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Asymptotic correlation in a queue

Published online by Cambridge University Press:  14 July 2016

B. D. Craven
B. D. Craven*
Affiliation:
University of Melbourne and University of Sheffield
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Extract

Let Xt denote the waiting time of customer t in a stationary GI/G/1 queue, with traffic intensity τ; let ρn denote the correlation between Xt and Xt+n. For a rational GI/G/1 queue, in which the distribution of the difference between arrival and service intervals has a rational characteristic function, it is shown that, for large n, ρn is asymptotically proportional to n–3/2e–βn, where β and the factor of proportionality are calculable. The asymptotic law n–3/2e–βn applies also to the approach of the waiting-time distribution to the stationary state in an initially empty rational GI/G/1 queue, and to the correlations in the queueing systems recently analysed by Cohen [1]. Its more general applicability is discussed.

Type
Research Papers
Information
Journal of Applied Probability , Volume 6 , Issue 3 , December 1969 , pp. 573 - 583
Copyright
Copyright © Applied Probability Trust 

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References

[1] Cohen, J. W. (1967) On two integral equations of queueing theory. J. Appl. Prob. 4, 343–355.CrossRefGoogle Scholar
[2] Copson, E. T. (1935) Theory of Functions of a Complex Variable. Oxford University Press.Google Scholar
[3] Craven, B. D. (1965) Serial dependence of a Markov process. J. Aust. Math. Soc. 5, 299–314.CrossRefGoogle Scholar
[4] Craven, B. D. (1963) Asymptotic transient behaviour of the bulk service queue. J. Aust. Math. Soc. 3, 503–512.CrossRefGoogle Scholar
[5] Daley, D. J. (1968) The serial correlation coefficients of waiting times in a stationary single server queue. J. Aust. Math. Soc. 8, 683–699.CrossRefGoogle Scholar
[6] Keilson, J. (1964) An alternative to Wiener-Hopf methods for the study of bounded processes. J. Appl. Prob. 1, 85–120.CrossRefGoogle Scholar
[7] Kingman, J. F. C. (1966) On the algebra of queues. J. Appl. Prob. 3, 285–326.CrossRefGoogle Scholar
[8] Lindley, D. V. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–295.CrossRefGoogle Scholar
[9] Smith, W. L. (1953) Distribution of queueing times. Proc. Camb. Phil. Soc. 49, 449–463.CrossRefGoogle Scholar