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Single server queue with uniformly bounded virtual waiting time

Published online by Cambridge University Press:  14 July 2016

J. W. Cohen
J. W. Cohen*
Affiliation:
Technological University, Delft
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Summary

In a previous paper [4] the author studied the stochastic process {wn, n = 1,2, …}, recursively defined by with K a positive constant, τ1, τ2, … σ1, σ2, …, independent, nonnegative stochastic variables. τ1,τ2…, are identically distributed, and σ1,σ2,…, are also identically distributed variables. For this process the generating function of the Laplace-Stieltjes transforms of the joint distribution of Wn, σ2 + … + σn and τ1 + … + τn−1 has been obtained. Closely related to the process {wn, n = 1, 2,…} is the process {un, n = 1, 2,…} with {un = K + [wn + τn − K]−, n = 1,2,…; these are dual processes.

In the present paper we study the stationary distributions of the processes {wn, n= 1,2, …} and {un, n = 1,2, …}, and the distributions ot the entrance times and return times of the events “wn, n = 0” and “un = K” for some n, for discrete as well as for continuous time. For these events various taboo probabilities are also investigated. The mathematical descri ption of the processes {wn, n = 1,2, …} and {un, n= 1,2, …} gives all the necessary information about the time-dependent behaviour for the general dam model with finite capacity K, since the process {wn, n= 1,2, …} is the basic process for such dam models. In Sections 5, 6 and 7 the general theory is applied to the models M/G/1 and G/M/1. Complete explicit solutions are obtained for these models.

The present theory also leads to new and important results for the queueing system or dam model G/G/1 with infinite capacity. For instance the joint distribution of the busy period (or wet period) and of the supremum of the dam content dunng this period is obtained.

Type
Research Papers
Information
Journal of Applied Probability , Volume 5 , Issue 1 , April 1968 , pp. 93 - 122
Copyright
Copyright © Sheffield: Applied Probability Trust 

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References

[1] Prabhu, N. U. (1965) Queues and Inventories. John Wiley and Sons, New York.Google Scholar
[2] Gaver, D. P. and Miller, R. G. (1962) Limiting distributions for some storage problems. Studies in Applied Probability and Management Science. Stanford U. Press.Google Scholar
[3] Daley, D. J. (1964) Single server queueing systems with uniformly limited queueing time. J. Aust. Math. Soc. 4, 489–505.CrossRefGoogle Scholar
[4] Cohen, J. W. (1967) On two integral equations of queueing theory. J. Appl. Prob. 4, 343–355.Google Scholar
[5] Doob, J. L. (1953) Stochastic Processes. John Wiley and Sons, New York.Google Scholar
[6] Cohen, J. W. (1968) Extreme value distributions for the M/G/1 and the G/M/1 queueing system. Ann. Inst. Henri Poincaré 4, 83–98.Google Scholar
[7] Takács, L. (1964) Application of ballot theorems in the theory of queues. Proc. Symp. Congestion Theory, Chapel Hill. U. of North Carolina Press, Chapel Hill.Google Scholar
[8] Cohen, J W and Greenberg, I (1968) Distributions of crossings of level K in a busy period for the M/G/1 queueing system. Ann. Inst. Henri Poincaré 4, 75–81.Google Scholar