CONVERGENCE OF DEPARTURES IN TANDEM-NETWORKS OF CENTER-DOT GI/INFINITY QUEUES/

We consider an infinite series of independent and identical ./GI/infin ity queues fed by an arbitrary stationary and ergodic arrival process, A(1). Let A(i) be the arrival process to the ith node, and let nu(i) be the law of A(i). Denote by T(.) the input-output map of the ./GI/in finity node; that is, nu(i+1) = T(nu(i)). It is known that the Poisson process is a fixed point for T. In this paper, we are interested in t he asymptotic distribution of the departure process from the nth node, nu(n+1) = T-n(nu(1)), as n --> infinity). Using couplings for random walks, this limiting distribution is shown to be either a Poisson proc ess or a stationary nu-Poisson process, depending on the joint distrib ution of A(I) and the service process. This generalizes a result of Ve re-Jones (1968, Journal of the Royal Statistical Society, Series B 30: 321-333) and is similar in flavor to Mountford and Prabhakar (1995, A nnals of Applied Probability 5(1): 121-127), where Poisson convergence is established for departures from a series of exponential server que ues using coupling methods.