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We define the concept of a virtual customer to study the sojourn time of a processor
sharing queue. If the random process that models the total number of jobs in the system
can be scaled to have both fluid and diffusion limits, then we can derive similar limits
for the sojourn time of a virtual customer for this queue.
By specializing to systems such as the M/M/1 queue with time varying rates, we show
that the natural scale invariance property for the processor sharing service discipline corresponds
directly to the uniform acceleration scaling limits necessary for its queueing fluid and diffusion limits..
We use these asymptotic results to obtain strong law of large numbers and central limit theorems
for the virtual sojourn time of any given job size for the M/M/1 queue with time varying rates. This
gives us simple fluid and diffusion approximations for the sojourn times of virtual customers.
This is joint research with Mor Harchol-Balter of Carnegie Mellon University and Robert C.Hampshire
of Princeton University.
