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 $\left\{{\hat{Y}}^t\right\}$ 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.