Sleep Scheduling Algorithm for the $k$k-Coverage Problem in 3D Heterogenous BF-WSNs

Battery-free Wireless Sensor Networks (BF-WSNs) have emerged as an essential part of Internet of Things (IoT) systems. Although two-dimensional (2D) BF-WSNs have been researched, three-dimensional (3D) ones are more indicative of real-world applications. Achieving <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-coverage in this scenario is a greater challenge and has yet to be investigated. This paper presents an optimization problem in heterogeneous 3D BF-WSNs to maximize <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-coverage quality while considering the recharging and sampling rates. We prove the problem is NP-Hard and decouple it into two subproblems. The first optimizes <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-coverage quality for each time slot without energy constraints. The second ensures that nodes scheduled for operation in each time slot can be immediately replenished with sufficient energy. We prove the near-optimal and optimal solutions to the two subproblems composes the near-optimal solution to the original problem. Following the development of Distributed Iterative Grouping (DIG) algorithm and Adaptive Sampling (AS) method to address two subproblems each, we propose a sleep scheduling algorithm to integrate them and solve the original problem. Simulation results verify the effectiveness and efficiency of the proposed algorithm.