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Open networks of queues: their algebraic structure and estimating their transient behavior

Published online by Cambridge University Press:  01 July 2016

William A. Massey
William A. Massey*
Affiliation:
Bell Laboratories
*
*Postal address: Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, U.S.A.
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Abstract

We develop the mathematical machinery in this paper to construct a very general class of Markovian network queueing models. Each node has a heterogeneous class of customers arriving at their own Poisson rate, ultimately to receive their own exponential service requirements. We add to this a very general type of service discipline as well as class (node) switching. These modifications allow us to model in the limit, service with a general distribution. As special cases for this model, we have the product-form networks formulated by Kelly, as well as networks with priority scheduling. For the former, we give an algebraic proof of Kelly's results for product-form networks. This is an approach that motivates the form of the solution, and justifies the various needs of local and partial balance conditions.

For any network that belongs to this general model, we use the operator representation to prove stochastic dominance results. In this way, we can take the transient behavior for very complicated networks and bound its joint queue-length distribution by that for M/M/1queues.

Keywords

FOCK SPACESFREE SEMIGROUPSSTOCHASTIC DOMINANCEKELLY NETWORKSTENSOR GEOMETRIC DISTRIBUTION
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 1 , March 1984 , pp. 176 - 201
Copyright
Copyright © Applied Probability Trust 1984 

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References

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[5] Massey, W. A. (1984) An operator analytic approach to the Jackson network. J. Appl. Prob. 21 (2).CrossRefGoogle Scholar