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Approximate versions of Melamed's theorem


Barbour, A D; Brown, T (1996). Approximate versions of Melamed's theorem. Journal of Applied Probability, 33(2):472-489.

Abstract

In 1979, Melamed proved that, in an open migration process, the absence of 'loops' is necessary and sufficient for the equilibrium flow along a link to be a Poisson process. In this paper, we prove approximation theorems with the same flavour: the difference between the equilibrium flow along a link and a Poisson process with the same rate is bounded in terms of expected numbers of loops. The proofs are based on Stein's method, as adapted for bounds on the distance of the distribution of a point process from a Poisson process in Barbour and Brown (1992b). Three different distances are considered, and illustrated with an example consisting of a system of tandem queues with feedback. The upper bound on the total variation distance of the process grows linearly with time, and a lower bound shows that this can be the correct order of approximation.

Abstract

In 1979, Melamed proved that, in an open migration process, the absence of 'loops' is necessary and sufficient for the equilibrium flow along a link to be a Poisson process. In this paper, we prove approximation theorems with the same flavour: the difference between the equilibrium flow along a link and a Poisson process with the same rate is bounded in terms of expected numbers of loops. The proofs are based on Stein's method, as adapted for bounds on the distance of the distribution of a point process from a Poisson process in Barbour and Brown (1992b). Three different distances are considered, and illustrated with an example consisting of a system of tandem queues with feedback. The upper bound on the total variation distance of the process grows linearly with time, and a lower bound shows that this can be the correct order of approximation.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:1996
Deposited On:08 Apr 2010 15:13
Last Modified:05 Apr 2016 13:27
Publisher:Applied Probability Trust
ISSN:0021-9002
Publisher DOI:https://doi.org/10.2307/3215072
Related URLs:http://www.jstor.org/stable/3215072

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