Recent studies have numerically demonstrated the possible advantages of the asynchronous non-orthogonal multiple access (ANOMA) over the conventional synchronous non-orthogonal multiple access (NOMA). The ANOMA makes use of the oversampling technique by intentionally introducing a timing mismatch between symbols of different users. Focusing on a two-user uplink system, for the first time, we analytically prove that the ANOMA with a sufficiently large frame length can always outperform the NOMA in terms of the sum throughput. To this end, we derive the expression for the sum throughput of the ANOMA as a function of signal-to-noise ratio (SNR), frame length, and normalized timing mismatch. Based on the derived expression, we find that users should transmit at full powers to maximize the sum throughput. In addition, we obtain the optimal timing mismatch as the frame length goes to infinity. Moreover, we comprehensively study the impact of timing error on the ANOMA throughput performance. Two types of timing error, i.e., the synchronization timing error and the coordination timing error, are considered. We derive the throughput loss incurred by both types of timing error and find that the synchronization timing error has a greater impact on the throughput performance compared to the coordination timing error.