Average rate in a M/M/1 processor-sharing queue

We investigate the average rate per job in an open M/M/1 processor-sharing queue. We introduce three definitions of average rate as seen by the system and by each job, and present expressions for each in terms of the system rate and load. We compare the three measures and prove that they are strictly ordered over all loads. We next consider the system rate required to achieve a minimum average rate per job, and prove that it is increasing and convex. Finally, we consider the system rate required to achieve a minimum tail probability on the average rate per job, and present expressions illustrating when the system rate is constrained by the required tail probability.