Sum of product forms solutions to MSCCC queues with job type dependent processing times

Queueing models with simultaneous resource possession can be used to model production systems at which the production process occupies two or more resources(machines, operators, product carriers etc.) at the same time. A special class of these queueing models is the class of MSCCC queues, for which the stationary distribution has a product form. This was shown by Berezner et al. whose result depends on one special characteristic of MSCCC queues, being the processing times are job type independent exponentially distributed. However in many production situations processing times are job type dependent. Therefore we examined MSCCC queues with job type dependent exponentially distributed processing times. We determined the equilibrium probabilities of two special models using a detailed state description for which a solution using an aggregated state description is known. Comparing these two solutions we gained more insight in the structure of the solution to more general models for which such an aggregated state description no longer has the Markov property