On the existence of invariant measures for networks with string transitions

On the existence of invariant measures for networks with string transitions

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Article ID: iaor2004846
Country: Netherlands
Volume: 38
Issue: 3
Start Page Number: 257
End Page Number: 285
Publication Date: Jul 2001
Journal: Queueing Systems
Authors:
Keywords: networks: path, markov processes
Abstract:

Recently a new class of Markov network processes was introduced, characterized by so-called string transitions. These are continuous-time Markov processes on a discrete state space. It is known that they possess an invariant measure of a special form, called a product-form, provided that a certain system of so-called traffic equations possesses a solution. Little is known about the existence of solutions of the traffic equations. The present paper deals with this question, focusing on the most important special case of unit vector string transitions. It is shown for open networks with unit vector string transitions of bounded lengths that the traffic equations possess a solution. Furthermore, it is shown for a prominent example of a network featuring signals and batch services that the traffic equations possess a solution.

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