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The stationary tail asymptotics in the GI/G/1-type queue with countably many background states

Part of: Computer system organization Special processes

Published online by Cambridge University Press:  01 July 2016

Masakiyo Miyazawa  and
Yiqiang Q. Zhao
Masakiyo Miyazawa*
Affiliation:
Tokyo University of Science
Yiqiang Q. Zhao*
Affiliation:
Carleton University
*
∗ Postal address: Department of Information Sciences, Tokyo University of Science, Noda City, Chiba 278-8510, Japan. Email address: miyazawa@is.noda.sut.ac.jp
∗∗ Postal address: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada.
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Abstract

We consider the asymptotic behaviour of the stationary tail probabilities in the discrete-time GI/G/1-type queue with countable background state space. These probabilities are presented in matrix form with respect to the background state space, and shown to be the solution of a Markov renewal equation. Using this fact, we consider their decay rates. Applying the Markov renewal theorem, it is shown that certain reasonable conditions lead to the geometric decay of the tail probabilities as the level goes to infinity. We exemplify this result using a discrete-time priority queue with a single server and two types of customer.

Keywords

GI/G/1-type queueinfinite background statedecay ratestationary distributionMarkov additive processWiener-Hopf factorizationdual processpriority queue

MSC classification

Primary: 60K25: Queueing theory 60K15: Markov renewal processes, semi-Markov processes
Secondary: 68M20: Performance evaluation; queueing; scheduling
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 4 , December 2004 , pp. 1231 - 1251
Copyright
Copyright © Applied Probability Trust 2004 

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