Consider a single server queue with renewal arrivals and i.i.d. service times in which the server operates under a processor sharing service discipline. To describe the evolution of this system, we use a measure valued process that keeps track of the residual service times of all jobs in the system at any given time. From this measure valued process, one can recover the traditional performance processes, including queue length and workload. We propose and study a critical fluid model (or formal law of large numbers approximation) for a heavily loaded processor sharing queue. The fluid model state descriptor is a measure valued function whose dynamics are governed by a nonlinear integral equation. Under mild assumptions, we prove existence and uniqueness of fluid model solutions. Furthermore, we justify the critical fluid model as a first order approximation of a heavily loaded processor sharing queue by showing that, when appropriately rescaled, the measure valued processes corresponding to a sequence of heavily loaded processor sharing queues converge in distribution to a limit that is almost surely a fluid model solution.
"The fluid limit of a heavily loaded processor sharing queue." Ann. Appl. Probab. 12 (3) 797 - 859, August 2002. https://doi.org/10.1214/aoap/1031863171