  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.
