A NUMERICAL METHOD FOR THE STEADY-STATE PROBABILITIES OF A G1/G/C QUEUING SYSTEM IN A GENERAL CLASS

Abstract

A numerical method is proposed for solving the balance equations of the steady-state probabilities of a G~/G/c queueing system in a general class. The method is based on an iterative calculation of conditional prob­ abilities, instead of absolute probabilities, conditioned by the number of customers in the system. By skillfully exploiting a convergence property of the conditional probabilities, it provides a fairly accurate solution of the balance equations with relatively little computational burden. In this paper, a numerical method is proposed for solving the balance equa­ tions of the steady-state probabilities of a GI/G/c queueing system in a general class. The method is a direct application of the (modified) lumping method introduced in (6) for the stationary distribution of a Markov chain. It is based on an iterative calculation of conditional probabilities of the queueing system conditioned by the number of customers in the system. By using the conditional probabilities, rather than absolute probabilities, the system of linear equations of the steady-state probabilities is di,'ded into a set of smaller systems of linear equations, and it can be solved with less computational burden by exploiting convergence property of the conditional probabilities. Furthermore, errors included in the solution become fairly small. The computational time required for solving the balance equations by our method is nearly independent of the value of the utilization factor p. Hence, our method is effective even if p is near to l.