs of Section 4. We recall the definition stated in the introductory part of the paper. Let (, F , P) be a probability space. Given a real-valued function ξ on , the outer expectation E∗ ξ of ξ is defined as the infimum of Eζ over all random variables ζ on (, F , P) such that ζ ≥ ξ a.s. and Eζ exists. Let S be a metric space made into a measurable space by endowing it with the Borel σ -algebra B (S). Given a sequence Xn of maps from  to S and a measurable map X from (, F ) to (S, B (S)), we say that the Xn converge to X in distribution if (D.1) lim E∗ f (Xn ) = Ef (X) n→∞ for all bounded continuous functions f on S. We also define E∗ f (Xn ) = −E∗ (−f (Xn )). The next result is the continuous mapping principle, which is Theorem 1.3.6 in van der Vaart and Wellner [23]. T HEOREM D.1. Let S ′ be a metric space and function f : S → S ′ be Borel and continuous at all points of a set S0 ⊂ S. If the Xn converge in distribution to X in S and X assumes values in S0 , then the f (Xn ) converge in distribution to f (X) in S ′ . Let τ denote the topology on S. C OROLLARY D.1. Suppose that τ ′ is a metric topology on S which is coarser than τ and that open balls in (S, τ ) are Borel sets in (S, τ ′ ). If the Xn converge in distribution to X for topology τ ′ , a subset S0 of S is such that convergence to the elements of S0 in τ ′ implies convergence in τ , and X assumes values in S0 , then the Xn converge in distribution to X for topology τ . The next theorem relaxes the requirement of the separability of the range of X in the extended continuous mapping principle of van der Vaart and Wellner [23], cf., Theorem 1.11.1 in [23]. The proof is modelled on that of Lemma 3.1.13 in Puhalskii [20]. T HEOREM D.2. Let fn and f be maps from S to a metric space S ′ . Suppose that f is measurable and that the fn (xn ) converge to f (x) whenever the xn converge to x and x ∈ S0 , where S0 is a Borel subset of S. If the Xn converge in distribution to X and X assumes values in S0 , then the fn (Xn ) converge in distribution to f (X). ON MANY-SERVER QUEUES IN HEAVY TRAFFIC 191 P ROOF. Note that f (X) is a random element of S ′ , so convergence in distribution of the f (Xn ) to f (X) is well defined. Let g be a continuous bounded function from S ′ to R. Let Bδ (x) denote the closed ball of radius δ about x ∈ S and introduce hk (x) = inf sup sup g(fn (y)). δ>0 y∈Bδ (x) n≥k The hk are bounded upper semicontinuous functions on S and hk (x) ≥ g(fn (x)) for n ≥ k, so, for arbitrary k, envoking the Portmanteau theorem (Theorem 1.3.4 in van der Vaart and Wellner [23]) lim sup E∗ g(f (Xn )) ≤ lim sup E∗ hk (Xn ) ≤ Ehk (X). n→∞ n→∞ The convergence hypotheses on the fn and the continuity of g imply that the hk (x) monotonically decrease to g(f (x)) as k → ∞ for all x ∈ S0 . By bounded convergence, limk Ehk (X) = Eg(f (X)), and we conclude that lim sup E∗ g(f (Xn )) ≤ Eg(f (X)). n→∞ Applying this inequality to the function −g, we have that lim inf E∗ g(f (Xn )) ≥ Eg(f (X)). n→∞ Since E∗ g(f (X n )) ≥ E∗ g(f (Xn )), the proof is over.  We now review the extensions of Prohorov’s theorem. We say that the sequence Xn is asymptotically measurable if (D.2) lim E∗ f (Xn ) − E∗ f (Xn ) = 0  n→∞  for all bounded continuous functions on S. Define P∗ (A) = E∗ 1A for A ⊂ . We say that the sequence Xn is asymptotically tight if for every ε > 0 there exists a compact set K such that lim sup P∗ (Xn ∈ S \ K δ ) ≤ ε n→∞ for all δ > 0, where K δ = {x ∈ S : m(x, K) < δ}, m being the metric on S. The next four results are copied from van der Vaart and Wellner [23]. T HEOREM D.3. If the sequence Xn converges in distribution to a random element with a tight probability law, then it is asymptotically tight. T HEOREM D.4. If the sequence Xn is asymptotically tight and f : S → S ′ is a continuous mapping, then the sequence f (Xn ) is asymptotically tight. 