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We consider certain variants of the additive-increase and multiplicative-decrease end-to-end congestion avoidance algorithms. The algorithms of concern are tailored to reflect the behavior of TCP congestion avoidance, in particular, the response to multiple congestion indications within a single round-trip time. We derive the limit mean ordinary differential equation (ODE) for each algorithm, which solution yields the limit throughput distribution. We focus on the fairness of the throughput distribution and bias against long round-trip time connections. The modeling by the ODE method is justified for an asymptotically small adaptation of the rate process, which corresponds to small additive-increase and multiplicative-decrease parameters. We verify, through numerical simulation, how well the limit mean ODE result matches the behavior of the system with a realistic non-asymptotic setting. On the basis of our results, we discuss conditions under which the limit mean ODE method is applicable.