Many-server diffusion limits for G/Ph/n + M queues J. G. Dai
1
and Tolga Tezcan
2
December 10, 2008 Abstract This paper studies many-server limits for G/P h/n+M queues, which have a phasetype service time distribution and exponential patience time distribution. We prove a pair of diffusion-scaled customer-count and server-allocation processes, properly centered, converges in distribution to a diffusion process as the number of servers n goes to infinity. When the queues are critically loaded, the diffusion limit generalizes a result by Puhalskii and Reiman (2000) for G/P h/n queues without customer abandonment. When the queues are overloaded, the diffusion limit provides a refinement to a fluid limit; a fluid model based on the fluid limit has been demonstrated to be effective for performance analysis for overloaded M/M/n + M queues in Whitt (2004). The proof techniques employed in this paper are innovative. First, a perturbed system is shown to be equivalent to the original system. Next, two maps are employed in both the fluidand diffusion-scalings. These maps allow one to prove the limit theorems by applying the standard continuous-mapping theorem and the random-time-change theorem.
Keywords: multi-server queues, customer abandonment, phase-type distribution, manyserver heavy traffic, Halfin-Whitt regime, quality- and efficiency-driven regime, efficiencydriven regime AMS Subject Classification (2000): 90B20, 68M20, 60J70
1
Introduction
This paper studies diffusion limits for multi-server queues. The limits are for a pair of diffusion-scaled customer-count and server-allocation processes in the G/P h/n + M model, where the service-time distribution is phase-type and the customer patience-time distribution is exponential. We consider two separate parameter regimes: one for the critically loaded system and the other for the overloaded system. As argued in the seminal paper of Halfin and Whitt (1981), in the critical regime, the system provides high quality of service and at the same time achieves high server utilization. Thus, this parameter regime is also known as the Quality- and Efficiency-Driven (QED) limiting regime or the Halfin-Whitt limiting regime. For the overloaded M/M/n + M model, Whitt (2004) demonstrates that a certain fluid approximation can be useful in predicting steady-state performance of the 1
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, Email:
[email protected]; research supported in part by NSF grants CMMI-0727400 and CMMI-0825840, and by an IBM Faculty Award 2 Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, Email:
[email protected]
1
multi-server system. He further demonstrates that a diffusion limit provides a refined approximation. In the critical regime, Halfin and Whitt (1981) is the first paper to establish a diffusion limit for the GI/M/n model. Puhalskii and Reiman (2000) establishes a diffusion limit for the GI/P h/n model, where service time distribution is phase-type. Garnett et al. (2002) proves a diffusion limit for the M/M/n + M model, which allows customer abandonment. Whitt (2005) generalizes the result to the G/M/n + M model. In the same paper, Whitt proves a stochastic-process limit for the G/H2∗ /n model; this limit is not a diffusion process. Our first result, Theorem 1 and Corollary 1, extends the result of Puhalskii and Reiman (2000) to the G/P h/n + M model, which allows customer abandonment. This result also extends the work of Garnett et al. (2002) and Whitt (2005) to allow phase-type service time distribution. Our limit process is a multi-dimensional diffusion process. For the overloaded G/P h/n + M model, we establish a diffusion limit in Theorem 2. This limit is also a multi-dimensional diffusion process. It provides a refinement to a fluid limit, and it generalizes Whitt (2005) to the G/P h/n + M model. In addition to the results in Theorems 1 and 2, the techniques used in the proofs of these two theorems are innovative. In our proofs we first establish a sample-path representation for our G/P h/n + M model. This representation then allows us to obtain the customer-count and server-allocation processes as a map of primitive processes with a random-time-change. These primitive processes are either assumed or known to satisfy the functional central limit theorems (FCLTs). Therefore, our diffusion limits follow from the standard continuous-mapping theorem and the standard random-time-change theorem; see, for example, Ethier and Kurtz (1986). Halfin and Whitt (1981), Garnett et al. (2002), and Whitt (2004) all use Stone’s theorem to prove diffusion limit theorems in the critical regime. Stone (1963) is set up for the convergence of Markov chains to a diffusion process. This setting makes the generalization to general arrival processes such as non-renewal processes difficult. Puhalskii and Reiman (2000) also uses the continuous-mapping approach for the G/P h/n model. They employ a different sample-path representation for the customer-count process. Their representation requires them to use extensively the martingale FCLTs in their proofs, whereas our approach uses the standard FCLTs for random walks and Poisson processes. Pang et al. (2007) reviews a number of sample-path representations and martingale proofs for manyserver heavy traffic limits, and Whitt (2007) surveys the proof techniques for establishing martingale FCLTs. Our proofs show that for multi-server queues with phase-type service time distribution, there is a general approach to proving limit theorems, without employing martingale FCLTs. Our sample-path representation is based on the equivalence of our multi-server system to a perturbed system as illustrated in Tezcan (2006). This representation has been used in Dai and Tezcan (2005), Tezcan and Dai (2008) and Dai and Tezcan (2008) for multi-serverpool systems when the service time and patience time have exponential distributions. The sample-path argument has been explored previously in the setting of Markovian networks 2
in Mandelbaum et al. (1998) for strong approximations and Mandelbaum and Pats (1998) for general state-dependent networks. In our continuous-mapping approach, we have heavily explored some maps from Dd to d D , where for an integer d > 0, Dd is the space of right continuous paths in Rd that have left limits in (0, ∞). Variants of these maps have been employed in the literature. See, for example, Mandelbaum et al. (1998), Reed (2007), Tezcan and Dai (2008), Dai and Tezcan (2008), and Pang et al. (2007). We explore these maps not just in diffusion-scaling but also in fluid-scaling. Using a map twice, one for each scaling, allows us to obtain diffusion limits as a simple consequence of the standard continuous-mapping theorem and randomtime-change theorem. In the seminal paper, Reiman (1984) proves a conventional heavy traffic limit theorem for generalized Jackson networks. Our approach resembles the work of Johnson (1983), who explores a multidimensional Skorohod map also twice and provides a significant simplification of Reiman’s original proof. For the G/GI/n model, Reed (2007) proves a many-server limit for the customer-count process in the critical regime; his assumption on service-time distribution is completely general, and his limit is not a Markov process. For the overloaded G/GI/n model, Puhalskii and Reed (2008) proves a finite-dimensional-distribution limit for the one-dimensional customer-count process. Jelenkovic et al. (2004) proves a limit theorem for the GI/D/n model. Gamarnik and Momcilovic (2007) studies a many-server limit of the steady-state distribution of the GI/GI/n model, where the service time distribution is lattice-valued with a finite support. When the service time distribution is general, measure-valued processes have been used to give a Markovian description of the system. Kaspi and Ramanan (2007) obtains a measure-valued fluid limit for the G/GI/n model. Kang and Ramanan (2008) and Zhang and Dai (2008) independently obtain measure-valued fluid limits for the G/GI/n + G model; their work justify the fluid model proposed in Whitt (2006). Dai and He (2008) extends the work in this paper to the G/P h/n + G queue in the Halfin-Whitt regime. In Section 2, we introduce the phase-type distributions and the G/P h/n + M queue. The main results, Theorems 1 to 4 and a corollary, are stated in Section 3. In Section 4 we introduce a perturbed system that is equivalent to the G/P h/n + M queue and derive the dynamical equations that the perturbed system must satisfy. Fluid limit result is stated in Theorem 5, which is proved in Section 5. It is needed to we prove the diffusion limits in Section 6.