192 A. A. PUHALSKII AND J. E. REED T HEOREM D.5. If the sequence Xn is asymptotically tight and asymptotically measurable, then it has a subsequence which converges in distribution to a random element with a tight probability law. T HEOREM D.6. Suppose that the sequence Xn is asymptotically tight and (D.2) holds for all functions f from a subalgebra of the algebra of bounded continuous functions which separates points in S. Then the sequence Xn is asymptotically measurable. The next group of results is concerned with joint convergence. Let Yn be a sequence of maps from  to a metric space S ′ . The following is Lemma 1.4.3 from van der Vaart and Wellner [23]. T HEOREM D.7. Each of the sequences Xn and Yn is asymptotically tight if and only if the sequence (Xn , Yn ) is asymptotically tight in S × S ′ . We use the following corollary of Example 1.4.6 in van der Vaart and Wellner [23]. T HEOREM D.8. Suppose that the Xn converge in distribution in S to a separable random element X and the Yn converge in distribution in S ′ to a separable random element Y . If Xn and Yn are independent for each n, then the (Xn , Yn ) converge in distribution in S × S ′ to (X, Y ), where X and Y are independent. The next result is usually referred to as Slutsky’s lemma; see Example 1.4.7 in van der Vaart and Wellner [23]. T HEOREM D.9. If the sequence Xn converges in distribution in S to a separable random element X and the sequence Yn converges in distribution in S ′ to a constant element c ∈ S ′ , then the sequence (Xn , Yn ) converges in distribution in S × S ′ to (X, c). The following result is in a similar vein. T HEOREM D.10. Suppose that Xnε , where ε > 0, converge in distribution in S to random elements Xε as n → ∞ and that the X ε converge in distribution in S to a random element X as ε → 0. If lim lim sup P∗ m(Xn , Xnε ) > δ = 0 ε→0 n→∞   for arbitrary δ > 0, then the Xn converge in distribution in S to X. 193 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC P ROOF. As it follows by the Portmanteau theorem (Theorem 1.3.4 in van der Vaart and Wellner [23]), it suffices to prove that (D.1) holds for any real-valued bounded uniformly continuous function f on S. We have |E∗ f (Xn ) − Ef (X)| ≤ |E∗ f (Xn ) − E∗ f (Xnε )| + |E∗ f (Xnε ) − Ef (X ε )| + |Ef (Xε ) − Ef (X)|. The hypotheses imply that lim sup lim sup |E∗ f (Xn ) − Ef (X)| ≤ lim sup lim sup |E∗ f (Xn ) − E∗ f (Xnε )|. ε→0 n→∞ ε→0 n→∞ By the triangle inequality for the outer expectation, |E∗ f (Xn ) − E∗ f (Xnε )| ≤ E∗ |f (Xn ) − f (Xnε )|. Given arbitrary γ > 0, let δ > 0 be such that |f (x) − f (y)| ≤ γ if m(x, y) < δ. Then |f (Xn ) − f (Xnε )| ≤ γ + 2 supx∈S |f (x)|1{m(Xn ,Xnε )>δ} . We conclude that lim sup lim sup |E∗ f (Xn ) − E∗ f (Xnε )| ε→0 n→∞ ≤ γ + 2 sup |f (x)| lim sup lim sup P∗ m(Xn , Xnε ) > δ = γ . x∈S ε→0 n→∞ Convergence (D.1) follows.    Convergence in distribution to separable random elements can be metrized. Let X and Y be mappings from  into S. Suppose that Y is measurable. We define ∗ dBL (X, Y ) = sup |E∗ f (X) − Ef (Y )|, f ∈BL1 where BL1 denotes the set of real-valued functions on S that are bounded in absolute value by 1 and admit a Lipshitz constant of 1. The next theorem appears in Dudley [7]; see also van der Vaart and Wellner [23], Section 12. T HEOREM D.11. The sequence Xn converges in distribution in S to a sepa∗ (X , X) = 0. rable random element X if and only if limn→∞ dBL n 1 Acknowledgments. The authors would like to thank the referees for their careful reading of the manuscript and helpful suggestions. In particular, the authors followed one of the referee’s advice to incorporate general initial conditions in Theorem 2.2. REFERENCES [1] B