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 resemble the ABR congestion avoidance, and apply 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.