We consider a sequence of single-server queueing models with renewal arrivals and general i.i.d. service times operating under a service policy that incorporates batches into processor sharing.
Each model is described by a measure-valued process that keeps track of the residual service times for all jobs present in the system and evolves according to a family of dynamic equations.
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.
We show that this limit process obeys periodic dynamics that are easy to describe as a function of the initial condition.