ILLINGSLEY, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York. MR1700749 [2] B OROVKOV, A. A. (1967). Limit laws for queueing processes in multichannel systems. Sibirsk. Mat. Ž. 8 983–1004. MR0222973 194 A. A. PUHALSKII AND J. E. REED [3] B OROVKOV, A. A. (1980). Asimptoticheskie Metody v Teorii Massovogo Obsluzhivaniya. Nauka, Moscow. MR570478 [4] B RÉMAUD , P. (1981). Point Processes and Queues. Springer, New York. MR636252 [5] D ELLACHERIE , C. (1972). Capacités et Processus Stochastiques. Springer, Berlin. MR0448504 [6] D OOB , J. L. (1984). Classical Potential Theory and Its Probabilistic Counterpart. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 262. Springer, New York. MR731258 [7] D UDLEY, R. M. (1990). Nonlinear functionals of empirical measures and the bootstrap. In Probability in Banach Spaces, 7 (Oberwolfach, 1988). Progress in Probability 21 63–82. Birkhäuser, Boston, MA. MR1105551 [8] D UDLEY, R. M. (1999). Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics 63. Cambridge Univ. Press, Cambridge. MR1720712 [9] E THIER , S. N. and K URTZ , T. G. (1986). Markov Processes. Wiley, New York. MR838085 [10] H ALFIN , S. and W HITT, W. (1981). Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29 567–588. MR629195 [11] I BRAGIMOV, I. A. and ROZANOV, Y. A. (1978). Gaussian Random Processes. Applications of Mathematics 9. Springer, New York. MR543837 [12] JACOD , J. and S HIRYAEV, A. N. (1987). Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin. MR959133 [13] J ELENKOVI Ć , P., M ANDELBAUM , A. and M OM ČILOVI Ć , P. (2004). Heavy traffic limits for queues with many deterministic servers. Queueing Syst. 47 53–69. MR2074672 [14] K ASPI , H. and R AMANAN , K. (2006). Fluid limits of many-server queues. Preprint. [15] K RICHAGINA , E. V. and P UHALSKII , A. A. (1997). A heavy-traffic analysis of a closed queueing system with a GI /∞ service center. Queueing Systems Theory Appl. 25 235–280. MR1458591 [16] L IPTSER , R. and S HIRYAYEV, A. (1989). Theory of Martingales. Kluwer, Dordrecht. [17] L OUCHARD , G. (1988). Large finite population queueing systems. I. The infinite server model. Comm. Statist. Stochastic Models 4 473–505. MR971602 [18] M ANDELBAUM , A. and M OMCILOVIC , P. (2008). Queues with many servers: The virtual waiting-time process in the QED regime. Math. Oper. Res. 33 561–586. [19] PANG , G., TALREJA , R. and W HITT, W. (2007). Martingale proofs of many-server heavytraffic limits for Markovian queues. Probab. Surv. 4 193–267. MR2368951 [20] P UHALSKII , A. (2001). Large Deviations and Idempotent Probability. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 119. Chapman & Hall/CRC, Boca Raton, FL. MR1851048 [21] P UHALSKII , A. A. and R EIMAN , M. I. (2000). The multiclass GI /P H /N queue in the Halfin–Whitt regime. Adv. in Appl. Probab. 32 564–595. MR1778580 [22] R EED , J. E. (2009). The G/GI /N queue in the Halfin–Whitt regime I: Infinite server queue system equations. Ann. Appl. Probab. 19 2211–2269. [23] VAN DER VAART, A. W. and W ELLNER , J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York. MR1385671 [24] W HITT, W. (2002). Stochastic-Process Limits. Springer, New York. MR1876437 ON MANY-SERVER QUEUES IN HEAVY TRAFFIC 195 [25] W HITT, W. (2005). Heavy-traffic limits for the G/H2∗ /n/m queue. Math. Oper. Res. 30 1–27. MR2125135 D EPARTMENT OF M ATHEMATICAL AND S TATISTICAL S CIENCES U NIVERSITY OF C OLORADO D ENVER P.O. B OX 173364, C AMPUS B OX 170 D ENVER , C OLORADO 80217-3364 USA AND IITP, M OSCOW B OLSHOY K ARETNY PER . 19 M OSCOW 127994 RUSSIA E- MAIL : [email protected] L EONARD N. S TERN S CHOOL OF B USINESS N EW YORK U NIVERSITY K AUFMAN M ANAGEMENT C ENTER 44 W EST 4 TH S TREET, KMC 8-79 N EW YORK , N EW YORK 10012 USA E- MAIL : [email protected]