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Approximate versions of Melamed's theorem

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

A. D. Barbour  and
Timothy C. Brown
A. D. Barbour*
Affiliation:
Universität Zürich
Timothy C. Brown*
Affiliation:
University of Melbourne
*
∗Postal address: Institut für Angewandte Mathematik, Universität Zürich, Winterthurerstr. 190, 8057 Zürich, Switzerland.
∗∗Postal address: Department of Statistics, University of Melbourne, Parkville, Victoria 3052, Australia.
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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.

Keywords

MELAMED'S THEOREMLOOP CONDITIONPOISSON APPROXIMATIONMIGRATION PROCESSNETWORK FLOWS

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 2 , September 1996 , pp. 472 - 489
Copyright
Copyright © Applied Probability Trust 1996 

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References

Asmussen, S. (1987) Applied Probability and Queues. Wiley, Chichester.Google Scholar
Baccelli, F. and Bremaud, P. (1987) Palm Probabilities and Stationary Queues. (Lecture Notes in Statistics 41). Springer, Berlin.Google Scholar
Barbour, A. D. and Brown, T. C. (1992a) The Stein-Chen method, point processes and compensators. Ann. Prob. 20, 1504–1527.Google Scholar
Barbour, A. D. and Brown, T. C. (1992b) Stein's method and point process approximation. Stoch. Proc. Appl. 43, 9–31.Google Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992) Poisson Approximation. Oxford University Press, Oxford.Google Scholar
Breiman, L. (1968) Probability. Addison-Wesley, New York.Google Scholar
Brémaud, P. (1981) Point Processes and Queues. Springer, New York.Google Scholar
Bremaud, P., Kannurpatti, R. and Mazumdar, R. (1992) Event and time averages: a review. Adv. Appl. Prob. 24, 377–411.CrossRefGoogle Scholar
Brown, T. C. and Donnelly, P. (1993) On conditional intensities and on interparticle correlation in nonlinear death processes. Adv. Appl. Prob. 25, 255–260.CrossRefGoogle Scholar
Brown, T. C. and Pollett, P. K. (1982) Some distributional approximations in Markovian queueing networks. Adv. Appl. Prob. 14, 654–671.CrossRefGoogle Scholar
Burke, P. J. (1956) The output of a queueing system. Operat. Res. 4, 699–704.CrossRefGoogle Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, Chichester.Google Scholar
Melamed, B. (1979) Characterizations of Poisson traffic streams in Jackson queueing networks. Adv. Appl. Prob. 11, 422–438.Google Scholar
Rachev, S. T. (1984) The Monge-Kantorovich mass transfer problem and its stochastic applications. Theory Prob. Appl. 29, 647–676.Google Scholar
Reich, E. (1957) Waiting times when queues are in tandem. Ann. Math. Statist. 28, 768–773.Google Scholar
Walrand, J. and Varaiya, P. (1981) Flows in queueing networks: a martingale approach. Math. Operat. Res. 6, 387–404.Google Scholar