Machine-type communications (MTC) technology, which enables direct communications among devices, plays an important role in realizing Internet-of-Things. However, a large number of MTC devices can cause severe collisions. As a result, the network throughput is decreased and the access delay is increased. To address this issue, a throughput-oriented non-orthogonal random access (NORA) scheme is proposed for massive machine-type communications (mMTC) networks. Specifically, by employing the technique of tagged preambles (PAs), multiple MTC devices (MTCDs) choosing the same PA can be distinguished and regarded as a non-orthogonal multiple access (NOMA) group, which enables multiple MTCDs to share the same physical uplink shared channel for transmissions by multiplexing in the power domain. The Sukhatme's classic theory and the characteristic function approach are adopted to formulate an optimization problem. The aim is to maximize the throughput subject to the constraints on the power back-off factor, the number of MTCDs included in a NOMA group, and the successful transmission probability. Based on the particle swarm optimization (PSO) algorithm, the formulated optimization problem is efficiently solved. The derived solution can be used to adjust the access class barring factor such that more MTCDs can obtain the access opportunities. Moreover, a lowcomplexity suboptimal solution is also developed, which can achieve near-PSO performance under high data rate requirement. Simulation results show that the proposed scheme can efficiently improve the network performance and comparison is made with the existing schemes. Index Terms-Massive machine-type communications (MTC) networks, non-orthogonal random access, power back-off.