This paper compares alternative approaches for computing the R matrix of Neuts in numerical solutions of queues of GI/PH/1 type. We consider the relative merits of using two algorithms proposed recently by Latouche, the state reduction procedure proposed in this journal by Kao, and an approach based on the work of Sengupta using the notion of matrix exponential. There are three areas of interest: efficiency, memory accesses, and accuracy. In the empirical investigation, we use a subset of PH/PH/1 problems studied by Ramaswami and Latouche and a set of problems having non-phase-type interarrival times. At the expense of incurring varying degrees of loss of accuracy, the two algorithms proposed by Latouche and the method based on the work of Sengupta call for a lesser amount of computation than that of Kao's. When compared with the three alternatives, this study shows empirically that the R matrices obtained by state reduction—even when the approach is extended to Markov chains with denumerable state spaces—are closer to their respective true but unknown values.
