Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T06:19:51.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Response time distributions in tandem G-networks

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Peter G. Harrison  and
Edwige Pitel
Peter G. Harrison*
Affiliation:
Imperial College, London
Edwige Pitel*
Affiliation:
Imperial College, London
*
Postal address for both authors: Department of Computing, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK.
Postal address for both authors: Department of Computing, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

The Laplace transform of the probability distribution of the end-to-end delay in tandem networks is obtained where the first and/or second queue are G-queues, i.e. they have negative arrivals. For the most general case the method is based on the solution of a boundary value problem on a closed contour in the complex plane, which itself reduces to the solution of a Fredholm integral equation of the second kind. We also consider the dependence or independence of the sojourn times at each queue in the two special cases where only one of the queues is a G-queue, the other having no negative arrivals.

Keywords

NEGATIVE CUSTOMERSQUEUEING NETWORKSSOJOURN TIMESEND-TO-END DELAYBOUNDARY VALUE PROBLEMSFREDHOLM INTEGRAL EQUATION

MSC classification

Primary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 1 , March 1995 , pp. 224 - 246
Copyright
Copyright © Applied Probability Trust 1995 

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

∗

Supported by the European Commission under ESPRIT BRA QMIPS n° 7269.

†

Supported by the European Commission under ESPRIT bursary n° ERBCHBICT920179.

References

Ahlfors, L. V. (1979) Complex Analysis. McGraw-Hill, New York.Google Scholar
Coffman, E. G. Jnr., Fayolle, G. and Mitrani, I. (1986) Sojourn times in a tandem queue with overtaking: Reduction to a boundary value problem. Commun. Statist. Stoch. Models 2, 43–65.Google Scholar
Cohen, J. W. (1969) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
Dubner, H. and Abate, J. (1968) Numerical inversion of Laplace transforms by relating them to the finite Fourier Cosine transform. J. Assoc. Comput. Mach. 15, 115–123.Google Scholar
Fayolle, G., Iasnogorodski, R. and Mitrani, I. (1983) The distribution of sojourn times in a queueing network with overtaking. Performance '83, ed. Agrawala, A. K. and Tripathi, S. K., pp. 477–486, North-Holland, Amsterdam.Google Scholar
Gakhov, F. D. (1976) Boundary Value Problems. Pergamon Press, Oxford.Google Scholar
Gelenbe, E. (1991) Product form networks with negative and positive customers. J. Appl. Prob. 28, 656–663.Google Scholar
Gelenbe, E. (1992) G-networks: Negative customers with batch removal. Performance Evaluation. Submitted.Google Scholar
Gelenbe, E. and Schassberger, R. (1990) Note on the stability of G-networks. Research Report 90-8. Université René Descartes, Paris.Google Scholar
Gelenbe, E., Glynn, P. and Sigman, K. (1991) Queues with negative arrivals. J. Appl. Prob. 28, 245–250.CrossRefGoogle Scholar
Harrison, P. G. and Pitel, E. (1993) Sojourn times in single server queues with negative customers. J. Appl. Prob. 30, 943–963.Google Scholar
Mitrani, I. (1985) Response time problems in communication networks. J. R. Statist. Soc. 47, 396–406.Google Scholar
Muskhelishvili, N. I. (1946) Singular Integral Equations. Noordhoff, Groningen.Google Scholar
Pitel, E. (1994) Queues with negative customers. , Computing Dept., Imperial College, London.Google Scholar
Reich, E. (1957) Waiting times when queues are in tandem. Ann. Math. Statist. 28, 768–773.CrossRefGoogle Scholar
Smirnov, V. I. (1964) A Course of Higher Mathematics, Vol. 4, Integral Equations and Partial Differential Equations. Pergamon Press, Oxford.Google Scholar