### Vsevolod Shneer (School of Mathematical and Computer Sciences, Heriot-Watt University)

#### Stability of Markov-modulated Markov chains

*Wednesday 30 November 2011 at 15.30, JCMB 6206*

##### Abstract

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 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 this chain is positive recurrent, then
so is the (X, Y)-chain. 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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