{
  "id": "math/0608258",
  "version": "v3",
  "published": "2006-08-10T12:53:14.000Z",
  "updated": "2009-01-16T06:57:35.000Z",
  "title": "A functional central limit theorem for the M/GI/$\\infty$ queue",
  "authors": [
    "Laurent Decreusefond",
    "Pascal Moyal"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/08-AAP518 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2008, Vol. 18, No. 6, 2156-2178",
  "doi": "10.1214/08-AAP518",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we present a functional fluid limit theorem and a functional central limit theorem for a queue with an infinity of servers M/GI/$\\infty$. The system is represented by a point-measure valued process keeping track of the remaining processing times of the customers in service. The convergence in law of a sequence of such processes after rescaling is proved by compactness-uniqueness methods, and the deterministic fluid limit is the solution of an integrated equation in the space $\\mathcal{S}^{\\prime}$ of tempered distributions. We then establish the corresponding central limit theorem, that is, the approximation of the normalized error process by a $\\mathcal{S}^{\\prime}$-valued diffusion. We apply these results to provide fluid limits and diffusion approximations for some performance processes.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2009-01-16T06:57:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F17",
      "60K25",
      "60B12"
    ],
    "keywords": [
      "functional central limit theorem",
      "functional fluid limit theorem",
      "point-measure valued process keeping track",
      "deterministic fluid limit",
      "corresponding central limit theorem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8258D"
    }
  }
}