Waiting time distributions for closed M/M/N processor sharing queues

The paper considers a multiple server processor sharing model with a finite number of terminals (customers). Each terminal can submit at most one job for service at any time. The think times of the terminals and the service time demands are independently exponentially distributed. The paper focuses attention on the exact detailed analysis of the waiting time distribution of a tagged job. It gives the Laplace-Stieltjes transform of the waiting time distribution conditioned on the job’s service time demand and the state of the other terminals and show that these transforms can be efficiently evaluated and inverted. Further results include the representation of conditoned waiting times as mixtures of a constant and several exponentially distributed components. The numerical precision of the present results is being compared with results from a discrete approximation of the waiting time distributions.