IMPROVED RESULTS ON POISSON PROCESS APPROXIMATION IN JACKSON NETWORKS

Abstract

Melamed (1979) proved that for an open migration process, a necessary and sufficient condition for the equilibrium flow along a link to be Poissonian is the absence of loops: no customer can travel along the link more than once. Barbour and Brown (1996) quantified the statement by allowing the customers a small probability of travelling along the link more than once and proved Poisson process approximation theorems analogous to Melamed's Theorem. Amongst the three bounds presented in Barbour and Brown (1996), the one in terms of the Wasserstein metric is of particular interest since it reveals more insightful information about the closeness between the process of flows and an approximating Poisson process, and it is small when the parameter of the system is small, except a logarithmic factor in terms of time in which the flows are considered. The bound was later improved by Brown, Weinberg and Xia (2000) who showed that the logarithmic factor in terms of time can be lifted at the cost of an extra parameter being introduced into the bound. In this paper, we present a new bound which simplifies and sharpens the bounds in the above-mentioned two papers and compare the performance of these bounds for a simple open migration process.