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Generalizations of the Pollaczek-Khinchin integral equation in the theory of queues

Published online by Cambridge University Press:  01 July 2016

Marcel F. Neuts
Marcel F. Neuts*
Affiliation:
The University of Arizona
*
∗Postal address: Dept of Systems and Industrial Engineering, University of Arizona, Tucson, A2 85721, USA
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Abstract

A classical result in queueing theory states that in the stable M/G/1 queue, the stationary distribution W(x) of the waiting time of an arriving customer or of the virtual waiting time satisfies a linear Volterra integral equation of the second kind, of convolution type. For many variants of the M/G/1 queue, there are corresponding integral equations, which in most cases differ from the Pollaczek–Khinchin equation only in the form of the inhomogeneous term. This leads to interesting factorizations of the waiting-time distribution and to substantial algorithmic simplifications. In a number of priority queues, the waiting-time distributions satisfy Volterra integral equations whose kernel is a functional of the busy-period distribution in related M/G/1 queues. In other models, such as the M/G/1 queue with Bernoulli feedback or with limited admissions of customers per service, there is a more basic integral equation of Volterra type, which yields a probability distribution in terms of which the waiting-time distributions are conveniently expressed.

For several complex queueing models with an embedded Markov renewal process of M/G/1 type, one obtains matrix Volterra integral equations for the waiting-time distributions or for related vectors of mass functions. Such models include the M/SM/1 and the N/G/1 queues, as well as the M/G/1 queue with some forms of bulk service.

When the service-time distributions are of phase type, the numerical computation of waiting-time distributions may commonly be reduced to the solution of systems of linear differential equations with constant coefficients.

Keywords

QUEUEING THEORYWAITING-TIME DISTRIBUTIONCOMPUTATIONAL PROBABILITY
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 4 , December 1986 , pp. 952 - 990
Copyright
Copyright © Applied Probability Trust 1986 

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