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Discrete queues with one server

Published online by Cambridge University Press:  14 July 2016

Lajos Takács
Lajos Takács*
Affiliation:
Case Western Reserve University
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Extract

In this paper mathematical methods are given for finding the distributions of the virtual waiting time and the occupation time for single-server queues in which the interarrival times and the service times are discrete random variables taking on positive or non-negative integral values. First we shall consider deterministic queues in which the interarrival times and the service times are numerical (non-random) quantities and deduce a fundamental relation for such queues. Then we shall consider random queues with recurrent input and general service times and study the time dependent behavior of such queues.

Type
Research Papers
Information
Journal of Applied Probability , Volume 8 , Issue 4 , December 1971 , pp. 691 - 707
Copyright
Copyright © Applied Probability Trust 1971 

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References

[1] Baxter, G. (1958) An operator identity. Pacific J. Math. 8, 649–663.CrossRefGoogle Scholar
[2] Baxter, G. (1960) An analytic problem whose solution follows from a simple algebraic identity. Pacific J. Math. 10, 731–742.Google Scholar
[3] Feller, W. (1959) On combinatorial methods in fluctuation theory. Probability and Statistics. The Harald Cramér Volume. Edited by Grenander, U. Almqvist and Wiksell, Stockholm. 79–91.Google Scholar
[4] Kingman, J. F. C. (1966) On the algebra of queues. J. Appl. Prob. 3, 285–326.Google Scholar
[5] Osgood, W. F. (1948) Functions of a Complex Variable. Hafner, New York.Google Scholar
[6] Pollaczek, F. (1952) Fonctions caractéristiques de certaines répartitions définies au moyen de la notion d'ordre. Application à la théorie des attentes. Comptes Rendus Acad. Sci. Paris 234, 2334–2336.Google Scholar
[7] Pollaczek, F. (1957) Problèmes stochastiques posés par le phénomène de formation d'une queue d'attente à un guichet et par des phénomènes apparentés. Mém. Sci. Math. 136, Gauthiers-Villars, Paris.Google Scholar
[8] Spitzer, F. (1956) A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82, 323–339.Google Scholar
[9] Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.Google Scholar
[10] Wendel, J. G. (1958) Spitzer's formula: A short proof. Proc. Amer. Math. Soc. 9, 905–908.Google Scholar
[11] Wendel, J. G. (1962) Brief proof of a theorem of Baxter. Math. Scand. 11, 107–108.CrossRefGoogle Scholar