A FUNCTIONAL CENTRAL LIMIT THEOREM FOR MARKOV ADDITIVE ARRIVAL PROCESSES AND ITS APPLICATIONS TO QUEUEING SYSTEMS

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1 A FUNCTIONAL CENTRAL LIMIT THEOREM FOR MARKOV ADDITIVE ARRIVAL PROCESSES AND ITS APPLICATIONS TO QUEUEING SYSTEMS HONGYUAN LU, GUODONG PANG, AND MICHEL MANDJES Abstract. We prove a functional central limit theorem for Marov additive arrival processes MAAPs where the modulating Marov process has the transition rate matrix scaled up by n α α > and the mean and variance of the arrival process are scaled up by n. It is applied to an infinite-server queue and a for-join networ with a non-exchangeable synchronization constraint, where in both systems both the arrival and service processes are modulated by a Marov process. We prove FCLTs for the queue length processes in these systems joint with the arrival and departure processes, and characterize the transient and stationary distributions of the limit processes. We also observe that the limit processes possess a stochastic decomposition property.. Introduction Marov additive arrival processes MAAP have been used to model the arrival processes of many stochastic systems, for example, telecommunication and service systems in random environments [2]. Its usefulness lies in capturing burstiness in the arrival processes, thus departing from the usual renewal-type assumptions. MAAPs are described by a couple A, X, where the process A is the counting process of arrivals, and the process X is a modulating Marov process. A popular model is the Marov-modulated Poisson process MMPP, which has been widely used to model a variety of relevant stochastic systems [2, 25]. For a broad range of queues with MMPP input analytical results have been derived, often employing the matrix computational approach. The main objective of this paper is to generate functional central limit theorems FCLTs for MAAPs, in particular, the counting process A, and their applications in specific, practically relevant, queueing systems. FCLTs for MMPPs have been studied in the literature under two types of scalings. In the first scaling, time is scaled up by a parameter n while space is scaled down by n, and thus the transition times of the modulating Marov process are implicitly accelerated by a factor n. Under this scaling, assuming that the modulating Marov process has a finite number of states and is irreducible, an FCLT can be proven for the scaled arrival process, where the limit process is a Brownian motion reviewed in This has been applied to prove heavy-traffic limits for single-server queueing networ models; see, e.g., [25, Ch. 9]. Under this same scaling, Steichen [23] considered an MAAP where the arrival process in each state can be non-poisson, and proved an FCLT with a Brownian motion limit. That result was also applied to study some single-server queueing networs in [23]. In the second scaling, time is not scaled, but the arrival rates in each state are scaled up by n and the space is scaled down by n, while at the same time the transition rates of the modulating Marov process are scaled up by n α for some α >. Under this scaling, an FCLT has recently been proved for the scaled arrival process in [], where the limit process is a Brownian motion reviewed in This is then applied in [] to prove an FCLT for the M/M/ queue with MMPP input. This scaling is useful in many-server systems, where the demand is relatively large but service times do not scale as the demand gets larger, and the modulating Marov process may speed up or slow down. Date: November 7, 25. Key words and phrases. Marov additive arrival process, functional central limit theorem, infinite-server queues, for-join networs with non-exchangeable synchronization, Gaussian limits, stochastic decomposition.

2 2 HONGYUAN LU, GUODONG PANG, AND MICHEL MANDJES FCLTs for MAAPs. MMPPs, in which the Poisson arrival rate jumps between several values, significantly generalize the traditional Poisson setting. Nonetheless, in many applications the assumption of the input being locally Poisson is not adequate. To remedy this we consider in this paper a general class of MAAPs, where the arrival process in each state can be a general stationary counting process, including renewal processes. We prove an FCLT for this class of MAAPs, in Theorem 2., under the second type of scaling and under three regimes of α values, i.e., < α <, α = and α >. The limit process is also a Brownian motion, whose variance coefficient compactly captures the variabilities in the interarrival times in each state as well as the variabilities in the modulating Marov process. We apply this FCLT to two queueing systems: a general infinite-server queue and a for-join networ with the non-exchangeable synchronization NES constraint. General infinite-server queue. Several recent papers have studied infinite-server queues with MMPP input. Exact analysis and related approximations have been derived for specific infinite-server queues in random environments Marov or semi-marov modulated in [3, 5, 9,, 2, 7, 9]. In [], an M/M/ queue with MMPP input is studied, leading to an FCLT for the queue length process under the second type of scaling mentioned above. [4] studies an M/GI/ queue with MMPP input and general service times depending on the state of the modulating Marov process upon arrival. The exact mean and variance formulas for the transient and stationary distributions of the queue length process are provided, and the asymptotic results are also obtained in the regime where the arrival rates are scaled up by n and the transition rates are scaled up by n +ɛ for some ɛ >. Central limit theorems are proved for the M/M/ queue with both the arrival and services modulated by a finite-state Marov process in [6, 7], where the arrival rates are scaled up by n. In Section 3, we establish an FCLT for the queue length process joint with the arrival and departure processes in the G/G/ queue where both the arrival process and the service time distributions are modulated by a Marov process applying Theorem 2., thus generalizing the existing literature substantially. The limiting queue-length and departure processes are continuous Gaussian processes, of which we characterize the transient and steady-state distributions. We also derive a stochastic decomposition property: the variabilities of the arrival process and modulating Marov process are captured in one limit component, while those of the service process are captured in a second independent limit component. For-join networ with NES. In our second application, we consider a for-join networ with NES, where both the arrival process and the joint service time distributions of the parallel tass of each job are modulated by a Marov process. In the networ, each job is fored into a fixed number of parallel tass, each of which is processed in a multi-server service station, and after service completion, each tas will join a buffer associated with the service station, waiting for synchronization. The NES constraint requires that synchronization occurs only when all the tass of the same job are completed. It is important to understand the joint dynamics of the service process as well as the waiting buffers for synchronization. Heavy-traffic limits are proved for a single-class multi-server for-join networ with NES, in the underloaded quality-driven QD regime [4] and the critically loaded quality-and-efficiency driven QED regime [6]. The setup considered is such that the arrival process is general satisfying an FCLT, whereas the service times of the parallel tass form i.i.d. random vectors that can be correlated. In addition, in [5], an infinite-server for-join networ with NES in a renewal alternating environment up-down cycles is studied, where the service vectors of parallel tass are correlated and the service processes are interrupted during the down periods. In this paper we study a multi-server for-join networ with NES in the QD regime, where both the arrival and service processes are modulated by a Marov process. We apply our FCLT for the MAAP to obtain a multi-dimensional Gaussian limit process for the processes representing the number of tass in service at each station and the numbers of tass in the waiting buffer for synchronization associated with each station, jointly with the arrival process and the process

