The Fluid Limit of an Overloaded Processor Sharing Queue

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.