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The M/G/1 queue with negative customers

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Peter G. Harrison  and
Edwige Pitel
Peter G. Harrison*
Affiliation:
Imperial College
Edwige Pitel*
Affiliation:
Imperial College
*
∗ Postal address: Department of Computing, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK.
∗ Postal address: Department of Computing, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK.
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Abstract

We derive expressions for the generating function of the equilibrium queue length probability distribution in a single server queue with general service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. For the case of first come first served queueing discipline for the positive customers, we compare the killing strategies in which either the last customer in the queue or the one in service is removed by a negative customer. We then consider preemptive-restart with resampling last come first served queueing discipline for the positive customers, combined with the elimination of the customer in service by a negative customer—the case of elimination of the last customer yields an analysis similar to first come first served discipline for positive customers. The results show different generating functions in contrast to the case where service times are exponentially distributed. This is also reflected in the stability conditions. Incidently, this leads to a full study of the preemptive-restart with resampling last come first served case without negative customers. Finally, approaches to solving the Fredholm integral equation of the first kind which arises, for instance, in the first case are considered as well as an alternative iterative solution method.

Keywords

G-QUEUESG-NETWORKSQUEUEING NETWORKSBOUNDARY VALUE PROBLEMSCOMPLEX ANALYSISFREDHOLM INTEGRAL EQUATION

MSC classification

Primary: 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 2 , June 1996 , pp. 540 - 566
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

The work of PGH is partly supported by the European Commission under ESPRIT BRA QMIPS no. 7269 and by the ESPRC under grant no. GR/H46244.

The work of EP is supported by the European Commission under ESPRIT bursary no. ERBCHBICT920179.

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