Luczak, Malwina J. and McDiarmid, Colin
(2006)
*Asymptotic distributions and chaos for the supermarket model.*
CDAM research report series (LSE-CDAM-2006-12).
Centre for Discrete and Applicable Mathematics, London School of Economics and Political Science, London, UK.

## Abstract

In the supermarket model there are n queues, each with a unit rate server. Customers arrive in a Poisson process at rate n, where 0 < < 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 ! 1. 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

Item Type: | Monograph (Report) |
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Official URL: | http://www.cdam.lse.ac.uk |

Additional Information: | © 2006 the authors |

Divisions: | Mathematics |

Subjects: | Q Science > QA Mathematics |

Date Deposited: | 10 Oct 2008 11:46 |

Last Modified: | 22 Sep 2021 23:13 |

URI: | http://eprints.lse.ac.uk/id/eprint/13801 |

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