Using a slight modification of the framework of Bramson  and Williams , we prove heavy traffic limit theorems for six families of multiclass queueing networks. The first three families are single-station systems operating under first-in-first-out (FIFO), generalized-head-of-the-line proportional processor sharing (GHLPPS) and static buffer priority (SBP) service disciplines. The next two families are reentrant lines that operate under first-buffer-first-serve (FBFS) and last-buffer-first-serve (LBFS) service disciplines; the last family consists of certain two-station, five-class networks operating under an SBP service discipline. Some of these heavy traffic limits have appeared earlier in the literature; our new proofs demonstrate the significant simplifications that can be achieved in the present setting.
"Heavy Traffic Limits for Some Queueing Networks." Ann. Appl. Probab. 11 (1) 49 - 90, February 2001. https://doi.org/10.1214/aoap/998926987