2
A G/P h/n + M queue
We first introduce a G/GI/n + M queue in Section 2.1. We then define the G/P h/n + M queue by restricting the service time distribution to be phase-type. Phase-type distributions are defined in Section 2.2.
3
2.1
A G/GI/n + M queue
A G/GI/n queue is a classical stochastic system that has been extensively studied in literature. In such a system, there are n identical servers. The service times {vi } are a sequence of independent, identically distributed (iid) random variables, where vi is the service time of the ith customer entering service. The service time distribution is general (the GI in the G/GI/n notational system), although for the rest of this paper we restrict it to phase-type service time distributions. The arrival process E = {E(t), t ≥ 0} is assumed to be general (the first G), where E(t) denotes the number of customer arrivals to the system by time t. Upon his arrival to the system, a customer gets into service immediately if there is an idle server; otherwise, he waits in a waiting buffer that holds a first-in-firstout (FIFO) queue. The buffer size is assumed to be infinite. When a server finishes a service, the server removes the leading customer from the waiting buffer and starts to serve the customer; when the queue is empty, the server begins to idle. Each customer has a patience time; when a customer’s waiting time in queue exceeds his patience time, the customer abandons the system without any service. Retrial is not modeled in this paper. We assume the patience times are a sequence of iid random variables that have an exponential distribution. Thus our model is a G/GI/n + M system, where +M signifies the exponential patience time distribution. We use α to denote the rate of the exponential patience distribution; thus 1/α is the mean patience time. Let m be the mean service time. Thus µ = 1/m is the mean service rate. We focus on systems when the arrival rate is high. Specifically, we consider a sequence of G/GI/n + M systems indexed by n. Thus, E n is the arrival process of the nth system, while the service time and patience time distributions do not change with n. We assume arrival rate λn → ∞ as the number of servers n goes to infinity. We further assume that λn = λ > 0, n→∞ n lim
lim
√
n→∞
n(λn /n − λ) = −βµ
for some β ∈ R,
(2.1)
and ˜n ⇒ E ˜ E
as n → ∞,
(2.2)
˜ is a Brownian motion, and where E ˜ n (t) = √1 E ˆ n (t), E n For future purposes, let ρn =
λn nµ
ˆ n (t) = E n (t) − λn t, E
and
ρ=
t ≥ 0.
λ . µ
Because customer abandonment is allowed, it is not necessary to assume ρn < 1 or ρ ≤ 1. Condition (2.1) implies that √ lim n(ρ − ρn ) = β. n→∞
4
When ρ = 1, the sequence of systems is critically loaded in the limit, and is said to be in the Quality- and Efficiency-Driven (QED) regime or Halfin-Whitt regime. When ρ > 1, the sequence of systems is overloaded, and is said to be in the Efficiency-Driven (ED) regime. Our focus is both the QED and ED regimes.
2.2
Phase-type distributions
In this section, we introduce a phase-type distribution. Such a distribution is assumed to have K ≥ 1 phases. The set of phases is assumed to be K = {1, . . . , K}. Each phase-type distribution has a set of parameters p, ν, and P , where p is a K-dimensional vector of nonnegative numbers whose sum is equal to 1, ν is a K-dimensional vector of positive numbers, and P is a K × K sub-stochastic matrix. We assume that the diagonals of P are zero, namely, Pii = 0 for i = 1, . . . , K, (2.3) and P is transient, namely, I − P is invertible.
(2.4)
(All vectors are envisioned as column vectors unless specified otherwise.) A (continuous) phase-type random variable v is defined as the time until absorption in an absorbing state of a continuous-time Markov chain. With p, ν, and P , the continuous-time Markov chain can be described as follows. It has K + 1 states, 1, 2, . . . , K, K + 1, with state K + 1 being the absorbing state. The rate matrix (or generator) of the Markov chain is F h , G= 0 0 where F = diag(ν)(P − I) is a K × K matrix, h = −F e is a K-dimensional vector, and e is the K-dimensional vector of ones. Definition 1. A continuous phase-type random variable v with parameters p, ν, and P , denoted as v ∼ Ph(p, ν, P ), is defined to be the first time until the continuous-time Markov chain with initial distribution p and rate matrix G to reach state K + 1. Note that given condition (2.3), the rate matrix G and (ν, P ) are uniquely determined from each other. It is well known (see, for example, Latouche and Ramaswami (1999)) that P{v ≤ x} = 1 − p0 exp(F x)e, x ≥ 0. For our purposes, we provide an alternative way to sample a Ph(p, ν, P ) random variable. We first sample a sequence of phases k1 , k2 , . . ., kL in {1, 2, . . . , K}
5
(2.5)
as follows. We sample phase k1 following distribution p on K. Assume k1 ∈ K, . . . , ki ∈ K have been sampled. Setting j = ki , sample a phase from {1, . . . , K, K + 1} following a distributionPthat is determined by the jth row of P ; the probability of getting phase K + 1 is 1 − K `=1 Pj` ≥ 0. The resulting phase is denoted by ki+1 . When ki+1 = K + 1, terminates the process and set L = i. Otherwise, continue the sampling process. Because the matrix P is assumed to be transient, one has L < ∞ almost surely. Let ξ1 , ξ2 , . . ., ξL be independently sampled from exponential distributions with rate νk1 , . . ., νkL , respectively. Then L X v= ξi . (2.6) i=1
3
Main results
We assume that the service times {vi } follow a phase-type distribution Ph(p, ν, P ). Each customer’s service time can be decomposed into a number of phases as in (2.6). Thus, when a customer is in service, it must be in one of the K phases of service. Let Zkn (t) denote the number of customers in phase k service in the nth system at time t; the service time in phase k is exponentially distributed with rate νk . We use Z n (t) to denote the corresponding Kdimensional vector. (All vectors are envisioned as column vectors unless stated otherwise, and for a vector v = (vk ) ∈ Rd , |v| = maxk |vk |.) We call Z n = {Z n (t), t ≥ 0} the serverallocation process. Let N n (t) denote the number of customers in the nth system at time t, either in queue or in service. Setting X n (t) = N n (t) − n
(3.1)
for t ≥ 0, we call X n = {X n (t), t ≥ 0} the customer-count process in the nth system. One can check that (X n (t))+ is the number of customers waiting in queue at time t, and (X n (t))− is the number of idle servers at time t, where for a real number a, a+ = max(a, 0) and a− = max(−a, 0). Clearly, e0 Z n (t) = n − (X n (t))−
(3.2)
for t ≥ 0, where, as before, e is the vector of ones. The processes X n and Z n describe the “state” of the system as time evolves. Hereafter, they are called state processes for the nth system. They satisfy constraint (3.2). Assume that ρ ≥ 1. Intuitively, when n is large, all n servers are 100% busy, and there should be nq customers on average waiting in the buffer, where q = (λ − µ)/α.
