The fluid limit of an overloaded processor sharing queue

This paper primarily concerns strictly supercritical fluid models, which arise 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 Gromoll et al. (2002) and Puha and Williams (2004). An important distinction between critical and strictly supercritical fluid models is that the total mass for a 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, this paper contains descriptions of each of the following: the distribution of the mass as it builds up from zero, the set of stationary solutions, and the limiting behavior of an arbitrary solution as time tends to infinity. In addition, a fluid limit result is proved that justifies strictly supercritical fluid models as first order approximations to overloaded processor sharing queues.