State-Dependent Rates and Semi-Product-Form via the Reversed Process

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We consider the problem of finding a decomposed solution to a queueing 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 shown in detail for two queues and shown to extend to larger systems. Although not all the results for semi-product-form solutions are entirely new, the method by which they are derived is both novel, intuitive and leads to generalisations.