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The use of Spitzer's identity in the investigation of the busy period and other quantities in the queue GI/G/1

Published online by Cambridge University Press:  09 April 2009

J. F. C. Kingmán
J. F. C. Kingmán
Affiliation:
Statistical Laboratory, University of Cambridge, England
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As an illustration of the use of his identity [10], Spitzer [11] obtained the Pollaczek-Khintchine formula for the waiting time distribution of the queue M/G/1. The present paper develops this approach, using a generalised form of Spitzer's identity applied to a three-demensional random walk. This yields a number of results for the general queue GI/G/1, including Smith' solution for the stationary waiting time, which is established under less restrictive conditions that hitherto (§ 5). A soultion is obtained for the busy period distribution in GI/G/1 (§ 7) which can be evaluated when either of the distributions concerned has a rational characteristic function. This solution contains some recent results of Conolly on the quene GI/En/1, as well as well-known results for M/G/1.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 2 , Issue 3 , February 1962 , pp. 345 - 356
Copyright
Copyright © Australian Mathematical Society 1962

References

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