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On a tandem queueing model with identical service times at both counters, I

Published online by Cambridge University Press:  01 July 2016

O. J. Boxma
O. J. Boxma*
Affiliation:
University of Utrecht
*
∗Postal address: Mathematical Institute, University of Utrecht, Budapestlaan 6, Utrecht 3508 TA, The Netherlands.
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Abstract

This paper considers a queueing system consisting of two single-server queues in series, in which the service times of an arbitrary customer at both queues are identical. Customers arrive at the first queue according to a Poisson process.

Of this model, which is of importance in modern network design, a rather complete analysis will be given. The results include necessary and sufficient conditions for stationarity of the tandem system, expressions for the joint stationary distributions of the actual waiting times at both queues and of the virtual waiting times at both queues, and explicit expressions (i.e., not in transform form) for the stationary distributions of the sojourn times and of the actual and virtual waiting times at the second queue.

In Part II (pp. 644–659) these results will be used to obtain asymptotic and numerical results, which will provide more insight into the general phenomenon of tandem queueing with correlated service times at the consecutive queues.

Keywords

QUEUEING THEORYTANDEM QUEUESSTATIONARITY CONDITIONSSOJOURN TIMEACTUAL WAITING TIMEVIRTUAL WAITING TIME
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 3 , September 1979 , pp. 616 - 643
Copyright
Copyright © Applied Probability Trust 1979 

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References

1. Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
2. Boxma, O. J. (1977) Analysis of Models for Tandem Queues. Ph.D. Thesis, University of Utrecht.Google Scholar
3. Boxma, O. J. (1978) On the longest service time in a busy period of the M/G/1 queue. Stock. Proc. Appl. 8, 93–100.CrossRefGoogle Scholar
4. Boxma, O. J. (1979) On a tandem queueing model with identical service times at both counters, II. Adv. Appl. Prob. 11, 644–659.Google Scholar
5. Cohen, J. W. (1969) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
6. Cohen, J. W. (1976) On Regenerative Processes in Queueing Theory. Springer Verlag, Berlin.Google Scholar
7. Doetsch, G. (1943) Theorie und Anwendung der Laplace Transformation. Dover, New York.Google Scholar
8. Jackson, J. R. (1957) Networks of waiting lines. Opns Res. 5, 518–521.Google Scholar
9. Kelly, F. P. (1976) Networks of queues. Adv. Appl. Prob. 8, 416–432.Google Scholar
10. Kleinrock, L. (1964) Communication Nets; Stochastic Message Flow and Delay. McGraw-Hill, New York.Google Scholar
11. Mitchell, C. R., Paulson, A. S. and Beswick, C. A. (1977) The effect of correlated exponential service times on single server tandem queues. Naval Res. Logist. Quart. 24, 95–112.CrossRefGoogle Scholar
12. Titchmarsh, E. C. (1952) The Theory of Functions. Clarendon Press, Oxford.Google Scholar