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On Generalized Networks of Queues with Positive and Negative Arrivals

Published online by Cambridge University Press:  27 July 2009

Xiuli Chao  and
Michael Pinedo
Xiuli Chao
Affiliation:
Division of Industrial and Management Engineering, Department of Industrial and Mechanical Engineering, New Jersey Institute of Technology, Newark, New Jersey 07102
Michael Pinedo
Affiliation:
Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027
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Abstract

Consider a generalized queueing network model that is subject to two types of arrivals. The first type represents the regular customers; the second type represents signals. A signal induces a regular customer already present at a node to leave. Gelenbe [5] showed that such a network possesses a product form solution when each node consists of a single exponential server. In this paper we study a number of issues concerning this class of networks. First, we explain why such networks have a product form solution. Second, we generalize existing results to include different service disciplines, state-dependent service rates, multiple job classes, and batch servicing. Finally, we establish the relationship between these networks and networks of quasi-reversible queues. We show that the product form solution of the generalized networks is a consequence of a property of the individual nodes viewed in isolation. This property is similar to the quasi-reversibility property of the nodes of a Jackson network: if the arrivals of the regular customers and of the signals at a node in isolation are independent Poisson, the departure processes of the regular customers and the signals are also independent Poisson, and the current state of the system is independent of the past departure processes.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 7 , Issue 3 , July 1993 , pp. 301 - 334
Copyright
Copyright © Cambridge University Press 1993

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