We develop a new class of asynchronous distributed algorithms for the explicit rate control of elastic sessions in an integrated packet network. Sessions can request for minimum guaranteed rate allocations (e.g., MCRs in the ATM context), and, under this constraint, we seek to allocate the max-min fair rates to the sessions. We capture the integrated network context by permitting the link bandwidths available to elastic sessions to be stochastically time varying. The available capacity of each link is viewed as some statistic of this stochastic process (e.g., a fraction of the mean, or a large deviations Equivalent Service Capacity (ESC)). For fixed available capacity at each link, we show that the vector of max-min fair rates can be computed from the root of a certain vector equation. A distributed asynchronous stochastic approximation technique is then used to develop a provably convergent distributed algorithm for obtaining the root of the equation, even when the link flows and the ...