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Product-form queueing networks with negative and positive customers

Published online by Cambridge University Press:  14 July 2016

Erol Gelenbe
Erol Gelenbe*
Affiliation:
Ecole des Hautes Etudes en Informatique
*
∗ Postal address: Ecole des Hautes Etudes en Informatique, Université René Descartes (Paris V), rue des Saints-Pères, 75006 Paris, France.
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Abstract

We introduce a new class of queueing networks in which customers are either ‘negative' or ‘positive'. A negative customer arriving to a queue reduces the total customer count in that queue by 1 if the queue length is positive; it has no effect at all if the queue length is empty. Negative customers do not receive service. Customers leaving a queue for another one can either become negative or remain positive. Positive customers behave as ordinary queueing network customers and receive service. We show that this model with exponential service times, Poisson external arrivals, with the usual independence assumptions for service times, and Markovian customer movements between queues, has product form. It is quasi-reversible in the usual sense, but not in a broader sense which includes all destructions of customers in the set of departures. The existence and uniqueness of the solutions to the (nonlinear) customer flow equations, and hence of the product form solution, is discussed.

Keywords

WORK CANCELLATIONNEGATIVE CUSTOMERS
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 3 , September 1991 , pp. 656 - 663
Copyright
Copyright © Applied Probability Trust 1991 

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Footnotes

This research was supported by the Distributed Algorithms Section of C3-CNRS (French National Program on Parallelism and Concurrency).

References

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