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Radio resource allocation strategy is one of the key issues in wireless CDMA cellular networks, which not only has to ensure that the system guarantees potentially different QoS requirements from diverse applications, but also fully utilizes the scarce wireless resources. Meanwhile, in CDMA cellular networks, successive interference cancellation (SIC) technique can be employed to reduce multiple access interference and improve system capacity. The possible implementation of radio resource allocation strategies in CDMA cellular networks with SIC, such as Rate Allocation (RA) algorithm, has not yet been extensively investigated. In this paper, we address the issue of the Rate Allocation (RA) algorithm in a DS-CDMA system with the constraint of the received power on the base station. As the problem of rate allocation to maximize the total utilities is a NP problem in the system with variable rate services, we develop two suboptimal Greedy Rate Allocation algorithms, namely upward greedy algorithm (UPGA) and downward greedy algorithm (DNGR). These two greedy algorithms have very low computational complexities. Extensive numerical results show that the proposed RA algorithms have better system performance than other ones with other decoding orders.