Connecting internally balanced quasi-reversible Markov processes

Abstract

We provide a general framework for interconnecting a collection of quasi-reversible nodes in such a way that the resulting process exhibits a product-form invariant measure. The individual nodes can be quite general, although some degree of internal balance will be assumed. Any of the nodes may possess a feedback mechanism. Indeed, we pay particular attention to a class of feedback queues, characterized by the fact that their state description allows one to maintain a record of the order in which events occur. We also examine in some detail the problem of determining for which values of the arrival rates a node does exhibit quasi-reversibility.