  This paper contains an asymptotic analysis of a fluid model for a heavily
loaded processor sharing queue. Specifically, we consider the behavior of
solutions of critical fluid models as time approaches \infty. The main theorems
of the paper provide sufficient conditions for a fluid model solution to
converge to an invariant state and, under slightly more restrictive
assumptions, provide a rate of convergence. These results are used in a related
work by Gromoll for establishing a heavy traffic diffusion approximation for a
processor sharing queue.
