# Lyapunov-Guided Energy Scheduling and Computation Offloading for Solar-Powered WSN
## 2 System Model and Assumption  
We consider a multiple terminal partial offload MEC network with solar power, as shown in Figure 1, where the hybrid access point HAP is equipped with a high computational power MEC server and a solar energy harvesting device with battery capacity \(B_{max}\) to perform wireless charging and offloading computations for the sensor node set \(\mathcal{K}=\{1,\ldots,|\mathcal{K}|\}\) covered by it. Assuming that the solar energy arrivals follow M-Markov Model [31], we investigate the independent task model where the task arrivals at a single sensor node and between different sensor nodes are independent of each other, and define the data arrivals at sensor node \(k\) in the time slot \(t\) to satisfy a Bernoulli distribution as \(D_{k}^{t}\geq 0\). The member nodes use RF energy harvesting techniques to harvest energy \(E_{k,har}^{t}\) and offload part of the computationally intensive tasks, with some of the tasks performed locally \(\Phi_{k,loc}^{t}\) and the remaining tasks \(\Phi_{k,off}^{t}\) offloaded to a remote MEC server. Due to the small amount of data in the calculation results, we ignore the calculation result return latency and energy consumption in this paper. To avoid data link interference between multiple nodes, the system uses the TDMA protocol to offload tasks to the HAP, defining the offload time of each node as \(\tau_{k}\), and considering that the system needs to complete the computation task within a given time period \(T\), the node offload time needs to satisfy \(\sum\limits_{k=1}^{K}\tau_{k}\leq T\). Assume that the data transmission link follows quasi-static fading and that the up and down transmission links are reciprocal and simultaneous through the orthogonal spectrum with the channel gain following a Rayleigh distribution [32]. In a solar powered MEC system, the initial available energy of the HAP at time slot \(E_{S}^{t-1}\) is assumed. The HAP harvests then stores solar energy throughout the time and broadcasts energy to the member nodes by wireless power transmission (WPT). The MEC server is used at the end of the time slot to compute and process the offloading data. Defining \(E_{S}^{t}\) as the available energy at the end of time slot \(t\), the energy buffer queue of the HAP can be expressed by  
\[E_{S}^{t}=\left[E_{S}^{t-1}-P_{S}^{t}\tau_{0}\right]^{+}+E_{S,har}^{t}-E_{MEC}^ {t} \tag{1}\]  
where \([x]^{+}=\max(x,0)\). \(E_{MEC}^{t}\) denotes the MEC server energy consumption, which can be calculated by \(E_{MEC}^{t}=\kappa_{mob,S}({f_{MEC}^{t}}^{3})^{3}\delta_{S}\Phi_{MEC}^{t}\). \(\kappa_{mob,S}\) denotes the power consumption parameter, which depends on the chip architecture [33]. The frequency \(f_{MEC}^{t}\) of the MEC  
Figure 1: The overview of solar-powered WSN based on hybrid multi-access. The SC consistently harvests and stores solar energy for charging a set of battery-free sensor nodes. The node does local computing and remote offloading based on the harvested energy. server is fixed and characterizes the CPU service capability. To avoid energy leakages, HAP energy available at the end of the battery needs to satisfy \(0\leq E_{S}^{t}\leq B_{\max}\). For simplicity, we consider a linear energy harvesting model where the energy harvested by the node in the time slot can be expressed as  
\[E_{k,har}^{t}=\eta_{k}P_{S}^{t}\tau_{0}^{t}h_{k} \tag{2}\]  
where \(\eta_{k}\in(0,1)\) denotes the energy harvesting efficiency. Each sensor node performs data processing using the harvested energy, part of which is used for local computation and part of which is offloaded to the MEC server in the HAP. The CPU frequency of a node \(k\) in the network in the time slot \(t\) is defined as \(f_{k}^{t}\), which characterizes the computational capacity of the node and can be dynamically tuned by adjusting the chip voltage and frequency scaling techniques [34]. \(\delta_{k}\) denotes the number of CPU cycles required to complete a task per unit of data size.