Waiting time distribution for the M∕M∕m queue

Abstract

A novel method is presented for the calculation of the waiting time distribution function for the M/M/m queue. It is shown that the conditional waiting time obeys an Erlang distribution with rate mμ, where μ is the service rate of a server. An explicit closed form solution is obtained by means of the probability density function of the Erlang distribution. The derivation of the result proved to be very simple. The significance of Khintchine's method and its close relation to the proposed method is pointed out. It is also shown that the waiting time distribution can be obtained from Takacs's waiting time distribution for the G/M/m queue as a special case. This reveals some insight into the significance of Takacs's more general, but rather complex, result.