(3.3)
(An intuitive explanation of the formula (3.3) will be given at the end of this section.) The
6
n customers in service are distributed among K phases following a distribution γ, where γ = µR−1 p,
(3.4)
0
(3.5)
R = (I − P ) diag(ν).
P One can check that K k=1 γk = 1, and one interprets γk to be the fraction of phase k load on the n servers. The preceding paragraph suggests the following centering for the state processes: ˆ n (t) = X n (t) − nq X
and Zˆ n (t) = Z n (t) − nγ
for t ≥ 0. Define the corresponding diffusion-scaled processes ˆ n (t) ˜ n (t) = √1 X X n
1 and Z˜ n (t) = √ Zˆ n (t) n
for t ≥ 0. We assume that ˜ n (0), Z˜ n (0)) ⇒ (X(0), ˜ ˜ (X Z(0))
as n → ∞
˜ ˜ for some random variables (X(0), Z(0)) that satisfy ( − ˜ −(X(0)) if q = 0, ˜ e0 Z(0) = 0 if q > 0.
(3.6)
(3.7)
˜ ˜ The random variables (X(0), Z(0)) are assumed to be defined on some probability ˜ ˜ ˜ ˜ G, ˜ S, ˜ Φ ˜ 0 , . . ., and Φ ˜K space (Ω, F, P), which is rich enough so that stochastic processes E, ˜ ˜ ˜ and G ˜ are are defined on this space and they are independent of (X(0), Z(0)). Here, E 0 K ˜ ˜ ˜ one-dimensional driftless Brownian motions, and S, Φ , . . ., and Φ are K-dimensional driftless Brownian motions. These Brownian motions, possibly degenerate, are mutually ˜ is λc2a for some constant c2a ≥ 0, the independent and start from 0; the variance of E ˜ ˜ Φ ˜ 0 , . . ., and Φ ˜ K are diag(ν), H 0 , variance for G is α, and the covariance matrices for S, K k . . ., and H , respectively, where the K × K matrix H is given by ( pki (1 − pkj ) if i = j, k (3.8) Hi,j = otherwise −pki pkj with p0 = p and pk being the kth column of P 0 . To state the main theorems of this paper, let ˜ (t) = X(0) ˜ ˜ − µβt + e0 M ˜ (t) − G(qt), ˜ U + E(t) ˜ ˜ 0 (µt) + (I − pe0 )M ˜ (t), V˜ (t) = (I − pe0 )Z(0) +Φ
7
(3.9) (3.10)
where ˜ (t) = M
K X
˜ k (νk γk t) − (I − P 0 )S(γt) ˜ Φ
(3.11)
k=1
˜ , V˜ ) is a (K + 1)-dimensional Brownian motion. Our first result for t ≥ 0. The process (U is for critically loaded systems. Theorem 1. Assume (2.1), (2.2), (3.6) and (3.7) hold. If ρ = 1, then ˜ n , Z˜ n ) ⇒ (X, ˜ Z), ˜ (X as n → ∞, where ˜ Z) ˜ = Φ(U ˜ , V˜ ), (X,
(3.12)
and Φ maps (u, v) ∈ D × DK to (x, z) ∈ D × DK via Z t Z t x(t) = u(t) − α (x(s))+ ds − e0 R z(s) ds, 0 0 Z t − 0 z(t) = v(t) − p(x(t)) − (I − pe )R z(s) ds
(3.13) (3.14)
0
for t ≥ 0. Our next result is for overloaded systems. Theorem 2. Assume (2.1), (2.2), (3.6) and (3.7) hold. If ρ > 1, then ˜ n , Z˜ n ) ⇒ (X, ˜ Z), ˜ (X as n → ∞, where ˜ Z) ˜ = Ψ(U ˜ , V˜ ), (X, and Ψ maps (u, v) ∈ D × DK to (x, z) ∈ D × DK via Z t Z t 0 x(t) = u(t) − α x(s) ds − e R z(s) ds, 0 0 Z t 0 z(t) = v(t) − (I − pe )R z(s) ds
(3.15)
(3.16) (3.17)
0
for t ≥ 0. ˜ Z) ˜ as (K + 1)-dimensional diffusion processes Equations (3.12) and (3.15) define (X, (possibly degenerate). The maps Φ and Ψ are well defined and they enjoy a number of nice properties as stated in the following lemma, which is a corollary of Lemma 4 in the appendix. 8
Lemma 1. (a) For each (u, v) ∈ D×DK , there exists a unique (x, z) ∈ D×DK that satisfies (3.13)-(3.14), and hence the map Φ is well defined. Similarly, for each (u, v) ∈ D × DK , there exists a unique (x, z) ∈ D × DK that satisfies (3.16)-(3.17), and hence the map Ψ is well defined. (b) Both maps Φ and Ψ are continuous when both the domain and the range D × DK are endowed with the standard Skorohod J1 -topology when they are viewed as the path space D(R+ , RK+1 ). (c) Both maps are Lipschitz continuous in the sense that for any t > 0, there exists a constant Ct such that sup |Φ(y 1 )(s) − Φ(y 2 )(s)| ≤ Ct sup |y 1 (s) − y 2 (s)|, 0≤s≤t
(3.18)
0≤s≤t
sup |Ψ(y 1 )(s) − Ψ(y 2 )(s)| ≤ Ct sup |y 1 (s) − y 2 (s)| 0≤s≤t
(3.19)
0≤s≤t
(3.20) for any y 1 , y 2 ∈ DK+1 . (d) Both maps are positively homogeneous in the sense that Φ(ay) = aΦ(y)
and
Ψ(ay) = aΨ(y)
(3.21)
for each a > 0 and each y ∈ DK+1 . Suppose that each customer, including those initial customers, samples his first service phase that is yet to enter following distribution p at his arrival time to the system. One then can stratify the customers in the waiting buffer according to their first service phases. We use Qnk (t) to denote the number of waiting customers at time t whose first service will be in phase k. If phase k is not a first service phase for any customer, Qnk (t) = 0 for t ≥ 0. Recall that (X n (t))+ is the number of waiting customers at time t. The following result says that for critically loaded systems, the waiting customers are distributed among phases following distribution p. It is known as the state state collapse (SSC) result. Theorem 3. Assume (2.1), (2.2), (3.6) and (3.7) hold. If ρ = 1, then sup n−1/2 |Qn (s) − p(X n (s))+ | ⇒ 0 0≤s≤t
as n → ∞. Let Fkn be the number of the phase k customers in the system at time t, either waiting or in service. Setting ˆ n (t) = Qn (t) − npk q, Fˆkn (t) = Fkn (t) − n(pk q + γk ), Q k k ˜ n (t) = √1 Q ˆ n (t), and F˜ n (t) = √1 Fˆ n (t) Q k k n k n k
(3.22) (3.23)
˜ n (t) + Z˜ n (t) for t ≥ 0. The following result is a corollary to for t ≥ 0, clearly, F˜kn = Q k k Theorems 1 and 3. When there is no customer abandonment and the arrival process is renewal, it is identical to Theorem 2.3 of Puhalskii and Reiman (2000). 9
Corollary 1. Assume (2.1), (2.2), (3.6) and (3.7) hold. If ρ = 1, then F˜ n ⇒ F˜ , where + ˜ ˜ F˜ (t) = p(X(t)) + Z(t)
(3.24)
for t ≥ 0. The process F˜ is a diffusion process satisfying ˜ 0 (t) + pE(t) ˜ +M ˜ (t) F˜ (t) = F˜ (0) − βµpt + Φ Z t Z t −R F˜ (s) ds + (R − αI)p (e0 F˜ (s))+ ds 0
for t ≥ 0.