3 AN FCLT FOR MARKOV ADDITIVE ARRIVAL PROCESSES AND ITS APPLICATIONS 3 representing the number of synchronized jobs. We characterize the transient and steady-state joint distributions of the limit queueing processes, as multivariate Gaussian distributions, and of the synchronized process, as a Gaussian distribution. We also observe a similar stochastic decomposition property as in the infinite-server queues above, where the two independent limit components capture the variabilities of the arrival and modulating Marov processes, and of the service processes separately... Organization of the paper. The rest of the paper is organized as follows. We finish this section below with a summary of notations used in the paper. In Section 2, we present the general MAAP, review the existing FCLTs for MAAPs, and state the new FCLT under the second type of scaling. In Section 3, we apply the FCLT for the MAAP to a general infinite-server queueing model with both arrival and service times modulated by a Marov process. In Section 4, we apply the FCLT for the MAAP to a for-join networ with both arrival and service processes being modulated by a Marov process. The proofs of these results are presented in Section 5. We mae some concluding remars in Section Notations. The following notations will be used throughout the paper. R and R + R d and R d +, respectively denote sets of real and real non-negative numbers d-dimensional vectors, respectively, d 2. For a, b R, we denote a b := mina, b. For any x R +, x is used to denote the largest integer no greater than x. We use bold letter to denote a vector, e.g., x := x,..., x N R N. A is used to denote the indicator function of a set A. For two real-valued functions f and g, we write fx = Ogx if lim sup x fx/gx <. All random variables and processes are defined on a common probability space Ω, F, P. For any two complete separable metric spaces S and S 2, we denote S S 2 as their product space, endowed with the maximum metric, i.e., the maximum of two metrics on S and S 2. S is used to represent -fold product space of any complete and separable metric space S with the maximum metric for N. For a complete separable metric space S, D[,, S denotes the space of all S-valued càdlàg functions on [,, and is endowed with the Sorohod J topology see, e.g., [8,, 25]. Denote D D[,, R. The space D[,, D, denoted as D D, is endowed with the Sorohod J topology, that is, both inside and outside D spaces are endowed with the Sorohod J topology. Let D[,, R D denote the space of all continuous from above with limits from below real-valued functions on [, with the generalized Sorohod J topology [8, 24] for 2. Wea convergence of probability measures µ n to µ will be denoted as µ n µ. 2. An FCLT for Marov Additive Arrival Processes Consider a Marov additive arrival process A, X. The process X = Xt : t } is a finite-state irreducible stationary Marov process with state space S =,..., I} and transition rate matrix Q = q ij i,j=,...,i. The process A = At : t } is a counting process modulated by the Marov process X, defined as follows. Let π = π,..., π I be the stationary distribution of the Marov process X. We assume that the process starts in stationarity at time. We introduce some auxiliary notations. Let Π be a matrix with each row being the steady-state vector π, and P t = P ij t i,j=,...,i be the transition matrix, that is, P ij t := P Xt = j X = i for each t. Let Z = Z ij i,j=,...,i be the fundamental matrix, given by Z ij := P ij t π j dt. It holds that Z = Π Q Π. When the process A is an MMPP, that is, arrivals follow a Poisson process with rate λ i when X = i, i S, FCLTs are proved for the process A in two different scalings. In the first scaling that

4 4 HONGYUAN LU, GUODONG PANG, AND MICHEL MANDJES was introduced in Section, both time and space are scaled by n, and the diffusion-scaled process à n = Ãn t : t } is defined by à n t := n Ant /2 π i λ i nt, t. 2. By Theorem in [26], one can show that à n à in D as n, 2.2 where à = Ãt : t } is a driftless Brownian motion with variance coefficient with and β := 2 σ 2 := λ + β, 2.3 λ := π i λ i, 2.4 j= λ i λ j π i Z ij. 2.5 See also the discussion in Example in [25]. Note that under this scaling, the transition rates of the modulating Marov process are scaled up by n. In the second scaling introduced in Section, time is not scaled, but the arrival rates λ are scaled by n and transition rate matrices are scaled by n α for α >. Namely, we consider a sequence of the processes A n, X n indexed by n, and write the corresponding quantities by a superscript n. Assume that λ n i /n λ i > for i S as n and Q n = n α Q for some α >. Note that the stationary distribution of X n remains the same, π. Define the diffusion-scaled process Ân = Ân t : t } by  n t := A n n δ t π i λ n i t, for δ >, t. 2.6 Then it is shown in [] that  n  in D as n, 2.7 where the limit process  = Ât : t } is a driftless Brownian motion with variance coefficient β, α <, δ = α/2, σ 2 α := λ + β, α =, δ = /2, 2.8 λ, α >, δ = /2, with λ and β being defined in 2.4 and 2.5, respectively. Remar 2.. When α =, the limit processes under both scalings in fact coincide, as the arrival process and the modulating Marov process are sped up at the same rate. When α >, the modulating Marov process is sped up at a faster rate than the arrival process in each state, and thus the variability in the limit comes only from the Poisson processes with the spatial scaling n /2. When < α <, the modulating Marov process is sped up at a slower rate than the arrival process in each state, and thus the variability in the limit comes only from the modulating Marov process with the spatial scaling n α/2. In this paper, we consider the second type of scaling and prove an FCLT for the diffusion-scaled processes Ân when the process A n is general, including renewal process, in each state of X n.

