The successive and parallel interference cancellation schemes are two well-known types of multi-stage interference cancellation schemes using the conventional correlator receivers as a basic building block, which has been known to significantly improve the performance of DS/CDMA system in the multiple access communication. Performance comparison between these two schemes is made strictly based on the analytical and it has been shown that the successive interference cancellation (SIC) scheme is more resistant to fading than the parallel interference cancellation (PIC) scheme. We further investigate the performance of the successive IC scheme subject to the delay constraint, which may be imposed typically on most of service applications with a real-time transmission requirement, including speech and video applications. Our analysis demonstrates that the performance may be significantly improved by the groupwise successive interference cancellation (GSIC) scheme, which can be properly optimized to meet the given delay constraint.