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This paper contains an asymptotic analysis of a fluid model for a heavilyloaded processor sharing queue. Specifically, we consider the behavior ofsolutions of critical fluid models as time approaches \infty. The main theoremsof the paper provide sufficient conditions for a fluid model solution toconverge to an invariant state and, under slightly more restrictiveassumptions, provide a rate of convergence. These results are used in a relatedwork by Gromoll for establishing a heavy traffic diffusion approximation for aprocessor sharing queue

Invariant states and rates of convergence for a critical fluid model of a processor sharing queue