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ON THE STRUCTURE OF THE SPACE OF GEOMETRIC PRODUCT-FORM MODELS

Published online by Cambridge University Press:  21 May 2002

Nimrod Bayer  and
Richard J. Boucherie
Nimrod Bayer
Affiliation:
Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel, E-mail: [email protected]
Richard J. Boucherie
Affiliation:
Faculty of Mathematical Sciences, University of Twente, 7500 AE Enschede, The Netherlands
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Abstract

This article deals with Markovian models defined on a finite-dimensional discrete state space and possess a stationary state distribution of a product-form. We view the space of such models as a mathematical object and explore its structure. We focus on models on an orthant [script Z]+n, which are homogeneous within subsets of [script Z]+n called walls, and permit only state transitions whose ∥ ∥∞-length is 1. The main finding is that the space of such models exhibits a decoupling principle: In order to produce a given product-form distribution, the transition rates on distinct walls of the same dimension can be selected without mutual interference. This principle holds also for state spaces with multiple corners (e.g., bounded boxes in [script Z]+n).

In addition, we consider models which are homogeneous throughout a finite-dimensional grid [script Z]n, now without a fixed restriction on the length of the transitions. We characterize the collection of product-form measures which are invariant for a model of this kind. For such models with bounded transitions, we prove, using Choquet's theorem, that the only possible invariant measures are product-form measures and their combinations.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 16 , Issue 2 , April 2002 , pp. 241 - 270
Copyright
© 2002 Cambridge University Press

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