Achieving fair rates with ingress policing

We study a simple ingress policing scheme for a stochastic queuing network that uses a round-robin service discipline, and derive conditions under which the flow rates approach a max-min fair share allocation. The scheme works as follows: Whenever any of a flow's queues exceeds a policing threshold, the network discards that flow's arriving packets at the network ingress, and does so until all of that flow's queues fall below their thresholds. To prove our results, we use previously known results relating the stability of a queuing system to the stability of its fluid limit and extend these results to relate the flow rates of the stochastic system to those of a corresponding fluid model. In particular, we consider the fluid limit of a sequence of queuing networks with increasing thresholds. Using a Lyapunov function derived from the fluid limits, we find that as the policing thresholds are increased the state of the stochastic system is attracted to a relatively smaller and smaller neighborhood surrounding the equilibrium of the fluid model. We then show how this property implies that the achieved flow rates approach the max-min rates predicted by the fluid model.