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Adaptive modulation and coding (AMC) is widely used in wireless and mobile communications systems. A system using AMC technique can be modeled as the system with time-varying capacity. We propose a generic packet transmission scheduling algorithm that can be applied to various types of wireless systems with time-varying capacity. The proposed scheduling algorithm decides packet(s) to be transmitted taking account of not only the available data rate but also the weight for each user. We show that there exists a weight vector with which the maximum system throughput can be achieved for given quality of service (QoS) requirements of users. An incremental subgradient method to obtain such a weight vector is also suggested. We provide an application example of the proposed scheduling algorithm to the 1xEV-DO systems.