In this letter, we address the problem of maximizing the minimum user rates known as rate adaptive (RA) for asyn-chronous cellular networks with multicarrier modulation such as filter-bank-based multicarrier and orthogonal frequency-division multiplexing. By exploiting the relationship between margin adap-tive (MA) and RA, we propose a general convergence criterion to solve the MA power control problem permitting to achieve maximum fairness among all users for the RA problem. Simulation results demonstrate a remarkable improvement in terms of higher minimum user data rates for the RA when the proposed convergence criterion is utilized over the traditional iterative waterfilling for the power control scheme of the MA problem. Index Terms-OFDM, FBMC, asynchronous networks, inter-cell interference management, resource allocation and power control.