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A service system with two stages of waiting and feedback of customers

Published online by Cambridge University Press:  14 July 2016

Osman M. E. Ali  and
Marcel F. Neuts
Osman M. E. Ali*
Affiliation:
Military Academy, Cairo
Marcel F. Neuts*
Affiliation:
University of Delaware
*
∗ Postal address: Department of Mathematics, Military Academy, Cairo, Egypt.
∗∗ Postal address: Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, U.S.A.
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Abstract

Customers initially enter a service unit via a waiting room. The customers to be served are stored in a service room which is replenished by the transfer of all those in the waiting room at the points in time where the service room becomes empty. At those epochs of transfer, positive random numbers of ‘overhead customers' are also added to the service room. Algorithmically tractable expressions for the stationary distributions of queue lengths and waiting times at various embedded random epochs are derived. The discussion generalizes an earlier treatment by Takács in several directions.

Keywords

QUEUEING THEORYMARKOV RENEWAL PROCESSGALTON–WATSON PROCESSCOMPUTATIONAL PROBABILITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 2 , June 1984 , pp. 404 - 413
Copyright
Copyright © Applied Probability Trust 1984 

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Footnotes

Research supported by the National Science Foundation under Grant No. ECS-8205404 at the University of Delaware.

References

[1] Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
[2] Neuts, M. F. (1980) Computational problems related to the Galton–Watson process. In Computational Probability, Proceedings of the 1975 Brown Actuarial Research Conference on Computational Probability, ed. Kahn, P. M., Academic Press, New York, 11–37.CrossRefGoogle Scholar
[3] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore, Md.Google Scholar
[4] Takács, L. (1977) A queuing model with feedback. Rev. Française Automat. Informat. Recherche Opérationnelle 11, 345–354.Google Scholar