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The well known cloud computing is being extended by the idea of fog with the computing nodes placed closer to end users to allow for task processing with tighter latency requirements. However, offloading of tasks (from end devices to either the cloud or to the fog nodes) should be designed taking energy consumption for both transmission and computation into account. The task allocation procedure can be challenging considering the high number of arriving tasks with various computational, communication and delay requirements, and the high number of computing nodes with various communication and computing capabilities. In this paper, we propose an optimal task allocation procedure, minimizing consumed energy for a set of users connected wirelessly to a network composed of FN located at AP and CN. We optimize the assignment of AP and computing nodes to offloaded tasks as well as the operating frequencies of FN. The considered problem is formulated as a Mixed-Integer Nonlinear Programming problem. The utilized energy consumption and delay models as well as their parameters, related to both the computation and communication costs, reflect the characteristics of real devices. The obtained results show that it is profitable to split the processing of tasks between multiple FNs and the cloud, often choosing different nodes for transmission and computation. The proposed algorithm manages to find the optimal allocations and outperforms all the considered alternative allocation strategies resulting in the lowest energy consumption and task rejection rate. Moreover, a heuristic algorithm that decouples the optimization of wireless …