Stability of a Markov-modulated Markov Chain, with application to a wireless network governed by two protocols

TL;DR

This paper analyzes the stability of a Markov-modulated Markov chain, providing conditions for positive recurrence, and applies these results to wireless networks governed by two protocols.

Contribution

It offers new conditions for chain stability and extends these results to multi-dimensional cases, with applications to wireless network protocols.

Findings

Conditions for positive recurrence of the auxiliary chain are established.

Sufficient conditions for the original chain's positive recurrence are proven.

Application to wireless network models demonstrates practical relevance.

Abstract

We consider a discrete-time Markov chain $(X^t,Y^t)$, $t=0,1,2,...$, where the $X$-component forms a Markov chain itself. Assume that $(X^t)$ is Harris-ergodic and consider an auxiliary Markov chain ${\hat{Y}^t}$ whose transition probabilities are the averages of transition probabilities of the $Y$-component of the $(X,Y)$-chain, where the averaging is weighted by the stationary distribution of the $X$-component. We first provide natural conditions in terms of test functions ensuring that the $\hat{Y}$-chain is positive recurrent and then prove that these conditions are also sufficient for positive recurrence of the original chain $(X^t,Y^t)$. The we prove a "multi-dimensional" extension of the result obtained. In the second part of the paper, we apply our results to two versions of a multi-access wireless model governed by two randomised protocols.