5 AN FCLT FOR MARKOV ADDITIVE ARRIVAL PROCESSES AND ITS APPLICATIONS 5 Let τ n be the th jump time of X n for =, 2, 3,... and τ n. For each i S, define [ A λ n n τ n i := E + s An τ n ] X n u = i for τ n s u τ n + s, 2.9 and ν n i [ A n τ n := E + s An τ n2 λ n i 2 s 2 ] X n u = i for τ n s u τ n + s. 2. Assume that λ n = λ n,..., λn I and νn = ν n,..., νn I are positive vectors. Note that when the process A n is Poisson in each state of X n, we have that λ n i = νi n, i =,..., I. Note also that if the arrival process is renewal in each state, the parameter νi n = λ n i cn a,i 2, where c n a,i is the coefficient of variation CV of the interarrival times when the Marov process X n is in state i. Then, we can write, for each t, and E[A n t X n s, s t] = V ar[a n t X n s, s t] = λ n X n s ds = ν n X n s ds = We mae the following assumption on the parameters. Assumption. The parameters λ n and ν n satisfy λ n n λ RI +, The transition rate matrix Q n = n α Q for some α >. ν n n ν RI + as n. λ n i X n s = ids, 2. ν n i X n s = ids. 2.2 We now state the main result of this section. Its proof, as well as the proofs of all results presented in Sections 3 and 4, are provided in Section 5. Theorem 2.. Under Assumption, for the diffusion-scaled process Ân in 2.6, 2.7 holds where the limit process  is a driftless Brownian motion with variance coefficient β, < α <, δ = α/2, σ 2 α := ν + β, α =, δ = /2, 2.3 ν, α >, δ = /2, with β being defined in 2.5 and ν := π i ν i. 2.4 Remar 2.2. When the modulating Marov process X n is in state i, if the process A n is renewal, we obtain ν = I π iλ i c 2 a,i, where λ i is the arrival rate and c a,i is the CV of the interarrival times in the limit and ν i = λ i c 2 a,i.

6 6 HONGYUAN LU, GUODONG PANG, AND MICHEL MANDJES 3. Application to Infinite-Server Queues In this section, we apply the FCLT of the MAAP process, Theorem 2., to G/GI/ queues with Marov modulated arrival and service processes. It is shown in [3] that an FCLT for the number of customers/jobs in a G/GI/ queue holds, provided that the arrival process satisfies an FCLT and the service times are i.i.d. with a general distribution. Our FCLT below extends the existing results in [] and [4] by allowing more general arrival process, in that in each state of the underlying Marov process, the arrival process can be a general stationary point process, including a renewal process. It also proves the joint convergence of arrivals, queue length and departure processes rather than just queue length. The limiting queue-length process is a continuous Gaussian process, and possesses a stochastic decomposition property. Our results also generalize [3] for G/G/ queues in a Marov random environment. Consider a sequence of G/G/ queues modulated by a Marov process X n, behaving as described in Section 2. The arrival process A n is an MAAP with τ n denoting the arrival time of job,. The service times η,i : } are i.i.d. with a general distribution F i independent of n when the underlying Marov process X n is in state i upon the customer s arrival. Namely, we assume that the service time distribution of a customer is determined at the epoch of the arrival time, according to the state of the underlying Marov process X n. We also assume that conditional on the modulating Marov process X n, the arrival and service processes are independent, and that the system starts empty. Let Fi c := F i, i =,..., I. Let Q n = Q n t : t } be the queue length process describing the evolution of the number of customers in the system. Let D n = D n t : t } be the departure process counting the number of completed jobs. We have the following balance equation: D n t = A n t Q n t, t. 3. Define the diffusion-scaled processes ˆQ n = ˆQ n t : t } and ˆD n = ˆD n t : t } by where ˆQ n t := n δ Q n t nqt, ˆDn t := n δ D n t ndt = Ân t ˆQ n t, t, 3.2 qt := λ i π i F c i sds, and dt := λ i π i F i sds, t. Theorem 3.. For the sequence of G/GI/ models with Marov modulated arrival and service processes described above, Ân, ˆQ n, ˆD n Â, ˆQ, ˆD in D 3 as n, where  is the arrival limit defined in Theorem 2., the process ˆD = ˆDt : t } is defined by ˆDt := Ât ˆQt, t, and ˆQ, δ = α/2, < α <, ˆQ := ˆQ + ˆQ 2, δ = /2, α. The limit process ˆQ = ˆQ t : t } is a continuous Gaussian process, defined by ˆQ t := σα π i Fi c t s dw s, t, where W is a standard Brownian motion and σ 2 α is defined in 2.3. The limit process ˆQ 2 = ˆQ 2 t : t } is a continuous Gaussian process defined by ˆQ 2 t := s + x i > td ˆK i π i λ i s, x i 3.3

