Explicit solutions for queues with Hypo-exponential service time and applications to product-form analysis

Queueing systems with Poisson arrival processes and Hypo-exponential service time distribution have been widely studied in literature. Their steady-state analysis relies on the observation that the infinitesimal generator matrix has a block-regular structure and, hence, matrix-analytic method may be applied. Let $\pi{nk}$ be the steady-state probability of observing the $k$-th stage of service busy and $n$ customers in the queue, with $1\leq k \leq K$, and $K$ the number of stages, and let $\pibn=(\pi{n1}, \ldots, \pi{nK})$. Then, it is well-known that there exists a rate matrix {\R} such that $\pib{n+1}=\pibn {\R}$. In this paper we give a symbolic expression for such a matrix {\R}. Then, we exploit this result in order to address the problem of approximating a {\MH} queue by a model with initial perturbations which yields a product-form stationary distribution. We show that the result on the rate matrix allows us to specify the approximations for more general models than those that have been previously considered in literature and with higher accuracy.