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

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.