Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-21T22:35:01.183Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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

References

[1] Baskett, F., Chandy, K. M., Muntz, R. R. and Palacios, F. G. (1975) Open, closed and mixed networks of queues with product form. J. Assoc. Comput. Mach. 22, 248–260.CrossRefGoogle Scholar
[2] Garcia, C. D. and Zangwill, W. I. (1981) Pathways to Solutions, Fixed Points, and Equilibria. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
[3] Gelenbe, E. and Mitrani, I. (1980) Analysis and Synthesis of Computer Systems. Academic Press, London.Google Scholar
[4] Gelenbe, E. and Pujolle, G. (1986) Introduction to Networks of Queues. Wiley, Chichester.Google Scholar
[5] Gelenbe, E., Glynn, P. and Sigman, K. (1991) Queues with negative arrivals. J. Appl. Prob. 28, 245–250.Google Scholar
[6] Walrand, J. (1988) An Introduction to Queueing Networks. Prentice Hall, Englewood Cliffs, NJ.Google Scholar