This paper studies the fluid approximation (also known as the functional strong law of large numbers) and the stability (positive Harris recurrence) for a multiclass queueing network. Both of these are related to the stabilities of a linear fluid model, constructed from the first-order parameters (i.e., long-run average arrivals, services and routings) of the queueing network. It is proved that the fluid approximation for the queueing network exists if the corresponding linear fluid model is weakly stable, and that the queueing network is stable if the corresponding linear fluid model is (strongly) stable. Sufficient conditions are found for the stabilities of a linear fluid model.
"Fluid Approximations and Stability of Multiclass Queueing Networks: Work-Conserving Disciplines." Ann. Appl. Probab. 5 (3) 637 - 665, August, 1995. https://doi.org/10.1214/aoap/1177004699