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Mobile Edge Computing (MEC) enables rich services in close proximity to the end users to provide high quality of experience (QoE) and contributes to energy conservation compared with local computing, but results in increased communication latency. In this paper, we investigate how to jointly optimize task offloading and resource allocation to minimize the energy consumption in an orthogonal frequency division multiple access-based MEC networks, where the time-sensitive tasks can be processed at both local users and MEC server via partial offloading. Since the optimization variables of the problem are strongly coupled, we first decompose the orignal problem into three subproblems named as offloading selection (PO ), transmission power optimization (PT ), and subcarriers and computing resource allocation (PS ), and then propose an iterative algorithm to deal with them in a sequence. To be specific, we derive the closed-form solution for PO , employ the equivalent parametric convex programming to cope with the objective function which is in the form of sum of ratios in PT , and deal with PS by an alternating way in the dual domain due to its NP-hardness. Simulation results demonstrate that the proposed algorithm outperforms the existing schemes.