0
When ρ > 1, there is no SSC result for Qn or F n . We can still prove F˜ n ⇒ F˜ for some continuous process F˜ , but F˜ is not a diffusion process. Probably, one can prove ˜ and the latter is a diffusion process. Compared with our state that (F˜ n , Z˜ n ) ⇒ (F˜ , Z), n n processes X and Z , F n is a less significant performance process. Thus, we choose not to further pursue its limit in this paper. Our final theorem is concerned with the virtual waiting time process W n = {W n (t), t ≥ 0}. Here, W n (t) is the potential waiting time of a hypothetical, infinitely patient customer who arrives at the queue at time t. When there is no customer abandonment and the arrival process is renewal, the theorem is implied by Corollary 2.3 and Remark 2.6 of Puhalskii and Reiman (2000). Theorem 4. Assume (2.1), (2.2), (3.6) and (3.7) hold. If ρ = 1, then √
nW n ⇒
˜ + (X) . µ
(3.25)
as n → ∞. All proofs of the results stated in this section will be proved in Section 6. We end this section by providing an intuitive explanation of (3.3); readers are referred to Whitt (2004) for further discussion. Intuitively, λ − µ is the number of customers per unit of time that the system must “delete” in order for the system to reach an equilibrium. While in equilibrium, each customer in queue abandons at rate α, and collectively all q customers in queue abandon at rate qα customers per unit of time. Thus, one should have qα = λ − µ, which leads to (3.3).
4
System representation
In this section, we first describe a perturbed system, and show that this perturbed system is equivalent to our original system. We then develop the dynamical equations that the perturbed system must satisfy. 10
4.1
A perturbed system
Now we describe a perturbation of the G/P h/n + M queue specified in Section 2. In the perturbed system, each phase has a service queue for customers “in service”. Only the leading customer in the service queue is actually in service; all others are waiting in the service queue, which is ordered according to FIFO discipline. We use Zk∗ (t) to denote the number of customers in phase k service queue at time t. (Star-version quantities are associated with the perturbed system; the corresponding quantities in the original system are denoted by the same symbols without the star.) Each customer in the service queue is attached to exactly one server. Thus, there are exactly Zk∗ (t) servers in phase k at time t. All these Zk∗ (t) servers simultaneously work on the leading customer in the service queue. The service effort received by the leading customer is additive, proportional to the number of servers working on the customer. Each customer has a service requirement for each phase that he visits, with phase k service requirement being exponentially distributed with mean 1/νk . When the total service effort spent on a customer reaches his service requirement in a phase, the service in the phase is completed. When a customer completes a phase k service, the customer immediately moves to the next phase following a sampling procedure to be specified below, taking with him the associated server. If the service queue in the new phase is not empty at their arrival, the server joins the service immediately, collaborating with other servers who are already in service to work on the leading customer in the service queue. The newly arrived customer joins the end of the service queue. If the new service queue is empty, the server works on her customer who is the only one in the new phase of service. When a customer finishes a phase k service, it uses the kth row of P to sample the next phase of service P to join among phases {1, . . . , K, K + 1}; the probability of selecting phase K + 1 is 1 − K `=1 Pk` . If ` ∈ {1, . . . , K} is selected, both the server and the customer move next to phase `. If K +1 is selected, the customer exits the system and the associated server is released. The released server checks the FIFO real queue to select the next customer to work on if the real queue is not empty. The selected customer is attached to the server until the customer exits the system. If the real queue is empty, the server becomes idle. At a customer’s arrival time to the system, if there is an idle server, the customer grabs a free server and is attached to the server until the customer exits the system. Together, they move into the customer’s first phase of service, which is selected according to distribution p. The service and waiting mechanism is identical to the one described in the previous paragraph. If all the servers are busy at the customer’s arrival time, the customer joins the end of the FIFO real queue. Only the last customer in the FIFO queue can abandon the system. After he joins the queue, if the queue size is k, his patience time is exponentially distributed with rate kα. All other customers become infinitely patient. For each n fixed, now we show that when the arrival process E n is a renewal process, the perturbed system and the original system are equivalent in a precise mathematical
11
sense. For that, recall that the waiting buffer in the perturbed system maintains a FIFO queue for waiting customers. Let Q∗ (t) = (i1 , i2 , . . . , iL∗ (t) ), where L∗ (t) is the total number of customers waiting in queue at time t, and i` is the first service phase that the `th customer is yet to enter. Let ξ(t) be the remaining interarrival time at time t. (ξ has no star because the arrival processes in the perturbed system and in the original system are identical.) It follows that, {(ξ(t), Q∗ (t), Z ∗ (t)), t ≥ 0} is a continuous-time Markov process living in state space ∞ is the space of finite sequences taking values R+ × {1, . . . , K}∞ × ZK + , where {1, . . . , K} in {1, . . . , K}. Let {(ξ(t), Q(t), Z(t)), t ≥ 0} be the corresponding process in the original system. The process is also a continuous-time Markov process. Because at any time t, the phase k service rate is Zk∗ (t)νk in the perturbed system and Zk (t)νk in the original system, one can check that the two Markov processes {(ξ(t), Q(t), Z(t)), t ≥ 0}
and
{(ξ(t), Q∗ (t), Z ∗ (t)), t ≥ 0}
have the same generator. Thus, when they have the same initial distribution, they have the same distribution for the entire processes. In the following, we always choose the initial condition of the perturbed system to be identical to the original initial condition. Even if the arrival process is not a renewal process, the perturbed system can still have the same distribution as the original system. The rest of the paper does not require the arrival process to be renewal, as long as it satisfies (2.1)-(2.2) and the requirement that the perturbed system has the same distribution as the original one. See Tezcan (2006) for more general treatment of perturbed systems.
