Heavy Traffic Approximation for the Stationary Distribution of a Generalized Jackson Network: The BAR Approach

Abstract

In the seminal paper of Gamarnik and Zeevi [17], the authors justify the steady-state diffusion approximation of a generalized Jackson network (GJN) in heavy traffic. Their approach involves the so-called limit interchange argument, which has since become a popular tool employed by many others who study diffusion approximations. In this paper we illustrate a novel approach by using it to justify the steady-state approximation of a GJN in heavy traffic. Our approach involves working directly with the basic adjoint relationship (BAR), an integral equation that characterizes the stationary distribution of a Markov process. As we will show, the BAR approach is a more natural choice than the limit interchange approach for justifying steadystate approximations, and can potentially be applied to the study of other stochastic processing networks such as multiclass queueing networks. Introduction.This paper considers open single-class queueing networks that have d service stations. Each station has a single server operating under the first-in-first-out (FIFO) service discipline. Upon completing service at a particular station, customers are either routed to another station, or exit the network. There is a single class of customers at each station, meaning that all customers are homogenous in terms of service times and routing. A customer entering the network will exit in finite time with probability one, hence the term open network. For each station, the external interarrival times (possibly null), service times, and routing decisions are assumed to follow three separate i.i.d. sequences of random variables; the three sequences are assumed to be independent. Furthermore, the interarrival times, service times and routing decisions are assumed to be independent between different stations. Such a network is hereafter referred to as a generalized Jackson network (GJN).

Optimal control of a large dam with compound Poisson input and costs depending on water levels

“…This problem is very hard in general (see discussion of a similar problem in Whitt [34]). For solution of the limits interchange problem in generalized Jackson networks in heavy traffic see Gamarnik and Zeevi [11] and Braverman, Dai and Miyazawa [6]. The further references can be found in [6].…”

A unified approach for large queue asymptotics in a heterogeneous multiserver queue

“…The above inequality is one of the key techniques in the present approach, and has been used in the proof of Lemma 3.2 in Appendix B. It also can be found in the last part of the proof of Lemma 4.7 of [4].…”

“…Clearly, this condition is equivalent to the convergence of the first moments of T (n) i to finite and positive constants. We need the following fact, which is essentially a special case of Lemma 4.4 of [4], but an extra condition is assumed in [4]. We show that it is not needed in Appendix B. i ; n ≥ 1} is uniformly integrable, then, for a sequence q n > 0 converging to 0, we have, for some positive constants c i (δ) for each δ > 0, lim sup…”

“…In particular, a queueing network is apparently more interesting for application. Braverman et al [4] showed that its heavy-traffic approximation can be studied in the same spirit (see Section 5.2 for more discussion). This paper comprises five sections.…”

Heavy-Traffic Analysis of Sojourn Time Under the Foreground–Background Scheduling Policy

“…One of the main insights from queueing theory is that the queue length and sojourn time are of the order 1/(1 − ρ), as the traffic intensity of the system ρ approaches 100 percent utilization. This insight dates back to Kingman [13] and Prokhorov [19] and, appropriately reformulated, remains valid for queueing networks and multiple server queues [7,11,25]. This picture can change dramatically when the scheduling policy is no longer First-In-First-Out (FIFO).…”