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With the limited wireless spectrum and the ever-increasing demand for wireless services, how to enlarge wireless network throughput is a pressing issue. To exploit the wireless spatial capacity, concurrent transmissions, if controlled appropriately, can lead to overall higher spectrum utilization and network throughput. The optimal scheduling and power control for concurrent transmissions in rate-adaptive wireless networks is a very challenging NP-hard problem. In this paper, we propose an efficient power allocation and scheduling algorithm for concurrent transmissions which can improve network throughput with fairness consideration. We first formulate the optimal power allocation and scheduling problem, and convert the original non-convex problem into a series of convex problems using a two-phase approximation technique. Then, we propose the power and channel allocation with fairness (PCAF) algorithm to solve the problem efficiently. Extensive simulation results show the remarkable improvement in terms of both network throughput and fairness, comparing to the previous scheduling algorithms.