4.2
System equations
From now on, we focus on the perturbed system and drop the stars attached to its quantities. In this section, we describe the dynamical equations that the system must obey. For k = 1, . . . , K, let φk = {φk (j) : j = 1, 2, . . .} be a sequence of iid “Bernoulli random vectors”. For each j, φk (j) takes vector e` with probability Pk` , where e` is the K-dimensional vector with `th component 1 and all other components 0. Similarly, let φ0 = {φ0 (j), j = 1, 2, . . . , } be a sequence of iid Bernoulli random vectors; the probability that φ0 (j) = e` is p` . For k = 0, 1, . . . , K, define the routing process Φk (N ) =
N X
φk (j),
N = 1, 2, . . . .
j=1
For each k = 1, . . . , K, let Sk be a Poisson process with rates νk , and let G be a Poisson process with rate α. We assume that X n (0), E n , G, S1 , . . ., SK , Φ0 , . . ., and ΦK are mutually independent. 12
(4.1)
Let Tkn (t) be the cumulative amount of service effort received by customers in phase k service by time t, B n (t) be the cumulative number of customers who have entered service in (0, t], and Dn (t) be the cumulative number of customers who have completed service in (0, t]. Clearly, Z t Zkn (s) ds, t ≥ 0. (4.2) Tkn (t) = 0 n Then Sk (Tk (t)) is the cumulative number of phase k service completions by time t. Also Rt G 0 (X n (s))+ ds) is the cumulative number of customers who have abandoned the system by time t. One can check that the processes X n and Z n satisfy the following dynamical
equations: for t ≥ 0, n
n
0
n
Z (t) = Z (0) + Φ (B (t)) +
K X
Φk (Sk (Tkn (t))) − S(T n (t)),
(4.3)
k=1
X n (t) = X n (0) + E n (t) − Dn (t) − G
Z
t
(X n (s))+ ds ,
(4.4)
0
Dn (t) =
K X
Sk (Tkn (t)) − e0 Φk (Sk (Tkn (t)))
k=1 0
= −e
K X
Φk (Sk (Tkn (t))) − S(T n (t)) ,
k=1 n
e0 Z n (t) = e0 Z (0) + B n (t) − Dn (t),
(4.5) (4.6)
where n S(T n (t)) = (S1 (T1n (t)), . . . , SK (TK (t)))0 .
4.3
State process representation
Define centered processes ˆ = S(t) − νt, S(t)
ˆ = G(t) − αt, G(t)
ˆ k (N ) = Φ
N X
φk (j) − pk ,
k = 0, 1, . . . , K,
j=1
for t ≥ 0 and N = 1, 2, . . ., where, p0 = p and pk is the kth column of P 0 , k = 1, . . . , K, and ν = (ν1 , . . . , νK )0 . Setting M n (t) =
K X
ˆ k (Sk (T n (t))) − (I − P 0 )S(T ˆ n (t)), Φ k
k=1
13
(4.7)
one then has K X
Φk (Sk (Tkn (t))) − S(T n (t)) = M n (t) − R
t
Z
Z n (s) ds,
0
k=1
where R is defined in (3.5). From (4.5), 0
n
t
Z
0
n
Z n (s) ds.
D (t) = −e M (t) + e R
(4.8)
0
It follows from (4.3)-(4.8) and (3.2) that ˆ 0 (B n (t)) − p(X n (t))− Z n (t) = Z n (0) + p(X n (0))− + Φ Z t 0 n 0 Z n (s) ds, + (I − pe )M (t) − (I − pe )R 0 Z t n n n n 0 n 0 ˆ Z n (s) ds X (t) = X (0) + E (t) + λ t + e M (t) − e R 0 Z t Z t ˆ −G (X n (s))+ ds − α (X n (s))+ ds. 0
0
Recall that Zˆ n (t) = Z n (t) − nγ. We then have ˆ 0 (B n (t)) − p(X n (t))− Zˆ n (t) = (I − pe0 )Zˆ n (0) + Φ Z t 0 n 0 + (I − pe )M (t) − (I − pe )R Zˆ n (s) ds, 0
ˆ n (t) + (λn − nµ)t + e0 M n (t) − e0 R X (t) = X (0) + E Z t Z t ˆ −G (X n (s))+ ds − α (X n (s))+ ds, n
n
0
Z
t
Zˆ n (s) ds
0
0
where we have used (3.4) and (3.5) in the derivations. Setting ˆ n (t) + (λn − nµ)t + e0 M n (t) − G ˆ U n (t) = X n (0) + E
Z
t
(X n (s))+ ds ,
(4.9)
0
ˆ 0 (B n (t)) + (I − pe0 )M n (t) V n (t) = (I − pe0 )Zˆ n (0) + Φ
(4.10) (4.11)
for t ≥ 0, we finally have n
n
Z
X (t) = U (t) − α
t n
+
0
t
Z
(X (s)) ds − e R 0
Zˆ n (s) ds,
(4.12)
Zˆ n (s) ds.
(4.13)
0
Zˆ n (t) = V n (t) − p(X n (t))− − (I − pe0 )R
Z 0
14
t
By Lemma 1, we have obtained the following representation for the state processes (X n , Zˆ n ) = Φ(U n , V n ).
5
(4.14)
Fluid limits
¯ n, D ¯ n, E ¯ n , T¯n , X ¯ n , and Z¯ n via Define the fluid-scaled processes B ¯ n (t) = 1 Dn (t), ¯ n (t) = 1 B n (t), D B n n 1 1 n n n ¯ (t) = X n (t), T¯ (t) = T (t), X n n
¯ n (t) = 1 E n (t), E n 1 n Z¯ (t) = Z n (t) n
for t ≥ 0. Theorem 5. Assume (2.1), (2.2) and (3.6). Then ¯ n, D ¯ n, E ¯ n , T¯n , X ¯ n , Z¯ n ) ⇒ (B, ¯ D, ¯ E, ¯ T¯, X, ¯ Z) ¯ (B ¯ = µt, D(t) ¯ ¯ = λt, T¯(t) = γt, X(t) ¯ ¯ = γ for as n → ∞, where B(t) = µt, E(t) = q, and Z(t) t ≥ 0. Proof. Let ¯ n (t) = n−1 M n (t), M
¯ n (t) = n−1 U n (t), U
V¯ n (t) = n−1 V n (t),
¯ n (t) = n−1 Zˆ n (t) L
for t ≥ 0. By (4.14) and the positively homogeneous property of the map Φ, one has ¯ n, L ¯ n ) = Φ(U ¯ n , V¯ n ). (X Setting ¯ (t) = q + (λ − µ)t and V¯ (t) = 0 U
(5.1)
¯ , V¯ ) = (X, ¯ 0). We are going to show that for t ≥ 0, one can check that Φ(U ¯ n, U ¯ n , V¯ n ) ⇒ (0, U ¯ , 0). (M
(5.2)
Assuming (5.2), we now complete the proof of the theorem. The continuity of the map Φ implies that ¯ n, L ¯ n ) = Φ(U ¯ n , V¯ n ) ⇒ Φ(U ¯ , V¯ ) = (X, ¯ 0). (X ¯ n (t) + γ for t ≥ 0, one has Z¯ n ⇒ Z, ¯ from which one has T¯n ⇒ T¯. From Since Z¯ n (t) = L (4.8), one has Z t ¯ n (t) = −e0 M ¯ n (t) + e0 R D Z¯ n (s) ds. 0
15
