A. Economou: a Characterization of Product -form Stationary Distributions

We consider a general model for continuous-time Markov chains representing queueing systems in random environment. First we study the relationship between its equilibrium distribution and the Palm (or embedded) distributions at certain environmental change epochs. The results enable one to obtain the equilibrium distribution of the continuous-time model in terms of the equilibrium distribution of a discrete-time process. This is useful in simulation studies where one can extract information for the continuous-time model by recording the state of the system only in environmental change epochs. Then, we obtain necessary and sufficient conditions that ensure a product-form stationary distribution for the model. The related topics of partial balance and the ESTA (Events See Time Averages) property are also studied. As an illustration, we apply the results to study the stationary distributions of Jackson networks in random environment. For models that do not satisfy the product-form conditions, we develop a product-form approximation, which is proved to be very good for models evolving in a slowly changing random environment. We justify this fact by proving an explicit error bound for this approximation.