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In this paper, non-orthogonal multiple access (NOMA) in amplify-and-forward relay systems with partial relay selection (PRS) is investigated. More specifically, new exact closed-form expressions for the outage probabilities at two users are derived, based on which an asymptotic analysis at high signalto-noise ratio (SNR) is carried out. Additionally, in order to investigate the performance gap between the NOMA and orthogonal multiple access (OMA) schemes, a closed-form approximate expression at high SNR for the sum rate is derived. Furthermore, relying on our results, the impact of the PRS on the sum rate and outage probability of the proposed NOMA scheme is examined. In particular, the derived asymptotic expressions show that the proposed scheme can improve over the traditional OMA not only the sum rate but also the user fairness. Finally, simulation results are presented to corroborate the analytical results. Index TermsNon-orthogonal multiple access (NOMA), spectrum efficiency, amplify-and-forward (AF) relay, partial relay selection, outage probability, sum rate.