Rt ¯ ds = µt for t ≥ 0, and thus by the continuous mapping One can argue that e0 R 0 Z(s) n ¯n ⇒ B ¯ and ¯ ⇒D ¯ as n → ∞. The convergence of D ¯ n and (4.6) imply that B theorem, D ¯ B satisfies ¯ ¯ − D(t) ¯ ¯ = e0 Z(0) + B(t) e0 Z(t) ¯ ¯ ¯ for t ≥ 0. Since Z(t) = Z(0) = γ, we conclude B(t) = µt for ≥ 0. By assumptions (2.1) and (2.2), for each t > 0, one has ˆ n | ⇒ 0, n−1 sup |E
(5.3)
0≤s≤t
¯ n ⇒ E. ¯ This proves the theorem when (5.2) holds. which implies that E It remains to prove (5.2). By functional strong law of large numbers (FSLLN) ˆ n−1 sup |G(ns)| ⇒ 0, 0≤s≤t
ˆ k (bnsc)| ⇒ 0, n−1 sup |Φ 0≤s≤t
ˆ n−1 sup |S(ns)| ⇒0
(5.4)
0≤s≤t
for k = 0, 1, . . . , K. As a consequence, {S¯n (t), n ∈ Z+ } is stochastically bounded. This, together with (4.7), (5.4) and the fact that T¯kn (s) ≤ s for s ≥ 0, implies that ¯ n (s)| ⇒ 0 sup |M
(5.5)
0≤s≤t
¯ n (t) ≤ (X ¯ n (0))+ + E ¯ n (t). By (3.6) and the convergence of E ¯ n, as n → ∞. Note that B ¯ n (t), n ∈ Z+ } is stochastically bounded. This and (5.4) imply that {B ˆ 0 (B n (s)) ⇒ 0. sup n−1 Φ
(5.6)
0≤s≤t
Condition (3.6) implies that n−1 Zˆ n (0) ⇒ 0. This, (5.5) and (5.6) imply V¯ n ⇒ 0 as n → ∞. ¯ n (s))+ ≤ (X ¯ n (0))+ + E ¯ n (t), one can argue similarly that U ¯n ⇒ U ¯ . Hence, Since sup0≤s≤t (X we have shown (5.2) holds, and thus proved the theorem.
6
Diffusion limits
˜ n , S˜n , In this section, we prove Theorems 1 to 4. Define the diffusion-scaled processes G 0 K ˜ , . . ., Φ ˜ via Φ ˜ n (t) = √1 G(nt), ˆ G n
1 ˆ S˜n (t) = √ S(nt), n
˜ k,n (t) = √1 Φ ˆ k (bntc), Φ n
k = 0, . . . , K.
By Donsker’s functional central limit theorems, one has ˜ n , S˜n , Φ ˜ 0,n , . . . , Φ ˜ K,n ) ⇒ (G, ˜ S, ˜ Φ ˜ 0, . . . , Φ ˜ K ), (G ˜ is a one-dimensional driftless Brownian motion, and S, ˜ Φ ˜ 0 , . . ., Φ ˜ K are Kwhere G ˜ is α, and the covariance matrix dimensional driftless Brownian motions. The variance of G 16
˜ k is H k , k = 0, . . . , K, where the K × K matrix H k is given by for S˜ is diag(ν), and for Φ (3.8). By FCLT assumption (2.2) for the arrival process E n , initial condition assumption (3.6), and the independence assumption (4.1), one has ˜ n (0), Z˜ n (0), E ˜ n, G ˜ n , S˜n , Φ ˜ 0,n , . . . , Φ ˜ K,n ) ⇒ (X(0), ˜ ˜ ˜ G, ˜ S, ˜ Φ ˜ 0, . . . , Φ ˜K) (X Z(0), E,
(6.1)
˜ G, ˜ S, ˜ Φ ˜ 0, . . . , Φ ˜ K ) are mutually independent, and as n → ∞ and the components of (E, ˜ ˜ they are independent of (X(0), Z(0)). n n ˆ ¯ Let U (t) = U (t) − nU (t), and define the diffusion-scaled processes ˜ n (t) = n−1/2 U ˆ n (t) U
and V˜ n (t) = n−1/2 V n (t)
¯ is defined in (5.1). We now have the following lemma. for t ≥ 0, where U Lemma 2. As n → ∞, ˜ n , V˜ n ) ⇒ (U ˜ , V˜ ), (U ˜ , V˜ ) is a (K + 1)-dimensional Brownian motion defined in (3.9)-(3.11). where (U Proof. By (4.9)-(4.10), ˜ n (t) = X ˜ n (0) + E ˜ n (t) + U
√ ˜ n (t) − G ˜n n(λn /n − λ) + e0 M
Z
t
¯ n (s))+ (X
0
˜ 0,n (B ¯ n (t)) + (I − pe0 )M ˜ n (t), V˜ n (t) = (I − pe0 )Z˜ n (0) + Φ where ˜ n (t) = n−1/2 M n (t) = M
K X
˜ k,n (S¯n (T¯n (t))) − (I − P 0 )S˜n (T¯n (t)) Φ k k
k=1
¯ where S(s) ¯ and S¯n (s) = n−1 S(ns) for s ≥ 0. By the FSLLN, S¯n ⇒ S, = νs for s ≥ 0. The lemma now follows from (6.1), Theorem 5, the continuous mapping theorem, and the random-time-change theorem. ¯ (t) = 0 for t ≥ 0. Proof of Theorem 1. Assume that ρ = 1. Then q = 0 and λ = µ. Hence, U It follows from the state process representation (4.14) and the positively homogeneous property of the map Φ that ˜ n , Z˜ n ) = Φ(U ˜ n , V˜ n ). (X The theorem now follows from Lemma 2 and the continuous mapping theorem. Before proving Theorem 2, we first establish a lemma. It says that when ρ > 1, the number of idle servers goes to zero under diffusion-scaling. Lemma 3. Assume that ρ > 1. Let I n (t) = (X n (t))− and I˜n (t) = n−1/2 I n (t) for t ≥ 0. Then I˜n ⇒ 0, as n → ∞. 17
Proof. It follows from (4.12)-(4.13) that n
−1/2
√ ¯ (t) + U ˜ n (t) − α X (t) = nU
Z
t
−1/2
0
t
Z
X (s)) ds − e R Z˜ n (s) ds, 0 0 Z t Z˜ n (t) = V˜ n (t) − p(n−1/2 X n (t))− − (I − pe0 )R Z˜ n (s) ds. n
(n
n
+
0
Therefore, by Lemma 1 ˜n + (n−1/2 X n , Z˜ n ) = Φ(U
√
¯ , V˜ n ). nU
By the Lipschitz continuity property (3.18) of the map Φ, for any t > 0, there exists a constant Ct > 0 such that √ √ ¯ , V˜ n )(s) − Φ( nU ¯ , 0)(s)| ≤ Ct sup |U ˜ n (s)| + |V˜ n (s)| ˜ n + nU sup |Φ(U 0≤s≤t
0≤s≤t
√ √ ¯ , 0) = ( nq, 0). Therefore, for all n and all sample paths. One can check that Φ( nU √ ˜ n (s)| + |V˜ n (s)| . inf n−1/2 X n (s) ≥ nq − Ct sup |U (6.2) 0≤s≤t
0≤s≤t
˜ n (s)| + |V˜ n (s)| is stochastically bounded in n, which together By Lemma 2, sup0≤s≤t |U with (6.2) and (3.2) implies that sup0≤s≤t I˜n (s) → 0 in probability. Since t is arbitrary, the lemma is proved. Proof of Theorem 2. It follows from (4.12)-(4.13) that ˜ n (t) = U ˜ n (t) − α X
Z
t
n
−1/2
n
−
t
Z
˜ n (s) ds − e0 R X
(X (s)) ds − α
0
0
Z˜ n (t) = V˜ n (t) − pn−1/2 (X n (t))− − (I − pe0 )R
Z
t
Z˜ n (s) ds,
0
Z
t
Z˜ n (s) ds.
