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MR/GI/1 queues by positively correlated arrival stream

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

R. Szekli ,
R. L. Disney  and
S. Hur
R. Szekli*
Affiliation:
Wroclaw University
R. L. Disney*
Affiliation:
Texas A & M University
S. Hur*
Affiliation:
Texas A & M University
*
∗ Postal address: Mathematical Institute, Wrocław University, 50-384 Wrocław, pl. Grunwaldzki 2/4, Poland.
∗∗ Postal address: Department of Industrial Engineering, Texas A & M University, College Station, TX 77843-3131, USA.
∗∗ Postal address: Department of Industrial Engineering, Texas A & M University, College Station, TX 77843-3131, USA.
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Abstract

The effects of dependencies (such as association) in the arrival process to a single server queue on mean queue lengths and mean waiting times are studied. Markov renewal arrival processes with a particular transition matrix for the underlying Markov chain are used which allow us to change dependency properties without at the same time changing distributional conditions. It turns out that correlations do not seem to be pure effects, and three main factors are studied: (a) differences in the mean interarrival times in the underlying Markov renewal process, (b) intensity in the Markov renewal jump process, (c) variability in the point processes underlying the Markov renewal process. It is shown that the mean queue length can be made arbitrarily large in the class of queues with the same interarrival distributions and the same service time distributions (with fixed smaller than one traffic intensity), by making (a) large enough and (b) small enough. The existence of the moments of interest is confirmed and some stochastic comparison results for actual waiting times are shown.

Keywords

MARKOV RENEWAL QUEUESCORRELATIONSASSOCIATIONSTOCHASTIC ORDERING

MSC classification

Primary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 2 , June 1994 , pp. 497 - 514
Copyright
Copyright © Applied Probability Trust 1994 

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