{
  "id": "0712.2091",
  "version": "v1",
  "published": "2007-12-13T18:17:03.000Z",
  "updated": "2007-12-13T18:17:03.000Z",
  "title": "Asymptotic distributions and chaos for the supermarket model",
  "authors": [
    "Malwina J. Luczak",
    "Colin McDiarmid"
  ],
  "comment": "Published in Electronic Journal of Probability",
  "journal": "EJP, vol. 12 (2007), 75--99",
  "categories": [
    "math.PR"
  ],
  "abstract": "In the supermarket model there are n queues, each with a unit rate server. Customers arrive in a Poisson process at rate \\lambda n, where 0<\\lambda <1. Each customer chooses d > 2 queues uniformly at random, and joins a shortest one. It is known that the equilibrium distribution of a typical queue length converges to a certain explicit limiting distribution as n -> oo. We quantify the rate of convergence by showing that the total variation distance between the equilibrium distribution and the limiting distribution is essentially of order n^{-1}; and we give a corresponding result for systems starting from quite general initial conditions (not in equilibrium). Further, we quantify the result that the systems exhibit chaotic behaviour: we show that the total variation distance between the joint law of a fixed set of queue lengths and the corresponding product law is essentially of order at most n^{-1}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-12-13T18:17:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "68R05",
      "90B22",
      "60K25",
      "60K30",
      "68M20"
    ],
    "keywords": [
      "supermarket model",
      "asymptotic distributions",
      "total variation distance",
      "quite general initial conditions",
      "equilibrium distribution"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0712.2091L"
    }
  }
}