## 2 System and Channel Model
### HDT Protocol  
The protocol uses the time division multiple access (TDMA) method, as shown in Figure 2. The time \(T\) is divided into \(K+1\) segments. During the first segment, the sensor nodes harvest energy from the BS, which then allocates \(K\) time slots to the sensor nodes for transmitting packets to the BS. The sum of energy harvesting time and packet transmission time is equal to \(T\), and the expression can be easily derived as  
\[\alpha_{0}T+\sum_{K=1}^{K}(1-\alpha)_{k}T=T. \tag{3}\]  
The HDT protocol involves two processes for the sensor nodes: first, they must harvest energy from the BS, and then, they can use the harvested energy to transmit packets to the BS. The time required for sensor node \(N_{k}\) to harvest energy from the BS is \(\alpha_{0}T\) and the energy obtained during this process can be expressed as:  
\[E_{k}=\eta_{k}Pd_{k}^{-\lambda}G_{k2}\alpha_{0}T,\quad\forall k\in\{1,2, \ldots,K\}, \tag{4}\]  
where \(P\) denotes the energy harvesting power when the sensor node \(N_{k}\) harvests energy from the BS, \(\eta_{k}\) is the energy harvesting efficiency index of the energy harvesting process, \(d\) is the distance between the sensor node \(N_{k}\) and the BS, and \(\lambda\) is the path loss exponent due to the influence of the randomness of the channel state on the energy harvesting path.  
After energy harvesting, the harvested energy is utilized to send packets to the BS, and the power required for sending packets can be expressed as follows:  
\[P_{k}=\frac{E_{k}}{(1-\alpha)_{k}T}=\frac{\alpha_{0}PG_{k2}\eta_{k}d_{k}^{- \lambda}}{(1-\alpha)_{k}}. \tag{5}\]  
The packet transmission time is the time required for a packet of size \(L\) bits to travel from sensor node \(N_{k}\) to the BS; in this case, \(L\) denotes \(L_{1}\) and \(L_{2}\). The expression is as follows:  
\[T_{k}^{(v)}=\frac{\mathcal{R}_{k}}{\ln(1+\mu_{k}\gamma_{k}^{(v)})}, \tag{6}\]  
where \(\mathcal{B}_{k}=\frac{L\ln(2)}{W}\), \(\mu_{k}=-\frac{1}{\log(5*BER_{k})}\)[22], \(W\) denotes the system bandwidth, while \(\mu_{k}\) denotes the bit error rate. We also obtain the expression for the signal-to-noise ratio (SNR), which is as follows:  
\[\gamma_{k}^{(v)}=\frac{B_{k}G_{k1}^{(v)}d_{k}^{-\lambda}}{W\mathcal{Q}_{0}}= \frac{\alpha_{0}G_{k1}^{(v)}G_{k2}\gamma_{0}\eta_{k}d_{k}^{-2\lambda}}{(1- \alpha)_{k}}, \tag{7}\]  
In order to accommodate different downlink energy harvesting methods, we make use of various channel gain assumptions. Specifically, the channel gain \(G_{k1}^{(v)}\) utilizes different  
Figure 2: The time required for a node to send a data packet is measured by \((1-\alpha)_{k}T\), while \(\alpha_{0}T\) represents the time that all nodes spend together collecting energy.  
technologies depending on the two gain modes, which are represented by \(G_{k1j}\) and \(G_{k2j}\) for MRC and SC technologies, respectively, and the expression is as follows:  
\[G_{k1}^{(v)}=\left\{\begin{array}{cc}G_{k1}^{( MRC)}&=\sum_{j=1}^{M}G_{k1j},\qquad v=MRC\\ G_{k1}^{(SC)}&=\max_{j\in\{1,2,\ldots,M\}}G_{k1j},\qquad v=SC,\end{array}\right. \tag{8}\]  
where \(\gamma_{0}=\frac{p}{WZ_{0}},\quad\forall v\in\{MRC,SC\}\).