On Product Connection Theorems for Markov Chains

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 ∏ ν π (ν) nν of steady state probabilities π (ν) nν related to ergodic Markov chains X (ν) = {X(ν)(t) : t ∈ T} represents the steady state probability p(n) = p(n1, n2, . . .) of an ergodic multidimensional Markov chain of random vectors X(ν)(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. AMS Subject Classification: 60J25, 60K37