MARKOV NETWORK PROCESSES WITH PRODUCT FORM STATIONARY DISTRIBUTIONS

This study concerns the equilibrium behavior of a general class of Mar kov network processes that includes a variety of queueing networks and networks with interacting components or populations. The focus is on determining when these processes have product form stationary distribu tions. The approach is to relate the marginal distributions of the pro cess to the stationary distributions of ''node transition functions'' that represent the nodes in isolation operating under certain fictitio us environments. The main result gives necessary and sufficient condit ions on the node transition functions for the network process to have a product form stationary distribution. This result yields a procedure for checking for a product form distribution and obtaining such a dis tribution when it exits. An important subclass of networks are those i n which the node transition rates have Poisson arrival components. In this setting, we show that the network process has a product form dist ribution and is ''biased locally balanced'' if and only if the network is ''quasi-reversible'' and certain traffic equations are satisfied. Another subclass of networks are those with reversible routing. We wea ken the known sufficient condition for such networks to be product for m. We also discuss modeling issues related to queueing networks includ ing time reversals and reversals of the roles of arrivals and departur es. The study ends by describing how the results extend to networks wi th multi-class transitions.