7 AN FCLT FOR MARKOV ADDITIVE ARRIVAL PROCESSES AND ITS APPLICATIONS 7 where the processes ˆK i = ˆK i s, x : s, x }, i =,..., I, are independent Kiefer processes with mean and covariance function Cov ˆK i s, x, ˆK i t, y = s tf i x y F i xf i y, s, t, x, y, for each i =,..., I. The processes W and ˆK i, i =,..., I, are independent, and thus, so are the processes ˆQ and ˆQ 2. Here the integrals in 3.3 are defined in the mean-square sense following [3]; see the precise definition in Remar 3.. We remar that there is a stochastic decomposition property in the limit process, as shown in the independence of ˆQ and ˆQ 2. Note that in the corresponding prelimit processes are evidently dependent because of the modulating Marov process. The limit process ˆQ captures the variabilities resulting from the arrival process, as well as those resulting from the modulating Marov process. The limit process ˆQ 2 captures the variabilities from the service process, while, perhaps surprisingly, it is not affected by the variabilities of the modulating Marov process other than the steady-state distribution π. This is also shown in the following characterization of the limit processes. Corollary 3.. Under the assumptions of Theorem 3., the limit process ˆQ is Gaussian, with mean and covariance function β t I π ifi cs I π ifi ds, cs + u δ = α/2, < α <, Cov ˆQt, ˆQt+u = σ 2 α t I π ifi cs I π ifi ds cs + u + I π iλ i Fi sfi cu + s ds, δ = /2, α, for t, u, where β and σ 2 α are defined in 2.5 and 2.3, respectively. Its stationary distribution has variance β I 2ds, π ifi cs δ = α/2, < α <, V ar ˆQ = I π iλ i m s,i + σ 2 α I 2ds π ifi cs I π iλ i F i cs2 ds, δ = /2, α, with m s,i being the mean service time associated with F i. In addition, the limit process ˆD is Gaussian, with mean and covariance function β t I π I if i s π if i s + u ds, δ = α/2, < α <, Cov ˆDt, ˆDt + u = σ 2 α t I π I if i s π if i s + u ds + I π iλ i Fi sfi cu + s ds, δ = /2, α, for t, u, and lim t t V ar ˆDt = σ 2 α. Remar 3.2. When F i, i =,..., I, are identical, our results establish an FCLT for G/GI/ queues with an MAAP and i.i.d. service times. Moreover, when the arrival process is an MMPP and the service times are exponential with rate µ independent of the modulating Marov process, our results reduce to those in [] for M/M/ queues. 4. Application to For-Join Networs In this section, we apply the FCLT for the MAAP to a many-server for-join networ with the non-exchangeable synchronization NES constraint, where both the arrival process and the joint service time distribution of the parallel tass are modulated by a Marov process.

8 8 HONGYUAN LU, GUODONG PANG, AND MICHEL MANDJES Consider a sequence of many-server for-join networs with NES indexed by n and let n. We assume that the systems are operating in the QD regime, which is asymptotically equivalent to systems with infinite-server service stations. There is a single class of customers. Let the arrival processes A n be an MAAP as described in Section 2. Let η l,i = η l,i,..., ηl,i K be the service times that customer l brings in for the K parallel tass when the underlying Marov process X n is in state i at the epoch of arrival. Assume that the service times η l,i : l } are i.i.d. with a continuous joint distribution function F i and marginals F i i, =,..., K and i =,..., I. Let F j, be the joint distribution of the service times of parallel tass j, for j, =,..., K and i =,..., I. Let F m i be the distribution of the maximum of the service times η l,i,..., ηl,i i K, i.e., F m x = P η,i j x, j for x. Denote G i i := F for =,..., K, and G i m := F m i, i =,..., I. Let Q n = Q n,..., Qn K be the numbers of tass in service at service stations =,..., K. Let Y n = Y n,..., Y K n be the numbers of tass that have completed service but are waiting for service at the waiting buffers for synchronization corresponding to the service stations =,..., K. Let S n = S n t : t } be the process counting the number of synchronized jobs. Define the diffusion-scaled processes ˆQ n = ˆQ n,..., ˆQ n K, Ŷ n = Ŷ n,..., Ŷ K n and Ŝn by and where and q t := ˆQ n t := n δ Qn t nq t, λ i π i Ŷ n t := n δ Y n t ny t, =,..., K, t, Ŝ n t := n δ Sn t nst, t, G i sds, y t := st := λ i π i λ i π i G i m s G i sds, =,..., K, t, F i m sds, t. Theorem 4.. For the for-join networs with NES and Marov modulated arrival and service processes described above, Ân, ˆQ n, Ŷ n, Ŝn Â, ˆQ, Ŷ, Ŝ in D2K+2 as n, where  is the arrival limit defined in Theorem 2., ˆQ = ˆQ,..., ˆQ K, Ŷ = Ŷ,..., ŶK and Ŝ are defined as follows: ˆQ,, δ = α/2, < α <, ˆQ := ˆQ, + ˆQ,2, δ = /2, α, Ŷ,, δ = α/2, < α <, Ŷ := Ŷ, + Ŷ,2, δ = /2, α, Ŝ, δ = α/2, < α <, Ŝ := Ŝ + Ŝ2, δ = /2, α. The limit processes ˆQ, = ˆQ, t : t }, Ŷ, = Ŷ,t : t } and Ŝ = Ŝt : t } are continuous Gaussian processes defined by t ˆQ, t := σα π i G i t s dw s, t, Ŷ, t := σα π i F i i t s F m t s dw s, t,

