Computing steady-state queueing-time distributions of single-server queues: GI/M/1.

This paper presents a computationally efficient method to find the steady-state distributions of actual queueing times of the first customer, as well as of a randomly selected customer, of an arrival group for the queueing system GI^X/M/1, and hence the queueing-time distribution of a customer for the system GI/EX/1. The distribution of virtual queueing time is also obtained. Approximate analysis based on one or more roots is also discussed. Though the exact detailed as well as approximate computations for a variety of interarrival-time distributions such as generalized Erlang, mixed generalized Erlang, hyperexponential, generalized hyperexponential, and deterministic have been carried out, only representative results in the form of tables have been appended. The results obtained should prove useful to queueing theorists, practitioners, and others.