0
Let n
Z
δ (t) = α
t
n−1/2 (X n (s))− ds and n (t) = pn−1/2 (X n (t))−
0
for t ≥ 0. By Lemma 1, ˜ n , Z˜ n ) = Ψ(U ˜ n − δ n , V˜ n − n ). (X By Lemma 3, (δ n , n ) ⇒ (0, 0). The theorem follows from Lemma 2, (6.3) and the continuity of the map Ψ.
18
(6.3)
Proof of Theorem 3. Note that (X n (0))+ + E n (t) is the total number of customers who have entered into the buffer in [0, t]. Let On (t) = (X n (0))+ + E n (t) − (X n (t))+ be the cumulative number of customers who have departed from the buffer in [0, t]. For those (X n (t))+ waiting customers, their distribution is (X n (0))+ +E n (t)
Qn (t) =
X
ψ 0 (`) = Ψ0 (X n (0))+ + E n (t) − Ψ0 (On (t))
`=On (t)+1
ˆ 0 (X n (0))+ + E n (t) − Ψ ˆ 0 (On (t)) + p(X n (t))+ , = Ψ where {ψ 0P (i) : i ≥ 0} is a sequence of iid Bernoulli random vectors having distribution p, 0 0 0 n n n + ˆ0 Ψ (N ) = N i=1 ψ (i), and Ψ (N ) = Ψ (N ) − N p. Setting η (t) = Q (t) − p(X (t)) and n −1/2 n n η˜ (t) = n η (t), to prove the theorem, it suffices to prove η˜ ⇒ 0 as n → ∞. One can check that ˆ 0 (X n (0))+ + E n (t) − Ψ ˆ 0 (On (t)). η n (t) = Ψ (6.4) Thus, ˜ 0,n (X ¯ n (0))+ + E ¯ n (t) − Ψ ˜ 0,n (O ¯ n (t)), η˜n (t) = Ψ ¯ n (t) = n−1 On (t) and Ψ ˜ 0,n (t) = n−1/2 Ψ ˆ 0 (bntc). By Theorem 5, O ¯ n ⇒ O(t), ¯ where O where + + 0,n 0 0 ¯ = (X(0)) ¯ ¯ ˜ ˜ and Ψ ˜ is a K-dimensional O(t) + λt − (X(t)) = λt for t ≥ 0. Since Ψ ⇒Ψ Brownian motion, thus by Theorem 5 and the random time change theorem, one has η˜n → η˜, where ˜ 0 (q + λt) − Ψ ˜ 0 (λt). η˜(t) = Ψ When ρ = 1, one has q = 0 and therefore, η˜(t) = 0 for t ≥ 0, which implies the theorem. Proof of Theorem 4. Let Ln (t) be the cumulative number of customers who have abandoned the queue in [0, t]. One can check X n (t) = X n (0) + E n (t) − Ln (t) − Dn (t) or equivalently Ln (t) = X n (0) − X n (t) + E n (t) − Dn (t) for t ≥ 0. It follows from Theorem 5 and (6.5) that ¯ n, E ¯ n, L ¯ n, X ¯ n ) ⇒ (D, ¯ E, ¯ L, ¯ X), ¯ (D ¯ n (t) = n−1 Ln (t) and L(t) ¯ = (λ − µ)t for t ≥ 0. Define as n → ∞, where L ˜ n (t) = √1 Ln (t) − nL(t) ¯ L n for t ≥ 0. Since ρ = 1, (6.5) implies that ˜ n (t) = X ˜ n (0) − X ˜ n (t) + E ˜ n (t) − D ˜ n (t). L 19
(6.5)
˜ n (t) = n−1/2 (Dn (t) − nµt) for t ≥ 0, it follows from (4.8) that Setting D Z t 0 n 0 ˜n ˜ Z˜ n (s) ds. D (t) = −e M (t) + e R 0
Thus, ˜ n, E ˜ n, L ˜ n, X ˜ n ) ⇒ (D, ˜ E, ˜ L, ˜ X) ˜ (D in D4 as n → ∞, where ˜ (t) + e0 R ˜ D(t) = −e0 M
Z
t
˜ ds, Z(s)
0
˜ = X(0) ˜ ˜ + E(t) ˜ − D(t). ˜ L(t) − X(t) ˜ E, ˜ L, ˜ X) ˜ The theorem follows then from Theorem 3.1 of Talreja and Whitt (2008) because (D, is continuous almost surely.
Acknowledgement We thank Shuangchi He for helpful discussions.
