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On steady-state queue size distributions of the discrete-time GI/G/1 queue

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Tao Yang  and
M. L. Chaudhry
Tao Yang*
Affiliation:
Technical University of Nova Scotia
M. L. Chaudhry*
Affiliation:
Royal Military College of Canada
*
∗ Postal address: Department of Industrial Engineering, Technical University of Nova Scotia, Halifax, Nova Scotia B3J 2X4, Canada.
∗∗ Postal address: Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, Ontario K7K 5L0, Canada.
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Abstract

In this paper, we present results for the steady-state system length distributions of the discrete-time GI/G/1 queue. We examine the system at customer arrival epochs (customer departure epochs) and use the residual service time (residual interarrival time) as the supplementary variable. The embedded Markov chain is of GI/M/1 type if the embedding points are arrival epochs and is of M/G/1 type if the embedding points are departure epochs. Using the matrix analytic method, we identify the necessary and sufficient condition for both Markov chains to be positive recurrent. For the GI/M/1 type chain, we derive a matrix-geometric solution for its steady-state distribution and for the M/G/1 type chain, we develop a simple linear transformation that relates it to the GI/M/1 type chain and leads to a simple analytic solution for its steady-state distribution. We also show that the steady-state system length distribution at an arbitrary point in time can be obtained by a simple linear transformation of the matrix-geometric solution for the GI/M/1 type chain. A number of applications of the model to communication systems and numerical examples are also discussed.

Keywords

GI/G/1 DISCRETE-TIME QUEUESMATRIX ANALYTIC TECHNIQUE

MSC classification

Primary: 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 4 , December 1996 , pp. 1177 - 1200
Copyright
Copyright © Applied Probability Trust 1996 

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