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 ∞. 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.
"Invariant states and rates of convergence for a critical fluid model of a processor sharing queue." Ann. Appl. Probab. 14 (2) 517 - 554, May 2004. https://doi.org/10.1214/105051604000000017