THE M-SERVER QUEUE WITH POISSON INPUT

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 stateprobabilities is derived and a unique solution of these equationls 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,= 0 CiF in, where m is the number of servers, and a method for findin g the Ci's and gi's is given. HE multiserver queuing systemi with Poisson input and exponienltial service time has been extensively anal.yzed i.n the literature. The unique properties of exponentially distributed interarrival times and service times that give rise to a queuing system in which the number of customlers in the system, at any sequence of times, constitutes an imbedded Markov chain. Thus, knowledge of this number of customers in the system at a given time is equivalent to complete knowledge of the state of the systemn, and differential equations involving the probabilities of the number of customers in the system are readily obtained. When the service times are other than exponentially distributed, the analysis of the queuing system is more difficult. Perhaps the simplest nonexponential service distributionl is the Gamma distribution of order two, and it is this distribution that is used here. Consider a multiserver queuinig system with m-servers in parallel. Suppose that customers arrive according to Poisson process with density X, and are serviced at once by one of the m-servers, if available; if not, the customers wait in line in the order of their arrival. Customers are then serviced, in turn, at the first server available. If Oi, i = 1, 2, . are the successive service times, then let 6i be independent and identically distributed random variables with