Fluid limits and the batched processor sharing model

We consider a sequence of single-server queueing models operating under a service policy that incorporates batches into processor sharing. As a processor sharing model is serving all jobs present simultaneously, the rate that it serves each job at is dependent on the number of jobs present in the system. For this reason, keeping track of the residual service times of each job is essential in a processor sharing model in order to be informed of significant impending changes in queue length, and therefore processing rate. We require a tool that will not only allow the recovery of characteristics such as the queue length process but also encodes the residual service time of each job. Each model is described by a measure-valued process that evolves according to a family of dynamic equations. This measure-valued process is defined by placing a unit mass at the residual service time for each job in the system, thereby encapsulating the characteristic that analyzing a processor sharing system requires.

Under mild conditions and a law-of-large-numbers scaling, we prove that the sequence of measure-valued processes converges in distribution to an essentially deterministic limit process. This result heralds back to the consequence of the Law of Large Numbers, where letting n tend to infinity in a sum of n random variables scaled by 1/n, we obtain a constant. Thus we can approximate the complicated sum by a simple number. In our setting, we show that the limit process obeys periodic dynamics that are easy to describe as a function of the initial condition.