Designing mobile edge computing (MEC) systems by jointly optimizing communication and computation resources, which can help increase mobile batteries' lifetime and improve quality of experience for computation-intensive and latency-sensitive applications, has received significant interest. In this paper, we consider energy-efficient resource allocation schemes for a multi-user mobile edge computing system with inelastic computation tasks and non-negligible task execution durations. First, we establish a mathematical model to characterize the offloading of a computation task from a mobile to the base station (BS) equipped with MEC servers. This computation-offloading model consists of three stages, i.e., task uploading, task executing, and computation result downloading, and allows parallel transmissions and executions for different tasks. Then, we formulate the weighted sum energy consumption minimization problem to optimally allocate the task operation sequence, the uploading and downloading time durations as well as the starting times for uploading, executing and downloading, which is a challenging mixed discrete- continuous optimization problem and is NP-hard in general. We propose a method to obtain an optimal solution and develop a low-complexity algorithm to obtain a suboptimal solution, by connecting the optimization problem to a three-stage flow-shop scheduling problem and utilizing Johnson's algorithm as well as convex optimization techniques. Finally, numerical results show that the proposed sub-optimal solution outperforms existing comparison schemes.