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The covariance function of the virtual waiting-time process in an M/G/1 queue

Published online by Cambridge University Press:  01 July 2016

Teunis J. Ott
Affiliation:
Case Western Reserve University

Abstract

Let R(t) be the covariance function of the stationary virtual waiting-time process of a stable M/G/1 queue. It is proven that if R(t) exists, i.e., if the service-times have a finite third moment, then R(t) is positive and convex on [0, ∞), with an absolutely continuous derivative R’ and a bounded, non-negative second derivative R″. Also, and R″ cannot be chosen monotone. Contrary to a finding by Beneš [1] it is proven that if and only if the service-times have a finite fourth moment.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1977 

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References

[1] Beneš, V. E. (1957) On queues with Poisson arrivals. Ann. Math. Statist. 28, 670677.CrossRefGoogle Scholar
[2] Cramèr, H. and Leadbetter, M. R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
[3] Feller, W. (1966) An Introduction to Probability Theory and its Applications, II. Wiley, New York.Google Scholar
[4] Keilson, J. (1963) The first passage time problem. Ann. Math. Statist. 34, 10031011.CrossRefGoogle Scholar
[5] Keilson, J. (1971) On the structure of covariance functions and spectral density functions for processes reversible in time. Center for System Science Report CSS 71–03, University of Rochester, Rochester, N.Y. Google Scholar
[6] Keilson, J. and Kooharian, A. (1960) On time-dependent queueing processes. Ann. Math. Statist. 31, 104112.CrossRefGoogle Scholar
[7] Pakes, A. G. (1971) The correlation coefficients of the queue-lengths of some stationary single server queues. J. Austral. Math. Soc. 12, 3546.CrossRefGoogle Scholar
[8] Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
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