The massive amounts of machine-type user equipments (UEs) will be supported in the future fifth generation (5G) networks. However, the potential large random access (RA) delay calls for a new RA scheme and for a detailed assessment of its performance. Motivated by the key idea of non-orthogonal multiple access, the non-orthogonal random access (NORA) scheme based on successive interference cancellation (SIC) is proposed in this paper to alleviate the access congestion problem. Specifically, NORA utilizes the difference of time of arrival to identify multiple UEs with the identical preamble, and enables power domain multiplexing of collided UEs in the following access process, while the base station performs SIC based on the channel conditions obtained through preamble detection. Our analysis show that the performance of NORA is superior to the conventional orthogonal random access (ORA) scheme in terms of the preamble collision probability, access success probability and throughput of random access. Simulation results verify our analysis and further show that our NORA scheme can improve the number of the supported UEs by more than 30%. Moreover, the number of preamble transmissions and the access delay for successfully accessed UEs are also reduced significantly by using the proposed random access scheme.