It is well known that a simple binary feedback rate--based congestion avoidance scheme cannot ensure a fairness goal of the Available Bit Rate (ABR) service, namely, max--min fairness. In this paper we show how the rates are distributed for the network consisting of the binary switches, and end--systems employing an additive--increase/multiplicative decrease rate control. The modeling assumptions fairly resembles the ABR congestion avoidance, and applies to an arbitrary network topology. The results are obtained on the basis of a stochastic modeling, upon which we obtain certain analytical results, and conduct a numerical simulation. We validate the stochastic modeling through a discrete--event simulation. We believe that modeling presented in this paper enlight the performance issues of the binary ABR schemes. Keywords ABR, ATM, congestion control, binary scheme, EFCI, fairness, max--min, proportional fairness, stochastic approximation, ODE, Lyapunov, Runge-Kutta.