9 AN FCLT FOR MARKOV ADDITIVE ARRIVAL PROCESSES AND ITS APPLICATIONS 9 Ŝ t := σα π i F i m t s dw s, t, where W is a standard Brownian motion with variance coefficient σ 2 α as defined in Theorem 2.. The limit processes ˆQ,2 = ˆQ,2 t : t }, Ŷ,2 = Ŷ,2t : t } and Ŝ2 = Ŝ2t : t } are continuous Gaussian processes defined by ˆQ,2 t := s + x > td ˆK i π i λ i s, x, t, Ŷ,2 t := Ŝ 2 t := R K + R K + s + x t s + x j t, j d ˆK i π i λ i s, x, t, R K + s + xj t, j d ˆK i π i λ i s, x, t, where ˆK i s, x are independent multiparameter Kiefer processes Gaussian random field with mean and covariance Cov ˆK i s, x, ˆK i t, y = s tf i x y F i xf i y for s, t and x, y R K +. The integrals in ˆQ,2 t, Ŷ,2t and Ŝ2t are defined in the mean squared sense. The Brownian motion W is independent from ˆK i, i =,..., I, and thus, ˆQ, and ˆQ j,2 are independent, and so are Ŷ, and Ŷj,2 for each, j =,..., K. Ŝ and Ŝ2 are also independent. Remar 4.. We remar that there is also a stochastic decomposition property as the one we have seen for the infinite-server queueing model. The variabilities in the arrival process and the Marov process are captured in ˆQ,, Ŷ, and Ŝ for each, while the variabilities in the service process are captured in ˆQ,2, Ŷ,2 and Ŝ2. Corollary 4.. Under the assumptions of Theorem 4., the limit process ˆQ, Ŷ is a multidimensional Gaussian process with mean zero and covariance functions: for j, =,..., K, t, t, Cov ˆQ j t, ˆQ t Cov = ˆQ j, t, ˆQ, t, δ = α/2, < α <, Cov ˆQ j, t, ˆQ, t + Cov ˆQ j,2 t, ˆQ,2 t, δ = /2, α, CovŶjt, Ŷt CovŶj,t, = Ŷ,t, δ = α/2, < α <, CovŶj,t, Ŷ,t + CovŶj,2t, Ŷ,2t, δ = /2, α, Cov ˆQ j t, Ŷt Cov = ˆQ j, t, Ŷ,t, δ = α/2, < α <, Cov ˆQ j, t, Ŷ,t + Cov ˆQ j,2 t, Ŷ,2t, δ = /2, α, where Cov ˆQ j, t, ˆQ, t = σ 2 α t t CovŶj,t, Ŷ,t = σ 2 α π i π i G i j t s F i j π i π i G i t s t s F i m t s F i t t Cov ˆQ j, t, Ŷ,t = σ 2 α π i G i j t s π i ds, t s F m i t s ds, F i t s F m i t s ds,

10 HONGYUAN LU, GUODONG PANG, AND MICHEL MANDJES Cov ˆQ j,2 t, ˆQ,2 t = CovŶj,2t, Ŷ,2t = Cov ˆQ j,2 t, Ŷ,2t = t π i λ i t π i λ i F i F i j, t s, t s F i j t sf i t s ds, F i j, t s, t s F i j t sf i t s j,m t s, t s + F i j t sf m i t s F i,m t s, t s +F i t sf m i t s + F m i t s t s F m i t sf m i t s t π i λ i F i j,m t s, t s F i j t sf m i t s j, t s, t s + F i j t sf i t s ds, F i with σ 2 α being defined in 2.3, and for j =,..., K and x, y, F j,m x, y := F z for z R K + satisfying z j = x y and z j = y for j j. In addition, the limit process Ŝ is a continuous Gaussian process with mean zero and covariance functions: for t, t, CovŜt, CovŜt, Ŝt = Ŝt, δ = α/2, < α <, CovŜt, Ŝt + CovŜ2t, Ŝ2t, δ = /2, α, where t t CovŜt, Ŝt = σ 2 α π i F m i t s π i F m i t s ds, t CovŜ2t, Ŝ2t = π i λ i F m i t s t s F m i t sf m i t s ds, with σ 2 α being defined in 2.3, and lim t t V arŝt = σ2 α. 5. Proofs 5.. Proof of Theorem 2.. In this section, we prove Theorem 2.. First of all, we write the process Ân as where and  n t = Ân t + Ân 2 t, t, 5.  n t := n δ A n t ds, λ n X n s ds, 5.2  n 2 t := t n δ λ n X n s ds π i λ n i t. 5.3 We now focus on proving the convergence of Ân. Without loss of generality, we pic state as the reference state. Let T n be the first time that Xn t reaches state from the initial state, and T n be the + th jump time of X n t reaching state i.e., the th excursion time. Define a counting process associated with the sequence : =, 2,...}: N n t := max : t, =,, 2,...}, t, and :=.

11 AN FCLT FOR MARKOV ADDITIVE ARRIVAL PROCESSES AND ITS APPLICATIONS Then we can decompose the process Ân where into three processes:  n t = Ân,t + Ân,2t + Ân,3t, t, 5.4  n,t := A n n δ t T n  n,2t := N n t n δ =  n,3t := n δ A n An A n t A n N n t λ n X n s ds N n t, 5.5 λ n X n s ds λ n X n s ds, We will prove the convergence of the three processes Ân,, Ân,2 and Ân,3 in the following lemmas. Before proving the convergence of the three processes Ân,, Ân,2 and Ân,3, we present some properties on the processes N n and the sequence : =,, 2,...}. Let T n := T n T n for =, 2,.... Then T n : =, 2,...} forms an i.i.d. sequence of random variables. Let γ n := E[ T n]. It is evident that γn < and there exists γ > such that γ n = n α γ, since X n has transition rate matrix Q n = n α Q. Thus, it follows from the FLLN for delayed renewal processes that n α N n γ e in D as n, 5.8 where et t for t. Lemma 5.. For any ɛ > and fixed T >, Proof. It suffices to show that lim P n lim E n sup t [,T ] [ sup t [,T ] By 5.5, we obtain the following upper bound: [ E  n,t ] [ n δ E sup t [,T ] sup A n t T n t [,T ]  n,t > ɛ  n,t ] ] + nδ E [ =. 5.9 =. 5. sup t [,T ] λ n X n s ds ] [ [E n δ E A n T T n [ ]] X n s : s T + T n δ E [ ] = 2 T n δ E λ n X n s ds 2 n δ max i S λn i λ n X n s ds ] E[ ]. 5. By Assumption, we have that n max i S λ n i max i S λ i < as n. Since Q n = n α Q, it is evident that E[ T n] = O/nα see, e.g., [2, pp ]. Thus it follows that 2 n δ max i S λn i E[ T n ] as n,

