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On queues with periodic Poisson input

Published online by Cambridge University Press:  14 July 2016

Austin J. Lemoine
Austin J. Lemoine*
Affiliation:
Systems Control, Inc.
*
∗ Postal address: Systems Control, Inc., 1801 Page Mill Rd., Palo Alto, CA 94304, U.S.A.
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Abstract

This paper is concerned with asymptotic results for a single-server queue having periodic Poisson input and general service-time distribution, and carries forward the analysis of this model initiated in Harrison and Lemoine. First, it is shown that a theorem of Hooke relating the stationary virtual and actual waiting-time distributions for the GI/G/1 queue extends to the periodic Poisson model; it is then pointed out that Hooke's theorem leads to the extension (developed in [3]) of a related theorem of Takács. Second, it is demonstrated that the asymptotic distribution for the server-load process at a fixed ‘time of day' coincides with the distribution for the supremum, over the time horizon [0,∞), of the sum of a stationary compound Poisson process with negative drift and a continuous periodic function. Some implications of this characterization result for the computation and approximation of the asymptotic distributions are then discussed, including a direct proof, for the periodic Poisson case, of a recent result of Rolski comparing mean asymptotic customer waiting time with that of a corresponding M/G/1 system.

Keywords

SINGLE-SERVER QUEUESPERIODIC POISSON INPUTWAITING TIMESERVER LOADASYMPTOTIC DISTRIBUTIONSCOMPOUND POISSON PROCESSPERIODIC FUNCTIONS

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 4 , December 1981 , pp. 889 - 900
Copyright
Copyright © Applied Probability Trust 1981 

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Footnotes

This work was sponsored by the National Science Foundation under Grant No. ENG–7824568.

References

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