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

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Author(s): Sergey Foss | Seva Shneer | Andrey Turlikov

Journal: Stochastic Systems
ISSN 1946-5238

Volume: 2;
Issue: 1;
Start page: 208;
Date: 2012;
Original page

Keywords: Stochastic stability | Markov-modulated Markov chain

ABSTRACT
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 Ŷt 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 Ŷ-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.
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