12 2 HONGYUAN LU, GUODONG PANG, AND MICHEL MANDJES and we have proved 5.. Lemma 5.2. For any ɛ > and fixed T >, lim P n sup t [,T ]  n,3t > ɛ Proof. For each =, 2, 3,..., define Ă n := sup n δ An T n + t An T n t = t λ n X n s ds. 5.3 To prove 5.2, it suffices to prove that lim E[ Ă n ] n NT + =. 5.4 By 5.3 and conditioning, we obtain that E [ Ă n ] N n T + n δ E [ A n N n T + An N n T ] + n δ E [ N n T + [ ] 2 n δ E N n T + λ n X n s ds N n T N n T λ n X n s ds ] 2 n δ max i S λn i E[ T n ] as n, where the convergence follows from Assumption and E[ ] = n α γ. Thus, the lemma is proved. Lemma 5.3.  n,2, δ = α/2, < α <, Â, δ = /2, α, 5.5 in D as n, where the limit process  is a driftless Brownian motion with variance coefficient ν defined in 2.4. To prove this lemma, we need the following lemma, whose proof follows from a direct generalization of Theorem 2.7 in [22]. Lemma 5.4. Let ξ n,i : i } be an i.i.d. sequence for each n and U n t := n α t ξ n,i for each t and any α >. Then U n U in D as n where U is a stochastic process with stationary independent increments if and only if U n t Ut in R for each t as n. Proof of Lemma 5.3. Define a process Ăn,2 = Ăn,2 t : t } by n α t/γ Ă n,2t := Ă n n δ λ n X n s ds, t, 5.6 where Ăn = is defined in 5.3. We first show that, for each t, Ă n, δ = α/2, < α <,,2t Ăt, δ = /2, α, 5.7 in R as n, where Ăt has a normal distribution with mean and variance νt, with ν defined in 2.4. This follows from applying CLT for doubly indexed sequences by noting that the summation terms in 5.6 are i.i.d. for each given n. It suffices to show that, as n, n α V ar Ă n n δ λ n X n s ds, δ = α/2, < α <, 5.8 νγ, δ = /2, α.

13 AN FCLT FOR MARKOV ADDITIVE ARRIVAL PROCESSES AND ITS APPLICATIONS 3 By conditioning, we obtain V ar Ă n n δ λ n X n s ds = V ar Ă n + V ar n δ λ n T n X n s ds 2Cov Ă n, n δ λ n T n X n s ds = E [ V ar Ă n X n s : T n s T n ] [Ăn + V ar E X n s : T n s T n ] [ + V ar λ n n2δ T n X n s ds 2 ] E n δ Ă n λ n T n X n s ds E [ [ ] Ă n ] E λ n T n X n s ds [ ] = n 2δ E ν n T n X n s ds + 2 V ar λ n n2δ T n X n s ds 2 V ar λ n n2δ T n X n s ds [ [ ] = n 2δ E ν n X n s ds ] = n 2δ ν n i E X n s = ids. 5.9 Under the assumption of the underlying Marov process X n, we obtain that for each i =,..., I and t, Since E[ T n] = n α γ, we obtain that as n, [ ] n α 2δ νi n E X n s = ids X n s = ids π i t as n. 5.2 Thus, we have proved 5.7. By Lemma 5.4, we obtain that Ă n, δ = α/2, < α <,,2 Â, δ = /2, α,, δ = α/2, < α <, νγ, δ = /2, α in D as n. Now by 5.8, Theorem.4.5 of [25] and the continuous mapping theorem, we can conclude the convergence in 5.5. Completing the Proof of Theorem 2.. Recall the representation of the process Ân in and By Lemmas 5. and 5.2, we obtain that Ân, and Ân,3 as n, respectively. By Lemma 5.3, we obtain that i Ân,2  in D as n, when δ = /2 and α, where  is a driftless Brownian motion with variance parameter ν, and ii Ân,2 in D as n, when δ = α/2 and < α <. By Proposition 3.2 in [], we obtain that  2, δ = α/2, < α <,  n 2  2, δ = /2, α =, 5.23, δ = /2, α >, in D as n, where the limit process Â2 = Â2t : t } is a Brownian motion with mean and variance coefficient β. Here, the Brownian motion  is independent of Â2. Thus the proof is complete.

14 4 HONGYUAN LU, GUODONG PANG, AND MICHEL MANDJES 5.2. Proofs for applications to infinite-server queues. Proof of Theorem 3.. We first note that the process Q n can be written as Q n t = = A n t = τ n + η,i > tx n τ n = i A n s s + x i > tx n s = id η,i x i, t = From this, we obtain the following representation for the diffusion-scaled process ˆQ n : ˆQn t = ˆQ n t + ˆQ n 2 t for t, where and ˆQ n t := ˆQ n 2 t := n /2 δ F c X n s t sdân s = We next prove the convergences of ˆQ n and ˆQ n 2. To prove the convergence of ˆQ n, we show that Note that P P sup t T t T lim P n sup +P F c i t sx n s = idân s, 5.25 s + x i > tx n s = id An s η,i x i F i x i. n sup t T ˆQn t ˆQ t > ɛ sup t T ˆQn t ˆQ t > ɛ = F c i t sx n s = id  n s Âs > ɛ F c 5.26 = i t s X n s = i π i d Âs > ɛ, 5.28 where  is the limit process of the arrivals Ân as given in Theorem 2.. The convergence to zero of the first term on the right-hand side of 5.28 follows from the convergence of  n  in Theorem 2.. To prove the convergence of the second term in 5.28, we first observe that the process ˆQ n,2 = ˆQ n,2 t : t } defined by ˆQ n,2t := F c i t s X n s = i π i d Âs, t, is a Marov process. It is easy to chec that the generators of the processes ˆQ n,2 converge to zero. Thus, by [, Ch. IV, Thm. 2.5], we obtain the convergence of the second term in To show the joint convergence Ân, ˆQ n Â, ˆQ in D 2 as n, 5.29

