Approximation method for M/PH/1 retrial queues with phase type inter-retrial times

Abstract

We obtain a stability condition and develop a method for approximating the stationary distribution and waiting time moments of an M/PH/1 retrial queue with phase type inter-retrial times. This extends a previous method applicable for general service and retrial times which approximates only the stationary queue length distribution. Our approximation replaces the complex non-renewal process of secondary arrivals from the orbit with a phase type renewal process which matches the first two or three interarrival time moments. The queue is modelled as a quasi birth–death process with subdiagonal blocks of rank one. The stationary distribution is obtained by applying a known result for generators of this form and moments of the waiting time are obtained numerically by applying block Gaussian elimination to a truncated system. The performance of the approximation is evaluated on a set of instances for which exact solutions are obtainable. We also describe how the method can be used to approximate the distribution of the waiting time and the number of retrials.