## 3 System Model and Problem Formulation  
As shown in Figure 1, this study considers a WPMEC system that consists of an AP, an MEC server, and \(N\) fixed UNs denoted by a set \(\text{N}=\{1,\ 2,\dots,\ N\}\). The AP supplies energy to the UNs, which then use the harvested energy for data offloading. Each UN may possess computation-intensive, latency-critical tasks to offload to the AP. To reduce interference from information transmission between UNs, a block-based time-division multiple access (TDMA) structure is employed. In each time block, the AP provides energy to all UNs in the downlink, and then the UNs use the harvested energy to offload tasks to the AP in the uplink. Once the MEC server completes the tasks, it sends the results back to the UNs. Since the results are typically small, the return time is generally negligible. Consequently, the harvest-then-transmit protocol proposed in Ref. [21] is used and is referred to as the harvest-then-offload protocol in WPMEC. In the UN cooperative transmission mode, the AP determines the data transmission path of each UN based on its energy status, assisting \"far\" UNs in transmitting information. Multi-hop transmission is employed to aid \"far\" UNs in offloading tasks. The objective is to minimize the total time, which includes both the energy harvesting time and the task offloading time. Some notations used in this paper are shown in Table 1.  
Each UN exhibits computation-intensive tasks \(I_{i}\ i\ \in\ \mathrm{N}\) and needs to offload tasks from the UN to the MEC server to meet the critical latency. Considering the impact of the \"near-far\" effect in WPMEC, UNs use a UN cooperation model to offload their own data, which helps to improve the latency performance. \(L_{ij}\) is used to denote the link formed between UN \(i\) and UN \(j\). The uplink and downlink transmission power gains of link \(L_{ij}\) are expressed as \(g_{ij}\) and \(h_{ij}\), respectively. For any single block, a time division structure is used, as shown in Figure 2. Data transmission is the main factor affecting latency. Compared to the transmission time, the computation time and the resulting feedback time are negligible [18; 19]. At the downlink energy harvesting time \(t_{0}\), the AP uses transmission power \(P_{0}\) to broadcast wireless power to all UNs in the downlink, assuming that all UNs have sufficient \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Symbol** & **Description** & **Symbol** & **Description** \\ \hline \(I_{i}\) & Data generated by UN \(i\) & \(t_{0}\) & Downlink energy harvesting time \\ \hline \(L_{ij}\) & Link formed between UN \(i\) and UN \(j\) & \(B_{ij}\) & Status of the feasible links during the uplink transmission \\ \hline \(g_{ij}\), \(h_{ij}\) & Uplink and downlink transmission power gains of link \(L_{ij}\) & \(P_{c}\) & Power loss caused by the extra electronic device \\ \hline \(P_{0}\) & Transmission power of AP & \(P_{i}\) & Transmission power of UN \(i\) \\ \hline \(\zeta\) & Energy conversion efficiency & \(E_{dice}\) & Energy required to receive 1 bit of information \\ \hline \(r_{ui}\) & Information received by UN \(i\) through links \(L_{ui}\) & \(W\) & Bandwidth \\ \hline \(\eta\) & Noise power & \(T_{\mathit{eff}}\) & Data offloading time from UNs to the AP \\ \hline \(I_{A}\) & Data received by the AP & \(t_{ij}\) & Active time of link \(L_{ij}\) \\ \hline \end{tabular}
\end{table} 
Table 1: List of model notations. Figure 1: A WPMEC system using UN cooperation. Figure 2: Harvest-then-offload transmission protocol. battery capacity to store the harvested energy. The energy harvested by each UN during the WPT is expressed as follows:  
\[E_{i}{=}\ \zeta P_{0}h_{Ai}t_{0}, \tag{1}\]  
where \(\zeta\) represents the energy conversion efficiency of each UN, which is \(0<\zeta\leq 1\). Each UN begins transmitting information after the energy harvesting period has ended. Self-interference occurs when the AP transmits energy and receives information simultaneously.