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

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

2011-05-02

arxiv.org/abs/1105.0270v1

Mathematics

Probability

21 pages

Scientific paper

We consider a discrete-time Markov chain $(X^n,Y^n)$, where the $X$ component forms a Markov chain itself. Assuming that $(X^n)$ is ergodic, we formulate the following "naive" conjecture. Consider an auxiliary Markov chain $\{\widehat{Y}^n\}$ 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. The conjecture is: if the $\widehat{Y}$-chain is positive recurrent, then the $(X,Y)$-chain is positive recurrent too. We first show that, under appropriate technical assumptions, such a general result indeed holds, and then apply it to two versions of a multi-access wireless model governed by two randomised protocols.

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