**On the Response Time Distribution of a Class of Limited Processor Sharing Queues**

Processor sharing queues are often used to study the performance of time-sharing systems. In such systems the total service rate depends on the number of jobs present in the system and there is a limit implemented, called the multi-programming level (MPL), on the number of jobs that can be served simultaneously. Prior work showed that under highly variable jobs sizes, setting the MPL beyond the point where the service rate is maximized may reduce the mean response time.

In this talk we focus on the response time distribution of the MAP/PH/LPS-k(m) queue. In such a queue jobs arrive according to a Markovian arrival process (MAP), have phase-type (PH) distributed sizes, at most k jobs are processed in parallel and the total service rate depends on the number of jobs being served. Jobs that arrive when there are k or more jobs present are queued.

We briefly outline the main computational steps to compute the Laplace transform of the response time distribution and numerically invert it to study the impact of the MPL k. Numerical results are discussed at length and illustrate to what extent increasing k affects the quantiles and tail probabilities of the response time distribution. We further identify cases where having more variable job sizes may rather unexpectedly reduce the mean response time.