{
  "id": "1105.0270",
  "version": "v2",
  "published": "2011-05-02T09:17:33.000Z",
  "updated": "2012-12-25T10:07:04.000Z",
  "title": "Stability of a Markov-modulated Markov Chain, with application to a wireless network governed by two protocols",
  "authors": [
    "Sergey Foss",
    "Seva Shneer",
    "Andrey Tyurlikov"
  ],
  "comment": "23 pages",
  "journal": "Stochastic Systems, 2012, Vol. 2, No. 1, 208-231",
  "doi": "10.1214/11-SSY30",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a discrete-time Markov chain $(X^t,Y^t)$, $t=0,1,2,...$, where the $X$-component forms a Markov chain itself. Assume that $(X^t)$ is Harris-ergodic and consider an auxiliary Markov chain ${\\hat{Y}^t}$ 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 $\\hat{Y}$-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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-12-25T10:07:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J05",
      "60K20"
    ],
    "keywords": [
      "markov-modulated markov chain",
      "wireless network",
      "application",
      "transition probabilities",
      "discrete-time markov chain"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.0270F"
    }
  }
}