## 4 The Design of PNLDC Approach
### The PNLDC Approach Design  
The core thought of the PNLDC approach is: take full advantage of the residual energy in non-hotspots while not affecting the network lifetime, and reduce the delay by selecting partial nodes with a duty cycle of 1. The forwarding nodes set of nodes is first defined here. The range of the forwarding nodes set is formed by intersecting the circular transmission range of the sender with an arc that is parallel to the arc passing through the center of the circle. Suppose the parallel distance between these two arcs is \(x\), and the transmission range of the sender is \(r\). As shown in Figure 12, nodes in the shadow region are the candidate forwarding nodes. The distance between the sink and candidate forwarding nodes is  
Figure 11: The delay for forwarding data under different distance. Figure 10: The delay for detection under different distance. closer to the distance from the sender node to the sink, so as to avoid selecting forwarders that are far from the sink. Because the node density in the network is \(\rho\), the number of the candidate forwarding nodes (denoted as \(\alpha\)) can be calculated by Equation (8). \[\alpha=\rho\cdot\left[(d-x)^{2}\cdot\left(\cos^{-1}\frac{d^{2}+(d-x)^{2}-r^{2}}{ 2d\cdot(d-x)}\right)+\left(\cos^{-1}\frac{r^{2}+d^{2}-(d-x)^{2}}{2 dr}\right) \cdot r^{2}-\sin\left(\cos^{-1}\frac{r^{2}+d^{2}-(d-x)^{2}}{2 dr}\right)\cdot rd\right] \tag{8}\]  
where the area of the range of the forwarding nodes set is calculated by the area of fan-shaped DBC plus the area of fan-shaped ABC minus the area of quadrilateral ABDC. Next, to the phase that is the focus of this paper. According to Theorem 1, the greater the distance between the sender node and sink, the more the remaining energy. Thus, the duty cycle of partial nodes in the non-hot area is set to 1 based on the remaining energy in this region. And the higher the proportion of the selected nodes with full duty cycle in the area farther away from the sink. **Theorem 1**. : _For the nodes with a distance of \(x\) from the sink, the distance between its forwarding nodes and the sink is \(x^{\prime}\), then the remaining energy can be calculated by:_  
\[\mathbb{E}_{REM}^{x}=\left(\epsilon_{R}^{\prime}-\epsilon_{R}^{x}\right)\cdot \xi_{R}^{x}+\left(\epsilon_{T}^{\prime}-\epsilon_{T}^{x}\right)\cdot\xi_{T}^{ x}+\left(\xi_{R}^{x}+\xi_{T}^{x}-\zeta_{R}^{\prime}-\zeta_{T}^{\prime}\right) \cdot\mathcal{T}_{COM} \tag{9}\]  
Proof. : The remaining energy is related to the transmission of the packets, the reception of the packets, the low power listening, and the sensing duration. According to [65], we can get: \(\mathbb{E}_{REM}^{x}=\mathbb{E}_{T}^{\prime}+\mathbb{E}_{R}^{\prime}+\mathbb{ E}_{EN}^{\prime}+\mathbb{E}_{LPL}^{x}-\mathbb{E}_{T}^{x}-\mathbb{E}_{EN}^{x}- \mathbb{E}_{LPL}^{x}=\left(\epsilon_{R}^{\prime}-\epsilon_{R}^{x}\right)\cdot \xi_{R}^{x}+\left(\epsilon_{T}^{\prime}-\epsilon_{T}^{x}\right)\cdot\xi_{T}^{ x}+\left(\xi_{R}^{x}+\xi_{T}^{x}-\zeta_{R}^{\prime}-\zeta_{T}^{\prime}\right) \cdot\mathcal{T}_{COM}\), where \(\epsilon_{T}^{x}\) denotes the number of packets sent by the node \(x\) away from the sink and \(\epsilon_{R}^{x}\) is the number of packets received by the node \(x\) away from the sink. The calculation of \(\epsilon_{T}^{x}\) and \(\epsilon_{R}^{x}\) will be given in the Section 5.1. At the stage of optimizing the duty cycle, the PNLDC approach selects partial nodes and sets their duty cycle to 1, so as to take full advantage of the residual energy and reduce the delay. The selection of nodes with full duty cycle in different regions is based on Theorem 2. Figure 12: The range of candidate forwarding nodes. **Theorem 2**.