The rapid growth of mobile devices and the increasing complexity of tasks have made energy efficiency a critical challenge in Multi-Access Edge Computing (MEC) systems. This paper explores energy-efficient offloading strategies in large-scale MEC systems with heterogeneous mobile users, diverse network components, and frequent task handovers to capture user mobility. The problem is inherently complex due to the system's scale, task and resource diversity, and the need to maintain real-time performance. Traditional optimization approaches are often computationally infeasible for such scenarios. To tackle these challenges, we model the offloading problem using the restless multi-armed bandit (RMAB) framework and develop two scalable online policies that prioritize resources based on their marginal costs. The proposed policies dynamically adapt to the system's heterogeneity and mobility while ensuring near-optimal energy efficiency. Through extensive numerical simulations, we demonstrate that the policies significantly outperform baseline methods in power conservation and show robust performance under non-exponentially distributed task lifespans. These results highlight the practical applicability and scalability of our approach in dynamic MEC environments.

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