NUMERICAL IMPLEMENTATION OF THE AUGMENTED TRUNCATION APPROXIMATION TO SINGLE-SERVER QUEUES WITH LEVEL-DEPENDENT ARRIVALS AND DISASTERS

Abstract

This paper considers the computation of the stationary queue length distribution in single-server queues with level-dependent arrivals and disasters. We assume that service times follow a general distribution and therefore, we consider the stationary queue length distribution via an imbedded Markov chain. Because this imbedded Markov chain has infinitely many states, level dependence, and bidirectional jumps of levels, it is hard to compute the solution of the global balance equation exactly. We thus consider the augmented truncation approximation. In particular, we focus on the computation of the truncated state transition probability matrix of the imbedded Markov chain, assuming that the underlying continuous-time absorbing Markov chain during a service time is not uniformizable. Under some stability conditions, we develop a computational procedure for the truncated transition probability matrix, where the upper bound of errors owing to truncation can be set in advance. We also provide some numerical examples and demonstrate that our procedure works well.