Mobile edge computing has emerged as a promising technology to augment the computational capabilities of mobile devices. For a multi-user network in which its users periodically compute their tasks with the help of an edge cloud, we investigate the network lifetime maximization problem based on present user task information. We pursue this objective via a minimum energy efficiency maximization (MEEM) strategy that jointly optimizes the fraction of user task computations offloaded to the cloud and the respective allocation of edge computing and network communication resources across the users. We also investigate the network lifetime maximization problem for the case when the user task information is available for all future time slots, as well. This setting represents an upper bound for the MEEM strategy. Optimal solutions for both investigated strategies are formulated via feasibility testing and geometric programming. We show that MEEM can achieve a 70% lifetime improvement over the state-of-the-art and 450% lifetime improvement over the case of local user task computation only.