In this paper, we introduce a distributed power allocation strategy for random access, that has the capabilities of multipacket reception (MPR) and successive interference cancellation (SIC). The proposed random access scheme is suitable for machine-to-machine (M2M) communication application in fifth-generation (5G) cellular networks. A previous study optimized the probability distribution for discrete transmission power levels, with implicit limitations on the successful decoding of at most two packets from a single collision. We formulate the optimization problem for the general case, where a base station can decode multiple packets from a single collision, and this depends only on the signal-to-interference-plus-noise ratio (SINR). We also propose a feasible suboptimal iterative per-level optimization process; we do this by introducing relationships among the different discrete power levels. Compared with the conventional power allocation scheme with MPR and SIC, our method significantly improves the system throughput; this is confirmed by computer simulations.