Convergence to equilibria for fluid models of head-of-the-line proportional processor sharing queueing networks

Fluid models have recently become an important tool for the study of open multiclass queueing networks. We are interested in a family of such models, which we refer to as head-of-the-line proportional processor sharing (HLPPS) fluid models. Here, the fraction of time spent serving a class present at a station is proportional to the quantity of the class there, with all of the service going into the ‘first customer’ of each class. To study such models, we employ an entropy function associated with the state of the system. The corresponding estimates show that if the traffic intensity function is at most 1, then such fluid models converge exponentially fast to equilibria. When the traffic intensity function is strictly less than 1, the limit is always the empty state and occurs after a finite time. A consequence is that generalized HLPPS networks with traffic intensity strictly less than 1 are positive Harris recurrent. Related results for FIFO fluid models of Kelly type were obtained in Bramson.