IJPAM: Volume 1, No. 2 (2002)

ON PRODUCT CONNECTION
THEOREMS FOR MARKOV CHAINS

Dieter Baum$^1$, Gurami Sh. Tsitsiashvili$^2$
$^1$University of Trier, Dept. IV
D-54286 Trier, GERMANY
e-mail: baum@uni-trier.de
$^2$Institute of Applied Mathematics
Far Eastern Branch of RAS
Vladivostok, RUSSIA
e-mail: guram@iam-mail.febras.ru


Abstract.This paper addresses the problem of constructing multidimensional Markov chains with product form steady state distribution. Product connection theorems are established which guarantee that the product $\prod_{\nu} \pi_{n_\nu}^{(\nu)}$ of steady state probabilities $\pi_{n_\nu}^{(\nu)}$ related to ergodic Markov chains ${\cal X}^{(\nu)}= \{X^{(\nu)}(t) : t \in T\}$ represents the steady state probability $p({\frk n}) = p(n_1, n_2, \ldots)$ of an ergodic multidimensional Markov chain of random vectors $X^{(\nu)}(t)$, irrespectively of dependency relations. Such results are closely related to statements about product form queueing networks. In fact, it is shown that the theorems of Jackson and Gordon-Newell fit into this framework, and the same is true with respect to BCMP-type queueing networks, although not explicitely discussed here. General product connection theorems, in principle, may form the basis for discovering product form solutions for a wider class of queueing networks.

Received: January 8, 2002

AMS Subject Classification: 60J25, 60K37

Key Words and Phrases: product connection of Markov chains, PF-queueing networks

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2002
Volume: 1
Issue: 2