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Fair allocation of bandwidth and maximization of channel utilization are two important issues when designing a contention-based wireless medium access control (MAC) protocol. However, achieving both design goals at the same time is very difficult, and has not yet been addressed elsewhere. We study this challenging problem, particularly for data communications in IEEE 802.11 wireless local area networks (WLANs). We propose a priority-based fair medium access control (P-MAC) protocol by modifying the distributed coordination function (DCF) of the IEEE 802.11 MAC. The key idea is that the contention window size for each wireless station is properly selected to reflect: (1) the relative weights among data traffic flows, so as to achieve the weighted fairness; (2) the number of stations contending for the wireless medium, so as to maximize the aggregate throughput. In P-MAC, our approximations to the optimal contention window sizes, which are based on a theoretical analysis, are evaluated numerically and shown to work well under different network configurations and traffic scenarios. Moreover, simulation results show that, with few changes to the original DCF, P-MAC performs significantly better in terms of both fairness and throughput.