Approximate Waiting Time Distribution in GI/G/1 Queues

This paper presents an approximation method for obtaining the probability the server is busy and the mean waiting time as seen by the nth arriving customer for the GI/G/1 queueing system. Thus, transient behavior is the key issue of the method. The approximation consists of a simultaneous pair of recursion formulae whose state variables are the probability of delay and the mean waiting time. It is assumed that the Oth arriving customer finds the queue in some prescribed state from which the successive states are computed. Naturally, for the nth arrival when n is large, the computations provide approximations to the corresponding eqilibrium quantities when equilibrium exists. The procedure, however, is not limited to queues possessing an equilibrium state. For methods specially adapted to approximating the equilibrium quantities, we refer to a paper by A. A. Fredericks,1 in which the approximation Ao,i of that paper essentially corresponds to the equilibrium results obtained here; we also refer to Fredericks for an application to computer systems.2 Transient analysis is of particular importance in studying the recovery of a system from temporary overload. This can occur after a short 2003