Diffusion approximation for a processor sharing queue in heavy traffic
H. C. Gromoll
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.
Annals of Applied Probability, 14 (2004), 555-611.
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Last updated July 25, 2007.