Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-17T16:38:01.880Z Has data issue: false hasContentIssue false
  • English
  • Français

The steady state of a multi-server mixed queue

Published online by Cambridge University Press:  01 July 2016

N. B. Slater  and
T. C. T. Kotiah
N. B. Slater
Affiliation:
University of Hull
T. C. T. Kotiah*
Affiliation:
University of Hull
*
∗ Now at Southern Illinois University, Edwardsville.
Get access Rights & Permissions [Opens in a new window]

Abstract

In a multi-server queueing system in which the customers are of several different types, it is useful to define states which specify the types of customers being served as well as the total number present. Analogies with some problems in statistical mechanics are found fruitful. Certain generating functions are defined in such a way that they satisfy a system of linear equations. Solution of the associated eigenvector problem shows that the steady-state probabilities for states in which all the servers are busy can be represented by a weighted sum of geometric probabilities.

Keywords

MULTI-SERVER QUEUEEIGENVECTOR METHODSGEOMETRIC DISTRIBUTIONSFACTORIZATION OF DISCRIMINANTS
Type
Research Article
Information
Advances in Applied Probability , Volume 5 , Issue 3 , December 1973 , pp. 614 - 631
Copyright
Copyright © Applied Probability Trust 1973 

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

The final version of this paper was prepared for the printer by Dr. J. W. Thompson after Professor Slater's death on 31 January 1973.

References

Ancker, C. J. Jr. and Gafarian, A. V. (1961) Queueing with multiple Poisson inputs and exponential service times. Operations Research 9, 321–327.CrossRefGoogle Scholar
Kiefer, J. and Wolfowitz, J. (1955) On the theory of queues with many servers. Trans. Amer. Math. Soc. 78, 1–18.Google Scholar
Kotiah, T. C. T. (1968) Some results in single-server and multi-server queues. , University of Hull.Google Scholar
Kotiah, T. C. T. and Slater, N. B. (1973) On two-server Poisson queues with two types of customer. Operations Research 21, 597–603.Google Scholar
Painvin, L. (1858) Sur un certain système d'équations linéaires. J. (de Liouville) de Math. (2) iii, 41–46. Quoted in Muir, T. (1911) The Theory of Determinants. Vol. II, 432–434. Macmillan, London.Google Scholar
Pollaczek, F. (1961) Théorie analytique des problèmes stochastiques relatifs à un groupe de lignes téléphonique avec dispositif d'attente. Mem. Sci. Math. 150.Google Scholar