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Non-orthogonal multiple access (NOMA) and ambient backscatter communication (AmBC) have gained a lot of interest as key enabling technologies in wireless communications systems. NOMA is a key player for enhancing spectrum utilization either by multiplexing multiple messages for the same user or allowing multi-users access, while AmBC shows great potential for enhancing both spectrum and energy efficiency for battery-limited devices. In this paper, the performance of a downlink NOMA multiplexing based symbiotic-radio ($SR$) AmBC system is analyzed over Nakagami-m fading channels. New closed-form expressions for the exact and asymptotic outage probabilities are derived. Moreover, we analyze the diversity order and the influence of the system parameters on the outage performances. Besides, we proposed a power allocation optimization technique to achieve an outage-optimal performance. Through representative Monte-Carlo simulations, we have verified the analytical results. Finally, we compared the performance of the proposed system against a benchmark OMA-based system. [ABSTRACT FROM AUTHOR]