Stability of multidimensional skip-free Markov modulated reflecting random walks: Revisit to Malyshev and Menshikov's results and application to queueing networks

TL;DR

This paper revisits stability conditions for multidimensional Markov modulated reflecting random walks, extending classical results to a more general setting and applying findings to queueing network stability analysis.

Contribution

It extends Malyshev and Menshikov's stability results to multidimensional skip-free Markov modulated reflecting random walks and applies these to queueing networks.

Findings

Derived new stability and instability conditions for MMRRWs.

Applied stability criteria to analyze two-station queueing networks.

Extended classical RRW stability results to modulated, multidimensional cases.

Abstract

Let $\{\boldsymbol{X}_n\}$ be a discrete-time $d$-dimensional process on $\mathbb{Z}_+^d$ with a supplemental (background) process $\{J_n\}$ on a finite set and assume the joint process $\{\boldsymbol{Y}_n\}=\{(\boldsymbol{X}_n,J_n)\}$ to be Markovian. Then, the process $\{\boldsymbol{X}_n\}$ can be regarded as a kind of reflecting random walk (RRW for short) in which the transition probabilities of the RRW are modulated according to the state of the background process $\{J_n\}$; we assume this modulation is space-homogeneous inside $\mathbb{Z}_+^d$ and on each boundary face of $\mathbb{Z}_+^d$. Further we assume the process $\{\boldsymbol{X}_n\}$ is skip free in all coordinates and call the joint process $\{\boldsymbol{Y}_n\}$ a $d$-dimensional skip-free Markov modulated reflecting random walk (MMRRW for short). The MMRRW is an extension of an ordinary RRW and stability of ordinary RRWs have been studied by Malyshev and Menshikov. Following their results, we obtain stability and instability conditions for MMRRWs and apply our results to stability analysis of a two-station network.