Computable Bounds on the Spectral Gap for Unreliable Jackson Networks

Abstract

The goal of this paper is to identify exponential convergence rates and to find computable bounds for them for Markov processes representing unreliable Jackson networks. First, we use the bounds of Lawler and Sokal (1988) in order to show that, for unreliable Jackson networks, the spectral gap is strictly positive if and only if the spectral gaps for the corresponding coordinate birth and death processes are positive. Next, utilizing some results on birth and death processes, we find bounds on the spectral gap for network processes in terms of the hazard and equilibrium functions of the one-dimensional marginal distributions of the stationary distribution of the network. These distributions must be in this case strongly light-tailed, in the sense that their discrete hazard functions have to be separated from 0. We relate these hazard functions with the corresponding networks' service rate functions using the equilibrium rates of the stationary one-dimensional marginal distributions. We compare the obtained bounds on the spectral gap with some other known bounds.