Product form solutions to production systems with simultaneous resource possession

Queueing models with simultaneous resource possession can be used to model production systems, in which sev:eral resources are needed simultaneously to process a job. With these models, performance characteristics of the production systems can be calculated. This is done by recognising the relevant Markov chain and calculating the equilibrium probabilities. However these Markov chains become high-dimensional and allow very large jumps, which makes it extremely hard to find the exact solution to the equilibrium equations. In this paper we study three relatively simple models with simultaneous resource possession in order gain insight into the solution of these models in general. We analyse the equilibrium equations of the relevant Markov chains and examine whether they allow a product form solution. We show that for these three models the behaviour on the horizontal boundary of the corresponding random walk is crucial for the existence of a product form solution. Two of the three models have such a solution. For the third one we construct a product form approximation, based on the solution found for the other two models.