Refinements to the So-Called Simple Approximations for the Bulk-Arrival Queues: M X /G/1

Abstract

For the M x /G/1 queue, this paper gives explicit closed-form expressions in terms of roots of the so-called characteristic equation (CE) and approximations for the tail of the steady-state queueing-time distribution of a random customer of an arrival group. It is shown that the approximation improves if more than one root, in decreasing order of the negative real parts, is used, leading eventually to exact results if all the roots are used. Further, it is shown that our approximations are much simpler and, in general, more efficient to implement numerically than those given by Van Ommeren. Numerical aspects have been tested for a variety of service-time distributions including generalized distributions such as Coxian-k (C k ) with complex phase rates. Samples of numerical computations are also included in the form of tables and graphs. It is hoped that the results obtained should prove beneficial to both practitioners and queueing theorists dealing with bounds, inequalities, approximations, and other aspects.