## Stochastic Systems

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

#### 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.

#### Article information

Source
Stoch. Syst., Volume 2, Number 1 (2012), 208-231.

Dates
First available in Project Euclid: 24 February 2014

https://projecteuclid.org/euclid.ssy/1393252044

Digital Object Identifier
doi:10.1214/11-SSY030

Mathematical Reviews number (MathSciNet)
MR3352978

Zentralblatt MATH identifier
1292.93139

#### Citation

Foss, Sergey; Shneer, Seva; Tyurlikov, Andrey. Stability of a Markov-modulated Markov chain, with application to a wireless network governed by two protocols. Stoch. Syst. 2 (2012), no. 1, 208--231. doi:10.1214/11-SSY030. https://projecteuclid.org/euclid.ssy/1393252044

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