Achievability Bounds for T-Fold Irregular Repetition Slotted ALOHA Scheme in the Gaussian MAC

We address the problem of uncoordinated massive random-access in the Gaussian multiple access channel (MAC). The performance of low-complexity T-fold irregular repetition slotted ALOHA (IRSA) scheme is investigated and achievability bounds are derived. The main difference of this scheme in comparison to IRSA is as follows: any collisions of order up to T can be resolved with some probability of error introduced by noise. In order to optimize the parameters of the scheme we combine the density evolution method (DE) proposed by G. Liva and a finite length random coding bound for the Gaussian MAC proposed by Y. Polyanskiy. As energy efficiency is of critical importance for massive machine-type communication (mMTC), then our main goal is to minimize the energy-per-bit required to achieve the target packet loss ratio (PLR). We consider two scenarios: (a) the number of active users is fixed; (b) the number of active users is a Poisson random variable.