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In this paper, we focus on the bandwidth allocation issue in wireless access networks, which are made up of Ethernet Passive Optical Network (EPON) and Worldwide Interoperability for Microwave Access (WiMAX) networks, i.e., Fiber-Wire (FiWi) networks. Since the bandwidth allocation scheme largely determines the performance of the entire wireless access network, in the past decades, researchers have dedicated much effort to design bandwidth allocation algorithms based on different criteria in order to satisfy various performance requirements. Various types of bandwidth allocation scheme based on Max-Min Fairness (MMF) or Proportional Fairness (PF) criteria have been developed to increase not only system throughput but also user fairness. However, in general, there is a tradeoff relationship between maximizing system throughput and increasing the fairness among users in throughput, and the users satisfaction in their Quality of Service (QoS) cannot always be maximized by adopting fair bandwidth allocation methods. To cope with this issue, we propose a bandwidth allocation method which improves the QoS satisfaction of all users while maintaining the system throughput similar to standard schemes, such as MMF and PF. In our method, users satisfaction is quantified by using utility functions which can be different among users according to their applications and services. By transferring portions of bandwidth from fully filled users to others so as not to decrease the system throughput, the proposed scheme is able to eventually converge to a compromised point. The results of performance evaluation through computer simulations have demonstrated that our proposed scheme can successfully enhance the performance of wireless access networks.