The M-Server Queue with Poisson Input and Gamma-Distributed Service of Order Two

Analysis is made of the multiserver-queuing systems with Poisson input and service times distributed according to a second order gamma distribution. A set of difference equations involving the time-invariant state-probabilities is derived and a unique solution of these equations is found. The method used is that of treating the gamma-distributed service times or order two as the sum of two independent and identically distributed service times, which are exponentially distributed. It is shown that the time invariant probability of n customers being in the system is of the form, P n = ∑ i=0 i=m C i β i n , where m is the number of servers, and a method for finding the C i 's and β i 's is given.