Correlation formulas for Markovian network processes in a random environment

TL;DR

This paper analyzes the correlation structure over time and space in Markovian network processes influenced by a random environment, providing formulas for differences in correlations and insights into spectral gaps and variances.

Contribution

It introduces explicit formulas for correlation differences in Markovian networks in random environments, enhancing understanding of their temporal and spatial dependencies.

Findings

Correlation differences have a simple structure despite complex absolute values.

Comparison formulas enable analysis of spectral gaps and asymptotic variances.

Results facilitate understanding of how environmental changes affect network process dynamics.

Abstract

We consider Markov processes, which describe e.g. queueing network processes, in a random environment which influences the network by determining random breakdown of nodes, and the necessity of repair thereafter. Starting from an explicit steady state distribution of product form available in the literature, we notice that this steady state distribution does not provide information about the correlation structure in time and space (over nodes). We study this correlation structure via one step correlations for the queueing-environment process. Although formulas for absolute values of these correlations are complicated, the differences of correlations of related networks are simple and have a nice structure. We therefore compare two networks in a random environment having the same invariant distribution, and focus on the time behaviour of the processes when in such a network the environment changes or the rules for traveling are perturbed. Evaluating the comparison formulas we compare spectral gaps and asymptotic variances of related processes.