Response Time Distribution of a Class of Limited Processor Sharing Queues

Abstract

Processor sharing queues are often used to study the performance of time-sharing systems. In such systems the total service rate ?(m) depends on the number of jobs m present in the system and there is a limit implemented, called the multi-programming level (MPL), on the number of jobs k that can be served simultaneously. Prior work showed that under highly variable jobs sizes, setting the MPL k beyond the value k = arg maxm ?(m) may reduce the mean response time. In order to study the impact of the MPL k on the response time distribution, we analyse the MAP/PH/LPSk( 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 derive an expression for the Laplace transform of the response time distribution and numerically invert it to study the impact of the MPL k. Numerical results illustrate to what extent increasing k beyond k? increases the quantiles and tail probabilities of the response time distribution. They further demonstrate that for bursty arrivals and larger MPL k values having more variable job sizes may reduce the mean response time.