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The demand for QoS has made its support an essential component in today's wireless networks. The use of directional antennas in multi-hop wireless networks is known to provide substantial increase in the utilization of the wireless medium. However its effectiveness in providing QoS assured routing is unexplored. In this paper, we study the problem of bandwidth guaranteed routing in contention-based wireless mesh networks with directional antennas. We present a detailed analysis for the interference pattern of such networks taking into account the challenges imposed by the use of directional antennas such as deafness. Based upon this analysis, we formulate the QoS routing problem as an optimization problem. We then present a routing and admission control heuristic to solve the problem. Our simulation results demonstrate the effectiveness of our routing scheme and the necessity of admission control for QoS support.