On the Correlation Structure of Closed Queueing Networks

Abstract

Closed networks of exponential queues are prototypes of systems which show strong negative dependence properties. We use the device of a test customer traveling in the network to evaluate the qualitative behavior of the network's correlation. The results are: negative association of successive sojourn times during a cycle; negative correlation of two successive cycle times; feedback-sojourn times are positively associated; correlation between cycles vanishes over long distances geometrically fast. Generalizations to networks with general topology are given. Additionally we prove for cyclic networks that the conditional cycle time stochastically increases when the initial joint queue length vector increases with respect to the partial sum ordering. This connects the space–time correlation with space–time monotonicity.

Key Words:

• Gordon-Newell network
• Cyclic queues
• Product form
• Sojourn times
• Cycle times
• Association
• Negative association
• Point process
• Palm measure

AMS (1991) Subject Classification:

• 60K25

Acknowledgment

We are grateful to the referee for his careful reading of the first version of the paper and for his helpful comments which improved the exposition of the paper.

Hans Daduna's work was supported by Deutscher Akademischer Austausch-dienst. Ryszard Szelki's work was supported by the Alexander von Humboldt Fellowship, and by KBN Grant 2P03A04915