By executing offloaded tasks from mobile users, edge computing augments mobile user equipments (UEs) with computing/communications resources from edge nodes (ENs), enabling new services (e.g., real-time gaming). However, despite being more resourceful than UEs, allocating ENs' resources to a given favorable set of users (e.g., closer to ENs) may block other UEs from their services. This is often the case for most existing approaches that only aim to maximize the network social welfare or minimize the total energy consumption but do not consider the computing/battery status of each UE. This work develops an energy-based proportional-fair framework to serve all users with multiple tasks while considering both their service requirements and energy/battery levels in a multi-layer edge network. The resulting problem for offloading tasks and allocating resources toward the tasks is a Mixed-Integer Nonlinear Programming, which is NP-hard. To tackle it, we leverage the fact that the relaxed problem is convex and propose a distributed algorithm, namely the dynamic branch-and-bound Benders decomposition (DBBD). DBBD decomposes the original problem into a master problem (MP) for the offloading decisions and multiple subproblems (SPs) for resource allocation. To quickly eliminate inefficient offloading solutions, MP is integrated with powerful Benders cuts exploiting the ENs' resource constraints. We then develop a dynamic branch-and-bound algorithm (DBB) to efficiently solve MP considering the load balance among ENs. SPs can either be solved for their closed-form solutions or be solved in parallel at ENs, thus reducing the complexity. The numerical results show that DBBD returns the optimal solution in maximizing the proportional fairness among UEs. DBBD has higher fairness indexes, i.e., Jain's index and min-max ratio, in comparison with the existing ones that minimize the total consumed energy.

翻译：通过执行移动用户卸载的任务，边缘计算利用边缘节点提供的计算/通信资源增强移动用户设备的能力，从而支持实时游戏等新型服务。然而，尽管边缘节点比用户设备资源更丰富，但将边缘节点资源分配给特定有利用户群体（例如更靠近边缘节点的用户）可能阻碍其他用户设备获得服务。现有大多数方法仅以最大化网络社会总福利或最小化总能耗为目标，并未考虑每个用户设备的计算/电池状态，因此常出现上述情况。本文提出一种基于能量的比例公平框架，在多层边缘网络中兼顾服务需求与能量/电池水平，为所有用户提供多任务服务。由此形成的任务卸载与资源分配问题为混合整数非线性规划，属于NP难问题。为解决该问题，我们利用松弛问题为凸函数的特性，提出分布式算法——动态分支定界Benders分解法。该方法将原问题分解为决定卸载决策的主问题和多个负责资源分配的子问题。为快速剔除低效卸载方案，主问题整合了利用边缘节点资源约束的强Benders割。随后我们开发动态分支定界算法，在考虑边缘节点负载均衡的情况下高效求解主问题。子问题可通过闭式解求解，或在边缘节点并行求解，从而降低复杂度。数值结果表明，动态分支定界Benders分解法能在最大化用户设备间比例公平性方面返回最优解。相比现有最小化总能耗的方法，该算法在Jain公平指数和最小-最大比等公平性指标上表现更优。