Stochastic models of proportionally fair congestion controllers

Design of congestion control mechanisms using deterministic fluid models has attracted a lot of attention recently. In this paper, we relax two critical assumptions that lead to these deterministic models. Firstly, we explicitly account for the fact that the congestion notification mechanism at the router is probabilistic in nature and secondly, we account for the unresponsive flows at the router which are modeled as stochastic noise. By taking these factors into account for a single bottleneck link accessed by multiple proportionally fair congestion controlled sources, we derive a suitable AR (auto-regressive) process model to describe the system. We calculate the variance of the input process at the router to provide a design rule for selecting the link capacity to ensure near loss-free operation. We compare our results with those obtained from simulations. An interesting conclusion from our numerical and simulation results is that the improvement in network performance that one can obtain from multiple bits of congestion information (as opposed to one-bit marking) is negligible.