Fluid and Diffusion Approximation of a Multiclass Proportional Processor Sharing Queue with Reneging and General Distributions

In this paper, we resolve the long open problem of the diffusion approximation for a generally distributed processor sharing queue with reneging. In particular, we first extend the known fluid limit for a single class processor sharing queue with reneging to the multiclass proportional processor sharing queue with reneging. We prove uniqueness of fluid model solutions and convergence of fluid scaled models to fluid model solutions. Next, we show that, under diffusion-scaling, the models are tight, and that subsequential limits can be uniquely characterized by a certain system of SDEs driven by a Gaussian sheet on ${\mathbb{R}}_{+}^2$ with a nontrivial covariation structure.