Computable bounds on the spectral gap for unreliable Jackson networks

TL;DR

This paper develops computable bounds for the spectral gap of unreliable Jackson networks, linking convergence rates to properties of marginal distributions and their hazard functions, and compares these bounds with existing ones.

Contribution

It introduces new bounds on the spectral gap for unreliable Jackson networks based on hazard and equilibrium functions, extending previous theoretical results.

Findings

Spectral gap is positive iff coordinate birth-death processes have positive spectral gaps.

Bounds depend on hazard functions of stationary distributions, requiring them to be strongly light-tailed.

Compared new bounds with existing bounds, showing their effectiveness.

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 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 zero. 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.