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We focus on an earlier proposed class of ALOHA-type retransmission policies intended to control the random accessing of a single-channel communication network by a large number of packet-transmitting, bursty users. These control policies use channel feedback information to adaptively adjust the level of the traffic intensity, and can maintain any input rate less than e-1, for an infinite-population Poisson arrival model. Our analysis focuses on a simplified recursive model for describing locally the dynamics of the traffic intensity. The convergence analysis of this local model is reduced to stability analysis of a deterministic differential equation, and reveals a speed of convergence versus steady-state accuracy tradeoff. We introduce measures for evaluating this tradeoff. We use the proposed measures to show how certain parameter choices might best be made, and to study the effect of feedback limitations and channel errors on the performance of the random-access system. Our results show that with proper parameter choices the considered policies can be made extremely insensitive to channel errors.