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WiMAX (worldwide interoperability for microwave access) is a new choice for last-mile Internet access, especially in crowded urban areas or desolate rural areas, where cable and DSL infrastructures are difficult to be deployed. In this paper we propose a downlink adaptive power allocation (APA) scheme for WiMAX access networks, which allows the transmitter to grant different power levels to different subcarriers according to the channel state information. We formulate the APA as an optimization problem, where the demands of both revenue and fairness are taken into account. To solve the optimization problem, we develop a fairness-constrained greedy revenue algorithm. Our simulation results show that, the proposed APA scheme can give a good trade-off between revenue and fairness considerations.