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Non-orthogonal multiple access (NOMA) and millimeter-wave (mmWave) communication are two promising techniques to increase the system capacity in the fifth-generation (5G) mobile network. The former can achieve high spectral efficiency by modulating the information in power domain and the latter can provide extremely large spectrum resources. Fluctuating two-ray (FTR) channel model has already been proved to accurately agree with the small-scale fading effects in mmWave communications in experiments. In this paper, the performance of NOMA-based communications over FTR channels in mmWave communication systems is investigated in terms of outage probability (OP) and ergodic capacity (EC). Specifically, we consider the scenario that one base station (BS) transmits signals to two users simultaneously under NOMA scheme. The BS and users are all equipped with a single antenna. Two power allocation strategies are considered: the first one is a general (fixed) power allocation scheme under which we derive the OP and EC of NOMA users in closed form; the other one is an optimal power allocation scheme that can achieve the maximum sum rate for the whole system. Under the second scheme, not only the closed-form OP and EC but also the upper and lower bounds of EC are derived. Furthermore, we also derive the asymptotic expression for the OP in high average SNR region to investigate the diversity order under these two schemes. Finally, we show the correctness and accuracy of our derived expressions by Monte-Carlo simulation.