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In this paper, we present a new energy-efficient bandwidth allocation scheme for wireless networks. First of all, we investigate the intrinsic relationship between the energy consumption and transmission rates of mobile terminals, in which transmission rate is determined through channel allocations. Then, we propose two schemes for connection admission control: victim selection algorithm (VSA) and beneficiary selection algorithm (BSA) with the intent to reduce energy consumption of each terminal. Moreover, we introduce an adjustment algorithm to statistically meet the demands for quality of service (QoS) during the resource allocation. The performance of the proposed schemes is evaluated with respect to energy consumption rate of each successfully transmitted bit, throughput and call blocking probabilities. An extensive analysis and simulation study is conducted for Poisson and self-similar, multi-class traffic.