15 AN FCLT FOR MARKOV ADDITIVE ARRIVAL PROCESSES AND ITS APPLICATIONS 5 by endowing the product space with the maximum metric, we see that the convergence of Ân by assumption and ˆQ n in 5.27, as well as the continuity of their limits  and ˆQ, imply that 5.29 holds. Next we will show the convergence of ˆQn 2. Define the sequential empirical processes ˆK i n = ˆK i n t, x : t, x } by ˆK i n t, x := nt η,i x F i x, t, x, n = for each i =,..., I. By [3, Lemma 3.] and the independence of ˆK i n, i =,..., I, we now ˆK i n ˆK i in D D as n, 5.3 where ˆK i, i =,..., I, are independent Kiefer processes defined in Theorem 3.. We let A n i = A n i t : t } be the process counting the number of arrivals whose service type is i, i.e., } A n i t := max j : τ n j X n τ n j = i t, t, 5.3 where := and τ n :=, for i =,..., I. Define the fluid-scaled processes Ān i := n A n i for each i =,..., I. Thus, Theorem 2. directly implies the functional wea law of large numbers FWLLNs for A n i, i.e., We can rewrite 5.26 as Ān,..., Ān I π λ e,..., π I λ I e in D I as n ˆQ n 2 t = n /2 δ where the processes ˆQ n 2,i := ˆQ n 2,i t : t } are defined by ˆQ n 2,it := ˆQ n 2,it, t, 5.33 s + x i td ˆK n i Ān i s, x i, t Tightness of the processes ˆQ n 2,i : n } in D follows directly from the tightness of the corresponding processes for the G/GI/ queues in [3], for i =,..., I. Thus, we obtain the processes ˆQ n 2 : n } are tight. We now focus on proving the joint convergence of finite-dimensional distributions of ˆQ n and ˆQ n 2. We only need to show the case δ = /2, since otherwise the limit ˆQ 2 vanishes. Define the process ˆQ 2,i = ˆQ 2,i t : t } by ˆQ 2,i t := s + x i td ˆK i π i λ i s, x i, 5.35 for t and i =,..., I. The integral ˆQ2,i in 5.35 is understood as a mean-square integral. Specifically, we define ˆQ 2,i t := l.i.m l ˆQ2,i,l t, t, 5.36 where l.i.m. represents mean-square limit, that is, and ˆQ 2,i,l t := [ lim E ˆQ2,i t ˆQ 2,i,l t ] 2 =, l l,t s, xd ˆK i π i λ i s, x i = l ˆKi π i λ i s l j, ; π i λ i s l j, t s l j, j=

16 6 HONGYUAN LU, GUODONG PANG, AND MICHEL MANDJES with l,t, defined by l,t s, x := s = x t + l s l j < s s l jx t s l j, j= with the points = s l < sl 2 < < sl l = t being chosen so that max j l s l j sl j as l, and for a a 2, b b 2 and i =,..., I, ˆKi a, b ; a 2, b 2 = ˆK i a 2, b 2 ˆK i a, b 2 ˆK i a 2, b + ˆK i a, b We define additional processes ˆQ n 2,i,l = ˆQ n 2,i,l t : t } and Q n 2,i,l = Q n 2,i,l t : t } by ˆQ n 2,i,l t := Q n 2,i,l t := l,t s, xd ˆK n i Ān i s, x i = l,t s, xd ˆK n i π i λ i s, x i = l ˆKn Ān s l i j, ; Ān s l j, t s l j, j= l ˆKn π i λ i s l i j, ; π i λ i s l j, t s l j, where ˆKn i is defined similar to ˆKi in 5.37 with ˆK i replaced by ˆK n i. By the wea convergence of ˆK n i to ˆK i in D D as n, we easily obtain that, for i =,..., I, Q n 2,i,l j= f.d.d. ˆQ 2,i,l as n, where f.d.d. stands for the convergence in finite-dimensional distributions. By noting that Q n 2,i,l, i =,..., I, and A n are independent from each other, together with 5.29, we have Ân, ˆQ n, Q n 2,,l,..., Q n 2,I,l f.d.d. Â, ˆQ, ˆQ 2,,l,..., ˆQ 2,I,l as n. In order to establish the joint convergence of Ân, ˆQ n and ˆQ n 2,i in finite-dimensional distributions, i =,..., I, i.e., Ân, ˆQ n, ˆQ n 2,,..., ˆQ n 2,I f.d.d. Â, ˆQ, ˆQ 2,,..., ˆQ 2,I as n, 5.38 it is sufficient to show the following: for any T > and ɛ >, lim P n sup ˆQ n 2,i,l t Q n 2,i,l t > ɛ =, i =,..., I, 5.39 t T and, for t > and ɛ >, lim lim P ˆQ n l n 2,i,l t ˆQ n 2,i > ɛ =, i =,..., I. 5.4 We can easily obtain 5.39 from 5.3 and 5.32, as well as the continuity of ˆKi, i =,..., I. Following the proof of [3, Lemma 5.3], we immediately see that 5.4 also holds. Therefore, we have shown By the continuous mapping theorem, together with 5.33 and δ = /2, we further have Ân, ˆQ n, ˆQ n 2 f.d.d. Â, ˆQ, ˆQ 2 as n. Since ˆQ n : n } and ˆQ n 2 : n } are tight as previously shown, we have established the wea convergence of ˆQ n joint with Ân when δ = /2 and α. Furthermore, by noting 3. and 3.2, as well as the continuous mapping theorem, we obtain the wea convergence of Ân, ˆQ n, ˆD n jointly. The case when δ = α/2 and < α < can be obtained analogously by noting that the limit ˆQ n 2 vanishes as n. Therefore, the proof of Theorem 3. is complete.

