Computing the R matrices in matrix-geometric solutions for a class of QBD queues: A phase substitution approach

In a matrix-geometric solution of a QBD queue (a queue whose underlying structure is a quasi-birth-and-death process), the R matrix is pivotal in computing various performance characteristics of the queue. This paper presents an alternative procedure for computing the R matrix. By exploiting the structure of the infinitesimal generator, the authors obtain the approach by an efficient adaptation of the state reduction method for computing the stationary probability vector of a continuous-time Markov chain. They give an example involving a message queue with secondary jobs considered by Ali and Neuts.