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2012 Stability of a Markov-modulated Markov chain, with application to a wireless network governed by two protocols
Sergey Foss, Seva Shneer, Andrey Tyurlikov
Stoch. Syst. 2(1): 208-231 (2012). DOI: 10.1214/11-SSY030

Abstract

We consider a discrete-time Markov chain $(X^{t},Y^{t})$, $t=0,1,2,\ldots$ , where the $X$-component forms a Markov chain itself. Assume that $(X^{t})$ is Harris-ergodic and consider an auxiliary Markov chain $\{\widehat{Y}^{t}\}$ 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.

We first provide natural conditions in terms of test functions ensuring that the $\widehat{Y}$-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.

Citation

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Sergey Foss. Seva Shneer. Andrey Tyurlikov. "Stability of a Markov-modulated Markov chain, with application to a wireless network governed by two protocols." Stoch. Syst. 2 (1) 208 - 231, 2012. https://doi.org/10.1214/11-SSY030

Information

Published: 2012
First available in Project Euclid: 24 February 2014

zbMATH: 1292.93139
MathSciNet: MR3352978
Digital Object Identifier: 10.1214/11-SSY030

Subjects:
Primary: 93E15
Secondary: 60K25, 68M20

Rights: Copyright © 2012 INFORMS Applied Probability Society

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Vol.2 • No. 1 • 2012
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