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On customer flows in jackson queueing networks

Part of: Limit theorems Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Sen Tan  and
Aihua Xia
Sen Tan*
Affiliation:
University of Melbourne
Aihua Xia*
Affiliation:
University of Melbourne
*
∗ Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3052, Australia.
∗ Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3052, Australia.
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Abstract

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Melamed's theorem states that, for a Jackson queueing network, the equilibrium flow along a link follows a Poisson distribution if and only if no customers can travel along the link more than once. Barbour and Brown (1996) considered the Poisson approximate version of Melamed's theorem by allowing the customers a small probability p of travelling along the link more than once. In this note, we prove that the customer flow process is a Poisson cluster process and then establish a general approximate version of Melamed's theorem that accommodates all possible cases of 0 ≤ p < 1.

Keywords

Jackson queueing networkPalm distributionPoisson cluster processStein's methodnegative binomial distribution

MSC classification

Primary: 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems 60E15: Inequalities; stochastic orderings
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 4 , December 2010 , pp. 1094 - 1101
Copyright
Copyright © Applied Probability Trust 2010 

References

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Brown, T. C. and Xia, A. (2001). Stein's method and birth-death processes. Ann. Prob. 29, 1373–1403.CrossRefGoogle Scholar
Kallenberg, O. (1983). Random Measures, 3rd edn. Academic Press, London.CrossRefGoogle Scholar
Melamed, B. (1979). Characterizations of Poisson traffic streams in Jackson queueing networks. Adv. Appl. Prob. 11, 422–438.CrossRefGoogle Scholar