Convergence of Departures in Tandem Networks of·/GI/ ∞ Queues

Abstract

We consider an infinite series of independent and identical -/GI/∞ queues fed by an arbitrary stationary and ergodic arrival process, A1. Let Ai be the arrival process to the ith node, and let νi be the law of Ai. Denote by (·) the input–output map of the ·/GI/∞ node; that is, νi+1 = (νi). It is known that the Poisson process is a fixed point for . In this paper, we are interested in the asymptotic distribution of the departure process from the nth node, νn+1 = n(ν1), as n ∞. Using couplings for random walks, this limiting distribution is shown to be either a Poisson process or a stationary ν-Poisson process, depending on the joint distribution of A1 and the service process. This generalizes a result of Vere-Jones (1968, Journal of the Royal Statistical Society, Series B 30: 321–333) and is similar in flavor to Mountford and Prabhakar (1995, Annals of Applied Probability 5(1): 121–127), where Poisson convergence is established for departures from a series of exponential server queues using coupling methods.