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

We consider a discrete-time Markov chain (Xᵗ,Yᵗ), t=0,1,2,\… , where the X-component forms a Markov chain itself. Assume that (Xᵗ) is Harris-ergodic and consider an auxiliary Markov chain \\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.¶ first provide natural conditions in terms of test functions ensuring that the -chain is positive recurrent and then prove that these conditions are also sufficient for positive recurrence of the original chain (Xᵗ,Yᵗ). 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.