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• Formulate a non-cooperative game with both UEs and MECs as players. • Prove the existence of and convergence to a Nash equilibrium. • Develop a set of algorithms to find the Nash equilibrium. • Demonstrate numerical examples of non-cooperative games with and without MECs' participation. In a newly emerged fog computing environment, various user equipments (UE) enhance their computing power and extend their battery lifetime by computation offloading to mobile edge cloud (MEC) servers. Such an environment is distributed and competitive in nature. In this paper, we take a game theoretical approach to computation offloading optimization in a fog computing environment. Such an approach captures and characterizes the nature of a competitive environment. The main contributions of the paper can be summarized as follows. First, we formulate a non-cooperative game with both UEs and MECs as players. Each UE attempts to minimize the execution time of its tasks with an energy constraint. Each MEC attempts to minimize the product of its power consumption for computation and execution time for allocated tasks. Second, we develop a heuristic algorithm for a UE to determine its "heuristically" best response to the current situation, an algorithm for an MEC to determine its best response to the current situation, and an iterative algorithm to find the Nash equilibrium. Third, we prove that our iterative algorithm converges to a Nash equilibrium. We demonstrate numerical examples of our non-cooperative games with and without MECs' participation. We observe that our iterative algorithm always quickly converges to a Nash equilibrium. The uniqueness of our non-cooperative games is that the strategy set of a player can be discrete and the payoff function of a player can be obtained by a heuristic algorithm for combinatorial optimization. To the best of the author's knowledge, there has been no such investigation of non-cooperative games based on combinatorial optimization for computation offloading optimization in a fog computing environment. [ABSTRACT FROM AUTHOR]

As the technology of the Internet of Things (IoT) and mobile edge computing (MEC) develops, more and more tasks are offloaded to the edge servers to be computed. The offloading strategy performs an essential role in the progress of computation offloading. In a general scenario, the offloading strategy should consider enough factors, and the strategy should be made as quickly as possible. While most of the existing model only considers one or two factors, we investigated a model considering three targets and improved it by normalizing each target in the model to eliminate the influence of dimensions. Then, grey wolf optimizer (GWO) is introduced to solve the improved model. To obtain better performance, we proposed an algorithm hybrid whale optimization algorithm (WOA) with GWO named GWO-WOA. And the improved algorithm is tested on our model. Finally, the results obtained by GWO-WOA, GWO, WOA, particle swarm optimization (PSO), and genetic algorithm (GA) are discussed. The results have shown the advantages of GWO-WOA. [ABSTRACT FROM AUTHOR]