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 show that under mild assumptions, including standard heavy traffic assumptions, the (suitably rescaled) measure valued processes corresponding to a sequence of processor sharing queues converge in distribution to a measure valued diffusion process. The limiting process is characterized as the image under an appropriate lifting map, of a one-dimensional reflected Brownian motion. As an immediate consequence, one obtains a diffusion approximation for the queue length process of a processor sharing queue.
"Diffusion approximation for a processor sharing queue in heavy traffic." Ann. Appl. Probab. 14 (2) 555 - 611, May 2004. https://doi.org/10.1214/105051604000000035