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In this paper, we investigate the performance of non-orthogonal multiple access (NOMA)-based full-duplex Internet-of-Things (IoT) relay systems with simultaneous wireless information and power transfer (SWIPT) over Nakagami-m fading channels to improve the performance of a cell-edge user under perfect and imperfect successive interference cancellation (SIC). Two scenarios, i.e., direct and non-direct links, between the source node and cell-edge user are examined. The exact closed-form analytical and approximate expressions for the outage probability, system throughput, energy efficiency, and ergodic capacities are derived and validated via Monte Carlo simulations to characterize the proposed system performance. To further improve the system performance, we also provide a low-complexity algorithm to maximize the system throughput over-optimizing the time-switching factor. The results show that our proposed NOMA system can achieve superior performance compared to its orthogonal multiple access (OMA) counterpart under perfect SIC and with a low-to-medium signal-to-noise ratio under imperfect SIC, according to the level of residual self-interference and the quality of links.