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Characterizations of Poisson traffic streams in Jackson queueing networks

Published online by Cambridge University Press:  01 July 2016

Benjamin Melamed
Benjamin Melamed*
Affiliation:
Northwestern University
*
∗Postal address: Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois 60201, U.S.A.
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Abstract

The equilibrium behavior of Jackson queueing networks (Poisson arrivals, exponential servers and Bernoulli switches) has recently been investigated in some detail. In particular, it was found that in equilibrium, the traffic processes on the so-called exit arcs of a Jackson network with single server nodes constitute Poisson processes—a result extending Burke's theorem from single queues to networks of queues.

A conjecture made by Burke and others contends that the traffic processes on non-exit arcs cannot be Poisson in equilibrium. This paper proves this conjecture to be true for a variety of Jackson networks with single server nodes. Subsequently, a number of characterizations of the equilibrium traffic streams on the arcs of open Jackson networks emerge, whereby Poisson-related stochastic properties of traffic streams are shown to be equivalent to a simple graph-theoretical property of the underlying arcs. These results then help to identify some inherent limitations on the feasibility of equilibrium decompositions of Jackson networks, and to point out conditions under which further decompositions are ‘approximately’ valid.

Keywords

QUEUEING NETWORKSTRAFFIC IN QUEUEING NETWORKSPOISSON PROCESSESQUEUEING NETWORKS—DECOMPOSITIONDECOMPOSITION OF QUEUEING NETWORKS
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 2 , June 1979 , pp. 422 - 438
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

This research was partially supported by NSF Grant ENG-75-20223, Air Force Office of Scientific Research Grant AFOSR-76-2903, and by the Office of Naval Research Contract N00014-75-C-0492 (NR 042-296).

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