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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 = {W 1, W 2, · ··} 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

Information

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

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