A
A continuous map
Let K > 0 be a fixed integer. Given three functions h1 : RK+1 → R, h2 : RK+1 → RK and g : R → R, we wish to define a map Υ that maps D × DK to D × DK . For each y = (y1 , y2 ) ∈ D × DK , Υ(y) is defined to be any x = (x1 , x2 ) ∈ D × DK that satisfies Z t x1 (t) = y1 (t) + h1 (x(s)) ds, (A.1) 0 Z t x2 (t) = y2 (t) + h2 (x(s)) ds + g(x1 (s)) (A.2) 0
for t ≥ 0. We assume that h1 , h2 and g are Lipschitz continuous. For a function f : Rd → Rm with integers d, m > 0, it is said to be Lipschitz continuous with Lipschitz constant c > 0 if |f (u) − f (v)| ≤ c|u − v| (A.3) for each pair u, v ∈ Rd , where for a vector a = (ak ), |a| = maxk |ak | denotes the maximum norm of a. The function f is said to be positively homogeneous if f (au) = af (u)
(A.4)
for any a > 0 and u ∈ Rd . For an x ∈ Dd , where d is some positive integer, set ||x||t = sup0≤s≤t |x(s)|. The following lemma establishes the existence and the continuity of the map Υ. 20
Lemma 4. Assume that h1 , h2 and g are Lipschitz continuous. (a) For each y = (y1 , y2 ) ∈ D × DK , there exists a unique x = (x1 , x2 ) ∈ D × DK that satisfies (A.1) and (A.2). (b) The map Υ: D × DK → D × DK is Lipschitz continuous in the sense that for each T > 0, there exists a constant CT such that kΥ(y) − Υ(˜ y )kT ≤ CT ky − y˜kT
(A.5)
for any y, y˜ ∈ D × DK . (c) The map Υ is continuous when the domain D × DK and the range D × DK are both viewed as the path space D(R+ , RK+1 ) that is endowed with the standard Skorohod J1 topology. (d) If, in addition, h1 , h2 and g are assumed to be positively homogeneous, then the map Υ is positively homogeneous in the sense that Υ(ay) = aΥ(y)
(A.6)
for each a > 0 and each y ∈ DK+1 . Proof. Assume that h1 , h2 and g are Lipschitz continuous with Lipschitz constant c > 0. Let y = (y1 , y2 ) ∈ D × DK be given. Let T > 0 be fixed for the moment. Define x0 = y ) be defined via , xn+1 and for each integer n ≥ 0, let xn+1 = (xn+1 2 1 xn+1 (t) 1 (t) xn+1 2
t
Z = y1 (t) −
h1 (xn (s))ds,
(A.7)
(t)) h2 (xn (s))ds + g(xn+1 1
(A.8)
0 t
Z = y2 (t) − 0
for t ∈ [0, T ]. Setting X (n) (t) = kxn+1 (·) − xn (·)kt , because xn+1 (t) 2
−
xn2 (t)
Z
t
h2 (xn (s)) − h2 (xn−1 (s)) ds 0 Z t n +g y1 (t) − h1 (x (s))ds 0 Z t n−1 −g y1 (t) − h1 (x (s)(s))ds
= −
0
for t ∈ [0, T ], one has X (n+1) (t) ≤ (c + c2 )
Z 0
21
t
X (n) (s)ds
for t ∈ [0, T ]. Then, by Lemma 11.3 in Mandelbaum et al. (1998), X n+1 (t) ≤ (c + c2 )
Tn sup X (0) (s) n! 0≤s≤t
for t ∈ [0, T ]. Therefore, similar to (11.22) in Mandelbaum et al. (1998), {xn (·)} is a Cauchy sequence under the uniform norm k · kT . Since (D([0, T ], RK+1 ), k · kT ) is a complete metric space, (being a closed subset of the Banach space of bounded functions defined from [0, T ] into RK+1 and endowed with the uniform norm,) {xn } has a limit x that is in D([0, T ], RK+1 ). One can check that x satisfies (A.1) and (A.2) for t ∈ [0, T ]. This proves the existence of the map Υ from D([0, T ], RK+1 ) to D([0, T ], RK+1 ). Now we prove that the map from D([0, T ], RK+1 ) to D([0, T ], RK+1 ) is Lipshitz continuity with respect to the sup norm. Assume that y, y˜ ∈ D([0, T ], RK+1 ). Let Υ(y) be any solution x such that x and y satisfy (A.1) and (A.2) on [0, T ]. Similarly, let Υ(˜ y ) be any solution associated with y˜. Setting x = (x1 , x2 ) = Υ(y) and x ˜ = (˜ x1 , x ˜2 ) = Υ(˜ y ), then for any t ∈ [0, T ], Z
t
|Υ(y)(s) − Υ(˜ y )(s)|ds,
|x1 (t) − x ˜1 (t)| ≤ |y(t) − y˜(t)| + c 0
|x2 (t) − x ˜2 (t)| ≤ (c + 1)|y(t) − y˜(t)| Z t + (c + c2 ) |Υ(y)(s) − Υ(˜ y )(s)|ds. 0
Hence, |Υ(y)(t) − Υ(˜ y )(t)| ≤ (c + 1)|y(t) − y˜(t)| + (c + c2 )
Z
t
|Υ(y)(s) − Υ(˜ y )(s)|ds 0
for t ∈ [0, T ]. By Corollary 11.2 in Mandelbaum et al. (1998) kΥ(y)(·) − Υ(˜ y )(·)kT ≤ (c + 1)ky(·) − y˜(·)kT exp((c + c2 )T ). Hence, Υ is Lipshitz continuous, which implies part (b) of the lemma. The Lipschitz continuity of Υ as a map from D([0, T ], RK+1 ) to D([0, T ], RK+1 ) shows that it is well defined on [0, T ]. Since T > 0 is arbitrary, Υ as a map from D(R+ , RK+1 ) to D(R+ , RK+1 ) is well defined. This proves part (a) of the lemma. Next we prove the continuity of Υ provided that DK+1 = D(R+ , RK+1 ) is endowed with the standard Skorohod J1 topology (see, for example, Section 3 of Whitt (2002)). Consider a sequence {y n } and y in DK+1 such that y n → y as n → ∞. Let xn = Υ(y n ) and x = Υ(y). Note that since x ∈ DK+1 there exists M > 0 such that kΥ(y)(·)kT < M.
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(A.9)
Since y n → y as n → ∞ in the standard J1 topology on DK+1 , for fixed T > 0, there exists λn : [0, T ] → [0, T ], a strictly increasing, continuous mapping of [0, T ] onto itself such that ky n (·) − y(λn (·))kT → 0
and kλn (·) − e(·)kT → 0
as n → ∞, where e(t) = t for t ≥ 0. By Proposition 5.5.3 in Ethier and Kurtz (1986), we can also assume that λn is absolutely continuous with respect to Lebesgue measure on [0, T ] having derivatives λ˙ n satisfying kλ˙ n (·) − 1kT → 0 as n → ∞. Note that, for i = 1, 2 Z t Z λn (t) hi (Υ(y(λn (s))) λ˙ n (s) ds. (A.10) hi (Υ(y(s)) ds = 0
0
Hence, by (A.1) |Υ(y n )(t) − Υ(y)(λn (t))| ≤ (c + 1) |y n (t) − y(λn (t))| Z t 2 +(c + c ) |Υ(y n )(s)ds − Υ(y(λn (s)))| ds 0
+(c + c )kλ˙ n (·) − 1kT kΥ(y(·))kT . 2
Given δ > 0, choose n large enough so that kλ˙ n (·) − 1kT < δ/2M, and ky n (·) − y(λn (·))kT < δ/(2(c + 1)). Then by Gronwall’s inequality (see, for example, Corollary 11.2 in Mandelbaum et al. (1998)), kΥ(y n )(·) − Υ(y)(λn (·))kT ≤ δ exp((c + c2 )T ). Hence, Υ(y n ) → Υ(y) as n → ∞ in D([0, T ], RK+1 ) under the standard J1 topology. Since T > 0 is arbitrary, we have proved Υ(y n ) → Υ(y) as n → ∞ in D(R+ , RK+1 ) under the standard J1 topology. This implies part (c) of the lemma. To prove part (d) of the lemma, for y ∈ DK+1 , assume that x and y satisfy (A.1) and (A.2). Then, for a > 0, one can check that ax and ay also satisfy (A.1) and (A.2) because of the positive homogeneity of h1 , h2 and g. Therefore, Υ(ay) = aΥ(y), which implies part (d) of the lemma.
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