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Non-orthogonal multiple access (NOMA) assisted semi-grant-free (SGF) transmission has recently received significant research attention due to its outstanding ability of serving grant-free (GF) users with grant-based (GB) users' spectrum, which greatly improves the spectrum efficiency and effectively relieves the massive access problem of 5G and beyond networks. In this paper, we first study the outage performance of the greedy best user scheduling SGF scheme (BU-SGF) by considering the impacts of Rayleigh fading, path loss, and random user locations. In order to tackle the admission fairness problem of the BU-SGF scheme, we propose a fair SGF scheme by applying cumulative distribution function (CDF)-based scheduling (CS-SGF), in which the GF user with the best channel relative to its own statistics will be admitted. Moreover, by employing the theories of order statistics and stochastic geometry, the outage performances of both BU-SGF and CS-SGF schemes are analyzed. Theoretical results show that both schemes can achieve full diversity orders only when the served users' data rate is capped, which severely limits the rate performance of SGF schemes. To further address this issue, we propose a distributed power control strategy to relax such data rate constraint, and derive analytical expressions of the two schemes' outage performances under this strategy. Finally, simulation results validate the fairness performance of the proposed CS-SGF scheme, the effectiveness of the power control strategy, and the accuracy of the theoretical analyses.