In this paper, we investigate a non-orthogonal multiple access (NOMA) based mobile edge computing (MEC) network, in which two users may partially offload their respective tasks to a single MEC server through uplink NOMA. We propose a new offloading scheme that can operate in three different modes, namely the partial computation offloading, the complete local computation, and the complete offloading. We further derive a closed-form expression of the successful computation probability for the proposed scheme. As part of the proposed offloading scheme, we formulate a problem to maximize the successful computation probability by jointly optimizing the time for offloading, the power allocation of the two users and the offloading ratios which decide how many tasks should be offloaded to the MEC server. We obtain the optimal solutions in the closed forms. Simulation results show that our proposed scheme can achieve the highest successful computation probability than the existing schemes.