We revisit random multiple access (RMAC) for wireless systems with successive interference cancellation (SIC) employed at the access point (AP). We consider an asymptotically large number of users that transmit over a large number of orthogonal sub-channels. In each transmission block, each user chooses a degree d, where d is a random variable that follows a predefined degree distribution Ω(x). Then, users transmit in d sub-channels chosen uniformly at random. Specifically, we consider signal to interference and noise ratio (SINR) based RMAC where it is assumed that a user's information can be recovered successfully at a given iteration of the SIC process when its updated SINR is above a predetermined threshold. In this paper, we develop a generalized analytical framework based on the codes-on-graph representation to track the evolution of error probabilities in each iteration of the SIC process. We compare our approach to the conventional RMAC employing SIC which assumes that only clean, interference-free transmissions can be recovered successfully. This clean packet model relies on having time slots with a single user's transmission at each iteration of the SIC process. It was shown to be analogous to the iterative recovery process of codes-on-graph for the binary erasure channel (BEC), thus, allowing the direct application of the AND-OR tree analysis. We show that the clean packet model is a special case of our more generalized tree-based analytical framework. Our numerical results show that our model can support more users under the same power requirements.
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