17 AN FCLT FOR MARKOV ADDITIVE ARRIVAL PROCESSES AND ITS APPLICATIONS 7 Proof of Corollary 3.. The covariance functions of ˆQ can be obtained similarly to [3, Lemma 5.] in combination with Itô isometry as well as the fact that the Kiefer processes ˆK i, with i =,..., I, and the arrival limit  are independent of each other. The covariance functions of ˆD can also be derived similarly. We omit the details here for brevity Proofs for applications to for-join networs. Proof Setch of Theorem 4.. We first note that the processes Q n, Y n An t Q n t = Y n = l= Ant t = = S n t = = l= A n t l= τ n l + ηl,i > txn τ n l = i R K + A n s s + x i > txn s = id η l,i x i, τ n l + ηl,i R K + l= t τ n l + ηl,i j t, jx n τ n l = i and S can be represented as A n s s + x i t s + xi j t, jx n s = id η l,i x i, τ n l + ηl,i j t, jx n τ n l = i R K + A n s s + x i j t, jx n s = id η l,i x i. l= l= Then we can obtain the representations for the diffusion-scaled processes ˆQ n, Ŷ n and Ŝn as follows: ˆQ n t = Ŷ n t = Ŝ n t = G i t sxn s = idân s + n /2 δ F i + n /2 δ R K + s + x i > td ˆK n i Ān i s, x i, 5.4 i t s F m t sx n s = idân s R K + s + x i t s + xi j t, jd ˆK n i Ān i s, x i, 5.42 F i m t sx n s = idân s + n /2 δ R K + s + x i j t, jd ˆK n i Ān i s, x i, 5.43

18 8 HONGYUAN LU, GUODONG PANG, AND MICHEL MANDJES where the processes A n i, i =,..., I, are defined in 5.3, and the multiparameter sequential empirical processes ˆK n i = ˆK i nt, x : t, x RK + } are defined by ˆK n i t, x := nt η l,i x F i x, t, x R K n +. l= The wea convergence of the first terms in follows analogously from the proof for 5.25 in Theorem 3.. Note from [4, Thm. 3.] and the independence of ˆK n i, i =,..., I, that ˆK n i ˆK i in D[,, D K as n, 5.44 where ˆK i, i =,..., I, are independent generalized Kiefer processes with covariance functions in Theorem 4.. With similar argument to the proof in Theorem 3. and [4, Section 6.2], we can also show the wea convergence of the second terms in , as well as the joint convergence of Â, ˆQ, Ŷ, Ŝ. The details are omitted for brevity. Proof of Corollary 4.. The covariance functions of ˆQ j t and Ŷt, and Ŝt and Ŝt are analogous to [4, Theorem 3.4] for j, =,..., K, and t, t, together with the fact that the generalized Kiefer processes ˆK i s are independent, i =,..., I. We omit the details for brevity. 6. Concluding Remars We have studied a large class of MAAPs that can capture more burstiness and variabilities than MMPPs. Under mild conditions on the parameters, we have established an FCLT for the MAAPs. The FCLT is applied to non-marovian infinite-server systems and for-join networs with NES. It can be also similarly applied to obtain two-parameter heavy-traffic limits for infinite-server systems as in [2]. The FCLT can be potentially applied to study large-scale service systems in Marov random environments, for example, queueing networs in which all stations are modulated by the same Marov process. The results can be also used to study resource allocation and system design problems for such queueing and networ models. It may be also interesting to study the sample-path large deviation problems for queueing systems with MAAPs. Acnowledgments. Hongyuan Lu and Guodong Pang acnowledge the support from the NSF grant CMMI Michel Mandjes acnowledges the support from Gravitation project NETWORKS, grant number , funded by the Netherlands Organization for Scientific Research NWO. References [] D. Anderson, J. Blom, M. Mandjes, H. Thorsdottir, and K. de Turc. 26 A functional central limit theorem for a Marov-modulated infinite-server queue. Methodology and Computing in Applied Probability. Vol. 8, No., [2] S. Asmussen. 23 Applied Probability and Queues. 2nd edition. Springer, Berlin. [3] M. Bayal-Gursoy and W. Xiao. 24 Stochastic decomposition in M/M/ queues with Marov modulated service rates. Queueing Systems. Vol. 48, No., [4] J. Blom, O. Kella, M. Mandjes, and H. Thorsdottir. 24 Marov-modulated infinite-server queues with general service times. Queueing Systems. Vol. 76, No. 4, [5] J. Blom, M. Mandjes, and H. Thorsdottir. 23 Time-scaling limits for Marov-modulated infinite-server queues. Stochastic Models. Vol. 29, No., [6] J. Blom, K. de Turc, and M. Mandjes. 25 Analysis of Marov-modulated infinite-server queues in the central-limit regime. Probability in the Engineering and Informational Sciences. Vol. 29, No. 3, [7] J. Blom, K. de Turc, and M. Mandjes. 26 Functional central limit theorems for Marov-modulated infiniteserver systems. Mathematical Methods of Operations Research. Vol. 83, No. 3, [8] P. Billingsley. 29 Convergence of Probability Measures. Wiley, New Yor. [9] B. D Auria. 27 Stochastic decomposition of the M/G/ queue in a random environment. Operations Research Letters. Vol. 35, No. 6,

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