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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: [email protected]
∗∗ Postal address: Department of Management Science and Information Systems, Smeal College of Business Administration, Pennsylvania State University, University Park, PA 16802, USA. Email address: [email protected]
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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.

References

Baccelli, F. (1985). Two parallel queues created by arrivals with two demands. Res. Rept 426, INRIA-Rocquencourt, France.Google Scholar
Baccelli, F., and Makowski, A. M. (1989). Multidimensional stochastic ordering and associated random variables, Operat. Res. 37, 478–487.CrossRefGoogle Scholar
Baccelli, F., Makowski, A. M., and Shwartz, A. (1989). The fork-join queue and related systems with synchronization constraints: stochastic ordering and computable bounds, Adv. Appl. Prob. 21, 629–660.CrossRefGoogle Scholar
Esary, J. D., Proschan, F., and Walkup, D. W. (1967). Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474.CrossRefGoogle Scholar
Flatto, L, and Hahn, S. (1984). Two parallel queues created by arrivals with two demands I. SIAM J. Appl. Math. 44, 1041–1053.CrossRefGoogle Scholar
Gershwin, S. B. (1994). Manufacturing Systems Engineering. Prentice Hall, Englewood Cliffs.Google Scholar
Glasserman, P., and Wang, Y. (1998). Leadtime-inventory trade-offs in assemble-to-order systems. Operat. Res. 46, 858–871.CrossRefGoogle Scholar
Hausman, W., Lee, H., and Zhang, A. (1998). Joint demand fulfillment probability in a multi-item inventory system with independent order-up-to policies. Eur. J. Operat. Res. 109, 646–659.CrossRefGoogle Scholar
Levy, H., and Sidi, M. (1991). Polling systems with simultaneous arrivals, IEEE Trans. Commun. 39, 823–827.CrossRefGoogle Scholar
Li, H., and Xu, S. H. (2000). Stochastic bounds and dependence properties of survival times in a multicomponent shock model. To appear in J. Multivar. Anal.Google Scholar
Nelson, R., and Tantawi, A. N. (1988). Approximate analysis of fork/join synchronization in parallel systems. IEEE Trans. Comput. 37, 739–743.CrossRefGoogle Scholar
Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and their Applications. Academic Press, New York.Google Scholar
Song, J. S. (1998). On the order fill rate in a multi-item, base-stock inventory system. Operat. Res. 46, 831–845.CrossRefGoogle Scholar
Song, J. S., Xu, S. H., and Liu, B. (1999). Order-fulfillment performance measures in an assemble-to-order system with stochastic leadtimes. Operat. Res. 47, 131–149.CrossRefGoogle Scholar
Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability. Springer, New York.CrossRefGoogle Scholar
Takagi, H. (1993). Queueing Analysis—Foundation of Performance Evaluation. Vol. 3: Discrete-Time Systems. North-Holland, Amsterdam.Google Scholar
Tchen, A. H. (1980). Inequalities for distributions with given marginals. Ann. Prob. 8, 814–827.CrossRefGoogle Scholar
Tong, Y. L. (1980). Probability Inequalities in Multivariate Distributions. Academic Press, New York.Google Scholar
Xu, S. H. (1999). Structural analysis of a queueing system with multi-classes of correlated arrivals and blocking. Operat. Res. 47, 264–276.CrossRefGoogle Scholar
Xu, S. H., and Li, H. (2000). Majorization of weighted trees–a new tool to study correlated stochastic systems. Math. Operat. Res. 25, 298–323.CrossRefGoogle Scholar