{
  "id": "1707.05475",
  "version": "v1",
  "published": "2017-07-18T05:32:22.000Z",
  "updated": "2017-07-18T05:32:22.000Z",
  "title": "Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process",
  "authors": [
    "Toshihisa Ozawa",
    "Masahiro Kobayashi"
  ],
  "comment": "54 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a discrete-time two-dimensional process $\\{(L_{1,n},L_{2,n})\\}$ on $\\mathbb{Z}_+^2$ with a supplemental process $\\{J_n\\}$ on a finite set, where individual processes $\\{L_{1,n}\\}$ and $\\{L_{2,n}\\}$ are both skip free. We assume that the joint process $\\{Y_n\\}=\\{(L_{1,n},L_{2,n},J_n)\\}$ is Markovian and that the transition probabilities of the two-dimensional process $\\{(L_{1,n},L_{2,n})\\}$ are modulated depending on the state of the background process $\\{J_n\\}$. This modulation is space homogeneous except for the boundaries of $\\mathbb{Z}_+^2$. We call this process a discrete-time two-dimensional quasi-birth-and-death (2D-QBD) process and, under several conditions, obtain the exact asymptotic formulae of the stationary distribution in the coordinate directions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-18T05:32:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60K25"
    ],
    "keywords": [
      "discrete-time two-dimensional qbd process",
      "exact asymptotic formulae",
      "stationary distribution",
      "discrete-time two-dimensional process",
      "discrete-time two-dimensional quasi-birth-and-death"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}