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IEEE 802.11, the standard of wireless local area networks (WLANs), allows the coexistence of asynchronous and time-bounded traffic using the distributed coordination function (DCF) and point coordination function (PCF) modes of operations, respectively. In spite of its increasing popularity in real-world applications, the protocol suffers from the lack of any priority and access control policy to cope with various types of multimedia traffic, as well as user mobility. To expand support for applications with quality-of-service (QoS) requirements, the 802.11E task group was formed to enhance the original IEEE 802.11 medium access control (MAC) protocol. However, the problem of choosing the right set of MAC parameters and QoS mechanism to provide predictable QoS in IEEE 802.11 networks remains unsolved. In this paper, we propose a polling with nonpreemptive priority-based access control scheme for the IEEE 802.11 protocol. Under such a scheme, modifying the DCF access method in the contention period supports multiple levels of priorities such that user handoff calls can be supported in wireless LANs. The proposed transmit-permission policy and adaptive bandwidth allocation scheme derive sufficient conditions such that all the time-bounded traffic sources satisfy their time constraints to provide various QoS guarantees in the contention free period, while maintaining efficient bandwidth utilization at the same time. In addition, our proposed scheme is provably optimal for voice traffic in that it gives minimum average waiting time for voice packets. In addition to theoretical analysis, simulations are conducted to evaluate the performance of the proposed scheme. As it turns out, our design indeed provides a good performance in the IEEE 802.11 WLAN's environment, and can be easily incorporated into the hybrid coordination function (HCF) access scheme in the IEEE 802.11e standard.