  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.
