Under fairly general assumptions on the arrival and service time processes, we prove fluid and heavy traffic limit theorems for the unfinished work, queue length, sojourn time and waiting time processes associated with a single station multiclass generalized processor sharing model. The fluid limit of the unfinished work process is characterized by the Skorokhod map associated with a Skorokhod problem formulation of the generalized processor sharing model, while the heavy traffic diffusion limit is characterized using the corresponding extended Skorokhod map. An interesting feature of the diffusion limits is that they may fail to be semimartingales.
"Fluid and heavy traffic diffusion limits for a generalized processor sharing model." Ann. Appl. Probab. 13 (1) 100 - 139, January 2003. https://doi.org/10.1214/aoap/1042765664