Semi-Product-Form Solution for Models with State-Dependent Rates

In this paper we consider the problem of finding a decomposed solution to a Markov model where the action rates may depend on the global state space. To do this we consider regular cycles in the underlying state space and show that a semi-product form solution exists when the functions describing the action rates have specific forms. The approach is illustrated with a simple queueing example although it clearly extends to more general cases. The results for semi-product form solutions are not entirely new, however the method by which they are derived is both novel and intuitive.