### Serguei Foss (School of Mathematical and Computer Sciences, Heriot-Watt University)

#### Stability of a Markov-modulated Markov chain, with applications to wireless networks governed by two protocols

*Joint work with Seva Shneer (EPFL, Lausanne) and Andrey Tyurlikov (St Petersburg).*

*Wednesday 3 November 2010 at 15.30, JCMB 6206*

##### Abstract

We consider a general discrete-time Markov chain (X(n),Y(n)), n=0,1,... where
where the X-component forms a Markov chain itself. Assuming that (X(n)) is
ergodic, we may formulate the following ``naive'' conjecture.

Consider an auxiliary Markov chain (Ŷ(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 Ŷ-chain is
positive recurrent, then the (X,Y)-chain should be 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 protocols.

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