Exact and approximate numerical solutions to steady-state single-server queues:M/G/1 — a unified approach

This paper presents a unified approach for the numerical solutions of anM/G/1 queue. On the assumption that the service-time distribution has a rational Laplace-Stieltjes transform (LST), explicit closed-form expressions have been obtained for moments, distributions of system length and waiting time (in queue) in terms of the roots of associated characteristic equations (c.e.'s). Approximate analyses for the tails of the distributions based on one or more roots are also discussed. Numerical aspects have been tested for a variety of complex service-time distributions including but not restricted to only mixed generalized Erlang and generalized hyperexponential. A sample of numerical computations is also included. It is hoped that the results obtained would prove to be beneficial to both practitioners and theorists dealing with bounds, inequalities, approximations, and other aspects.