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Idle period approximations and bounds for the GI/G/1 queue

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Ronald W. Wolff  and
Chia-Li Wang
Ronald W. Wolff*
Affiliation:
University of California, Berkeley
Chia-Li Wang*
Affiliation:
National Dong Hwa University
*
∗ Postal address: Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA 94720, USA.
∗∗ Postal address: Department of Applied Mathematics, National Dong Hwa University, Hualien, Taiwan, ROC. Email address: [email protected]
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Abstract

The average delay for the GI/G/1 queue is often approximated as a function of the first two moments of interarrival and service times. For highly irregular arrivals, however, it varies widely among queues with the same first two moments, even in moderately heavy traffic. Empirically, it decreases as the interarrival time third moment increases. For GI/M/1 queues, a heavy-traffic expression for the average delay with this property has been previously obtained. The method, however, sheds little light on why the third moment arises. We analyze the equilibrium idle-period distribution in heavy traffic using real-variable methods. For GI/M/1 queues, we derive the above heavy-traffic result and also obtain conditions under which it is either an upper or lower bound. Our approach provides an intuitive explanation for the result and also strongly suggests that similar results should hold for general service. This is supported by empirical evidence. For any given service distribution, it has been conjectured that the expected delay under pure-batch arrivals, where interarrival times are scaled Bernoulli random variables, is an upper bound on the average delay over all interarrival distributions with the same first two moments. We investigate this conjecture and show, among other things, that pure-batch arrivals have the smallest third moment. We obtain conditions under which this conjecture is true and present a counterexample where it fails. Arrivals that arise as overflows from other queues can be highly irregular. We show that interoverflow distributions in a certain class have decreasing failure rate.

Keywords

GI/G/1 queueidle periodapproximationboundIMRLDMRLIFRDFRequilibrium distributionbatch arrivaloverflow process

MSC classification

Primary: 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 3 , September 2003 , pp. 773 - 792
Copyright
Copyright © Applied Probability Trust 2003 

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