Approximations for the M/G/m Queue

We study the M / G / m queue and obtain difference-differential equations for the equilibrium joint distribution of the number of customers present and of the remaining holding times for services in progress. A study of these equations leads to the proposal of a simple approximation for the queue length distribution. Approximate waiting time and busy period distributions easily follow. Simple expressions for the means of the variables in question are obtained. Some numerical results on the mean waiting time indicate that the proposed approximation is rather accurate. The technique presented in this paper can also be applied to variants of the M / G / m queue. In particular, a short treatment of the queue with finite waiting room is included.