Product-form solutions for models with joint-state dependent transition rates

In the last few years some novel approaches have been developed to analyse Markovian stochastic models with product-form solutions. In particular RCAT [4] has proved to be a very powerful result capable to derive most of the well-known product-forms previously formulated in queueing theory or stochastic Petri net analysis contexts as well as new ones. The main idea is to define a joint-process as a cooperation among a set of models and give the condition for and the expression of the equilibrium probability distribution of the joint-states as product of the equilibrium distributions of each model considered in isolation. This paper aims to formulate an approach to deal with models whose transition rates depend on the resulting joint-states. In practice, we extend what has been introduced to solve the same problem for queueing networks [8,9] and stochastic Petri nets [5]. However, since RCAT is more general than the results that are derived for a specific model, we show that some conditions on the transition rate specification that are not present in the original formulation arise. Several examples are given to point out the application of this result and strength the intuition about the implications of the formulated conditions.