**
The Fluid Limit of an Overloaded Processor
Sharing Queue
**

**
A. L. Puha, A. L. Stolyar and R. J. Williams **

Abstract
The main topic of this paper is strictly supercritical fluid
models as functional law of large
numbers approximations for overloaded processor sharing queues. Analogous results for critical fluid
models associated with heavily loaded processor sharing queues are contained in [6, 13].
Following [6, 13] we use measure valued processes to describe overloaded processor sharing
queues. An important distinction between critical fluid models and strictly supercritical
fluid models is that the total mass for a fluid model solution that starts from zero grows with time
for the latter, but it is identically equal to zero for the former.
For strictly supercritical
fluid models, the paper contains a description of the
distribution of the mass as it builds up from
zero. Also, the set of stationary fluid model solutions
(solutions for which the shape does not
change with time) is identified.
In addition, the shape of any fluid model solution is shown to
converge to an invariant shape as time tends to infinity.
Finally, a fluid limit result is stated
and proved here that justifies strictly supercritical fluid models as
first order approximations to
overloaded processor sharing queues.
Published in
* Mathematics of Operations Research,
31 (2006), 316-350. *

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Last updated July 25, 2006.