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On the dependence structure and bounds of correlated parallel queues and their applications to synchronized stochastic systems

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Haijun Li  and
Susan H. Xu
Haijun Li*
Affiliation:
Washington State University
Susan H. Xu*
Affiliation:
Pennsylvania State University
*
∗ Postal address: Department of Pure and Applied Mathematics, Washington State University, Pullman, WA 99164, USA. Email address: lih@haijun.math.wsu.edu
∗∗ Postal address: Department of Management Science and Information Systems, Smeal College of Business Administration, Pennsylvania State University, University Park, PA 16802, USA. Email address: shx@psu.edu
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Abstract

This paper studies the dependence structure and bounds of several basic prototypical parallel queueing systems with correlated arrival processes to different queues. The marked feature of our systems is that each queue viewed alone is a standard single-server queuing system extensively studied in the literature, but those queues are statistically dependent due to correlated arrival streams. The major difficulty in analysing those systems is that the presence of correlation makes the explicit computation of a joint performance measure either intractable or computationally intensive. In addition, it is not well understood how and in what sense arrival correlation will improve or deteriorate a system performance measure.

The objective of this paper is to provide a better understanding of the dependence structure of correlated queueing systems and to derive computable bounds for the statistics of a joint performance measure. In this paper, we obtain conditions on arrival processes under which a performance measure in two systems can be compared, in the sense of orthant and supermodular orders, among different queues and over different arrival times. Such strong comparison results enable us to study both spatial dependence (dependence among different queues) and temporal dependence (dependence over different time instances) for a joint performance measure. Further, we derive a variety of upper and lower bounds for the statistics of a stationary joint performance measure. Finally, we apply our results to synchronized queueing systems, using the ideas combined from the theory of orthant and supermodular dependence orders and majorization with respect to weighted trees (Xu and Li (2000)). Our results reveal how a performance measure can be affected, favourably or adversely, by different types of dependencies.

Keywords

Synchronized queuesassemble-to-order systemstochastic comparisonpositive and negative orthant dependencesupermodular dependencemajorization with respect to weighted treesbounds

MSC classification

Primary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 4 , December 2000 , pp. 1020 - 1043
Copyright
Copyright © by the Applied Probability Trust 2000 

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Footnotes

Supported in part by the NSF grant DMI9812994.

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