Global fairness of additive-increase and multiplicative-decrease with heterogeneous round-trip times

Consider a network with an arbitrary topology and arbitrary communication delays, in which congestion control is based on additive--increase and multiplicative--decrease. We show that the source rates tend to be distributed in order to maximize an objective function called $F_A^h$ (``$F_A^h$ fairness`). We derive this result under the assumption of rate proportional negative feedback and for the regime of rare negative feedback. This applies to TCP in moderately loaded networks, and to those TCP implementations that are designed to interpret multiple packet losses within one RTT as a single congestion indication and do not rely on re-transmission timeout. This result provides some insight into the distribution of rates, and hence of packet loss ratios, which can be expected in a given network with a number of competing TCP or TCP-friendly sources. We validate our findings by analyzing the parking lot scenario, and comparing with previous results \cite{floyd-91-b,mathis-97-a}, and an extensive numerical simulation with realistic parameter settings. We apply $F_A^h$ fairness to gain a more accurate understanding of the bias of TCP against long round trip times.