## 5 Performance Analysis and Simulation Results
### Delay for Detection  
In this section, we analyze the delay for detection in the PNLDC approach. According to Ref. [65], for a sensor node with a distance of \(x\) from the sink, the limitation on the probability of a missed packet (denoted as \(\mathcal{P}_{mis}\)) can be expressed by the following formula. \[ \begin{split}\mathcal{P}_{mis}\geq\left\{1-\frac{R_{S}}{R}[(1- \delta_{SEN})\mathcal{P}\{A\}+\delta_{SEN}]\right\}^{\alpha}\\ \text{where }\mathcal{P}\{A\}=\left\{\begin{array}{cc}\frac{4R_{S}} {\pi\gamma_{SEN}(1-\delta_{SEN})},&if\ 2R_{S}/\gamma<\mathcal{T}_{SEN}(1-\delta_{SEN})\\ \frac{4R_{S}-2\sqrt{4R_{S}^{2}-(1-\delta_{SEN})^{2}\gamma^{2}}}{\pi\gamma_{SEN} (1-\delta_{SEN})}+1-\frac{2a\sin\left(\frac{\gamma(1-\delta_{SEN})}{2R_{S}} \right)}{\pi\gamma_{SEN}(1-\delta_{SEN})},&else\end{array}\right.\end{split}  \tag{30}\]  
where the radius of the network is \(R\), the sensing range of the node is \(R_{S}\), \(\alpha\) is the number of nodes in this range, \(\gamma\) is the speed of the packet and \(\theta\) is the proportion of nodes with full duty cycle. Equation (31) is based on the network that adopts the same duty cycle. In the PNLDC approach, the probability of a missed packet (\(\mathcal{P}_{mis}\)) will alter because the duty cycle of some nodes is set to 1. The change of \(\mathcal{P}_{mis}\) is related to the proportion of nodes with a full duty cycle. Therefore, the formula applicable to this paper is as follows. \[\mathcal{P}_{mis}\geq C_{\alpha}^{\theta_{\alpha}}\bigg{(}\frac{R_{S}}{R}\bigg{)} ^{2\theta_{\alpha}}\bigg{\{}1-\frac{R_{S}}{R}[(1-\delta_{SEN})\mathcal{P}\{A \}+\delta_{SEN}]\bigg{\}}^{\alpha-\theta_{\alpha}} \tag{31}\]  
Based on the above analysis, next considering the case of detecting multiple packets at the same time. First we analyze the probability that all packets are not detected (denoted as \(\mathcal{P}_{mis}\)). It can be calculated by:  
\[\mathcal{P}_{mis}=(\mathcal{P}_{mis})^{\alpha_{m}} \tag{32}\]  
where \(\alpha_{m}\) represents the number of packets that are sent to the node at the same time, and \(\mathcal{P}_{mis}\) is the probability of a missed packet in the case of only a packet entering the range of perception. So, \(1-\mathcal{P}_{mis}\) represents the probability that at least one packet of \(\alpha_{m}\) packets is detected. Figure 19: The delay for forwarding data under different distance. The probability that at least one packet is not detected (denoted as \(\mathcal{P}_{misp}\)) can be derived:  
\[\mathcal{P}_{misp}=1-\left(1-\mathcal{P}_{mis}\right)^{\alpha_{m}} \tag{33}\]  
Figure 20 shows the variation trend of \(\mathcal{P}_{mis}\) under the different sensing duty cycle. With the decrease of the duty cycle, \(\mathcal{P}_{mis}\) also decreases, which coincides with Equation (31). And with the narrowing of the sensing range, the probability of missed a packet (\(\mathcal{P}_{mis}\)) increases. The variation trend of \(\mathcal{P}_{misp}\) under the different sensing duty cycle is shown in Figure 21. Here, three values of the number of packets sent to the node (2, 4 and 6) are considered. It can be obtained from the figure: as the number of packets increases, \(\mathcal{P}_{misp}\) also increases. In the same way as Figure 20, \(\mathcal{P}_{misp}\) decreases as the sensing duty cycle increases, which is consistent with Equations (31) and (33). In the PNLDC approach, assuming that the distance between the sensor node and the sink is \(x\). The probability of missed packets can be calculated using the infinitesimal method [65]. A fan-shaped region with a width of \(dx\) and an angle of \(d\varphi\) is selected. Thus the area of this region is \(xdxd\varphi\). Then, the weighted probability that at least one packet is not detected and the weighted probability that all packets are not detected can be calculated by the following equations.