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Limit theorems for periodic queues

Published online by Cambridge University Press:  14 July 2016

J. Michael Harrison  and
Austin J. Lemoine
J. Michael Harrison
Affiliation:
Stanford University
Austin J. Lemoine
Affiliation:
Control Analysis Corporation, Palo Alto, California
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Abstract

Consider a single-server queue with service times distributed as a general random variable S and with non-stationary Poisson input. It is assumed that the arrival rate function λ (·) is periodic with average value λ and that ρ = λE(S) < 1. Both weak and strong limit theorems are proved for the waiting-time process W = {W1, W2, · ··} and the server load (or virtual waiting-time process) Z = {Z(t), t ≧ 0}. The asymptotic distributions associated with Z and W are shown to be related in various ways. In particular, we extend to the case of periodic Poisson input a well-known formula (due to Takács) relating the limiting virtual and actual waiting-time distributions of a GI/G/1 queue.

Keywords

NON-STATIONARY QUEUESPERIODIC POISSON PROCESSREGENERATIVE EVENTRENEWAL THEORYWAITING TIMESERVER LOAD
Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 3 , September 1977 , pp. 566 - 576
Copyright
Copyright © Applied Probability Trust 1977 

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References

[1] Beneš, V. E. (1963) General Stochastic Processes in the Theory of Queues. Addison-Wesley, Reading, Mass.Google Scholar
[2] Blumenthal, R. M. and Getoor, R. K. (1968) Markov Processes and Potential Theory. Academic Press, New York.Google Scholar
[3] Feller, W. (1968) An Introduction to Probability Theory and Its Applications , I, 3rd edn. Wiley, New York.Google Scholar
[4] Hasofer, A. M. (1964) On the single-server queue with non-homogeneous Poisson input and general service times. J. Appl. Prob. 1, 369–384.CrossRefGoogle Scholar
[5] Lemoine, A. J. (1974) On two stationary distributions for the stable GI/G/1 queue. J. Appl. Prob. 11, 849–852.Google Scholar
[6] Neveu, J. (1965) Mathematical Foundations of the Calculus of Probability. Holden-Day, San Francisco.Google Scholar
[7] Reich, E. (1958) On the integro-differential equation of Takács. I. Ann. Math. Statist. 29, 563–570.Google Scholar
[8] Reich, E. (1959) On the integro-differential equation of Takács. II. Ann. Math. Statist. 30, 143–148.Google Scholar
[9] Takács, L. (1955) Investigation of waiting-time problems by reduction to Markov processes. Acta. Math. Acad. Sci. Hungar. 6, 101–129.Google Scholar
[10] Takács, L. (1963) The limiting distribution of the virtual waiting time and the queue size for a single-server queue with recurrent input and general service times. Sankhya A 25, 91–100.Google Scholar