Convergence to equilibria for fluid models of FIFO queueing networks

The qualitative behavior of open multiclass queueing networks is currently a topic of considerable activity. An important goal is to formulate general criteria for when such networks possess equilibria, and to characterize these equilibria when possible. Fluid models have recently become an important tool for such purposes. We are interested here in a family of such models, FIFO fluid models of Kelly type. That is, the discipline is first-in, first-out, and the service rate depends only on the station. To study such models, we introduce 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 with fixed concentrations of customer types throughout each queue. 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 Kelly networks with traffic intensity strictly less than 1 are positive Harris recurrent, and hence possess unique equilibria.