{
  "id": "1307.6947",
  "version": "v1",
  "published": "2013-07-26T07:55:41.000Z",
  "updated": "2013-07-26T07:55:41.000Z",
  "title": "Buffer-overflows: joint limit laws of undershoots and overshoots of reflected processes",
  "authors": [
    "Aleksandar Mijatović",
    "Martijn Pistorius"
  ],
  "comment": "11 pages, no figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\tau(x)$ be the epoch of first entry into the interval $(x,\\infty)$, $x>0$, of the reflected process $Y$ of a L\\'evy process $X$, and define the overshoot $Z(x) = Y(\\tau(x))-x$ and undershoot $z(x) = x - Y(\\tau(x)-)$ of $Y$ at the first-passage time over the level $x$. In this paper we establish, separately under the Cram\\'{e}r and positive drift assumptions, the existence of the weak limit of $(z(x), Z(x))$ as $x$ tends to infinity and provide explicit formulae for their joint CDFs in terms of the L\\'{e}vy measure of $X$ and the renewal measure of the dual of $X$. We apply our results to analyse the behaviour of the classical M/G/1 queueing system at the buffer-overflow, both in a stable and unstable case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-26T07:55:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G51",
      "60F05",
      "60G17"
    ],
    "keywords": [
      "joint limit laws",
      "reflected processes",
      "buffer-overflow",
      "undershoot",
      "joint cdfs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.6947M"
    }
  }
}