TY - JOUR

T1 - On the structure of the space of geometric product-form models

AU - Bayer, Nimrod

AU - Boucherie, Richardus J.

N1 - The preliminary stage of this work was supported by European grant BRA-QMIPS of CEC DG XIII, which enabled the research of N. Bayer at CWI Amsterdam+ The research of R.J. Boucherie has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences.

PY - 2002

Y1 - 2002

N2 - 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 [parallel R: parallel] [parallel R: parallel][infty infinity]-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.

AB - 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 [parallel R: parallel] [parallel R: parallel][infty infinity]-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.

KW - EWI-17890

KW - METIS-206699

KW - IR-71430

U2 - 10.1017/S0269964802162073

DO - 10.1017/S0269964802162073

M3 - Article

VL - 16

SP - 241

EP - 270

JO - Probability in the engineering and informational sciences

JF - Probability in the engineering and informational sciences

SN - 0269-9648

IS - 2

M1 - 10.1017/S0269964802162073

ER -