Explicit solutions for queues with Hypo- or Hyper-exponential service time distribution and application to product-form approximations

Queueing systems with Poisson arrival processes and Hypo- or Hyper-exponential service time distribution have been widely studied in the literature. Their steady-state analysis relies on the observation that the infinitesimal generator matrix has a block-regular structure and, hence, the matrix analytic method may be applied. Let πnk be the steady-state probability of observing the kth phase of service and n customers in the queue, with 1≤K, and K the number of phases, and let πn=(πn1,πnK). Then, it is well-known that there exists a rate matrix R such that πn+1=πnR. In this paper, we give a symbolic expression for such a matrix R for both cases of Hypo- and Hyper-exponential queueing systems. Then, we exploit this result in order to address the problem of approximating a M/HypoK/1 queue by a product-form model. We show that the knowledge of the symbolic expression of R allows us to specify the approximations for more general models than those that have been previously considered in the literature and with higher accuracy.