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A family of bounds for the transient behavior of a Jackson network

Published online by Cambridge University Press:  14 July 2016

William A. Massey
William A. Massey*
Affiliation:
AT&T Bell Laboratories
*
∗Postal address: AT&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, USA.
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Abstract

Using operator methods, we derive a family of stochastic bounds for the Jackson network. For its transient joint queue-length distribution, we can stochastically bound it above by various networks that decouple into smaller independent Jackson networks. Each bound is determined by a distinct partitioning of the index set for the nodes. Except for the trivial cases, none of these bounds can be extended to a sample path ordering between it and the original network. Finally, we can partially order the bounds themselves whenever one partition of the index set is the refinement of another. These results suggest new types of partial orders for stochastic processes that are not equivalent to sample-path orderings.

Keywords

TENSOR REPRESENTATIONSTOCHASTIC ORDERING
Type
Short Communications
Information
Journal of Applied Probability , Volume 23 , Issue 2 , June 1986 , pp. 543 - 549
Copyright
Copyright © Applied Probability Trust 1986 

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References

[1] Goodman, J. B. and Massey, W. A. (1984) The non-ergodic Jackson network. J. Appl. Prob. 21, 860–869.Google Scholar
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[6] Tsoucas, P. and Walrand, J. (1984) A note on stochastic bounds for queueing networks. Adv. Appl. Prob. 16, 926–928.Google Scholar