Time-Dependent Solution of the Two-Server Queue Fed by General Arrival and Exponential Service Time Distributions

Abstract

In this paper the queuing system GI/M/2 has been studied by using the “Phase” method. The Laplace transform of the time-dependent probability generating function and the steady-state solutions have been obtained. When the arrival time distribution is exponential, the results correspond to those obtained by Saaty (Saaty, T. L. 1960. Time-dependent solution of the many-server Poisson queue. Opns Res. 8 755–772.) for the queuing process M/M/2.

Transient behavior of multi-server queues with recurrent input and exponential service times

“…[Also see Roes, P. B. M. (1962) and Shyu (1962)]. Some of the special cases studied for the time dependent behavior of the queue length process are M/M/s [Karlin and McGregor (1958), Saaty (1960), Jackson and Henderson (1966)J, GI/M/2 [Bhat (1965), Arora (1962) who considers a specialized arrival schemeJ and GI/M/3 [Bhat (1966)J. Mention can also be made of studies of a quite general nature, in terms of the waiting times of the nth arrival, like those of Kiefer and Wolfowitz (1955), Pollaczek (1961) and Presman (1965).…”