## 5 Performance Analysis and Simulation Results
### Delay for Detection
\[\mathcal{P}_{misp}^{x}=\int_{0}^{R}\int_{0}^{2\pi}x\cdot\mathcal{P}_{misp}dxd\varphi \tag{34}\]  
\[\mathcal{P}_{misa}^{x}=\int_{0}^{R}\int_{0}^{2\pi}x\cdot\mathcal{P}_{misal}dxd\varphi \tag{35}\]  
The probability of missed packets will affect the delay for detection. This delay can be measured by the time that the packet is first detected by the sensor node after entering the perception region of nodes. According to [65], the delay for detection can be calculated as:  
Figure 20: The \(\mathcal{P}_{mis}\) under different sensing duty cycle. \[\mathcal{Y}_{DET}=\left\{ \begin{array}{c}C_{\alpha}^{\theta\alpha}\left(\frac{R_{ \varsigma}}{K}\right)^{2\theta\alpha}\frac{\left(\pi R^{2}\right)^{\alpha- \theta\alpha+1}-\left(2\phi_{\gamma}R_{\varsigma}-\pi R^{2}\right)^{\alpha- \theta\alpha+1}}{2R_{\varsigma}\cdot\gamma\left(\pi R^{2}\right)^{\alpha- \theta\alpha}\left(-\theta\alpha+1\right)},\ if\ \frac{d}{\gamma}>h\\ \frac{\int_{0}^{\theta\alpha}\left[C_{\alpha}^{\theta\alpha}\left(\frac{R_{ \varsigma}}{K}\right)^{2\theta\alpha}\left(1-\frac{2\phi R_{\varsigma}}{\pi R^ {2}}\right)^{\alpha-\theta\alpha}+\sum_{i=1}^{\kappa}\mathcal{P}\left\{B \right\}\right]d\psi}{\gamma},\ else\end{array} \right. \tag{36}\]  
where \(d\) denotes the length of the intersection between the packet transmission path and the sensing region of the sensor node, \(\gamma\) is the speed of the packet transmission and \(h=\mathcal{T}_{\mathit{SEN}}(1-\delta_{\mathit{SEN}})\). \(\psi\) is the length of the packet transmission path in the sensing region, and \(\psi_{m}\) is the average maximum value of \(\psi\). And according to [65], \(\mathcal{P}\{B\}=\left( \begin{array}{c}\alpha\\ i\end{array} \right)\left(1-\frac{2\varphi R_{\varsigma}}{\pi R^{2}}\right)^{ \alpha-i}\left(\mathcal{P}\{C\}\cdot\frac{2\varphi R_{\varsigma}}{\pi R^{2}} \right)^{i}\), where \(\mathcal{P}\{C\}\) represents the probability of the path of a packet passes through a specific perception region of a sensor node that does not detect the packet, and \(i\) is the number of this kind of nodes. If \(d/\gamma>\mathcal{T}_{\mathit{SEN}}(1-\delta_{\mathit{SEN}})\), \(\mathcal{P}\{C\}=0\). Else if \(d/\gamma\leq\mathcal{T}_{\mathit{SEN}}(1-\delta_{\mathit{SEN}})\), \(\mathcal{P}\{C\}=\frac{R_{\varsigma}}{K}(1+\mathcal{P}\{A\})(1-\delta_{ \mathit{SEN}})\). Based on the above analysis, the weighted delay for detection can be calculated by Equation (37). \[\mathcal{Y}_{DET}^{x}=\left\{ \begin{array}{c}\mathcal{Y}_{DET},if\ \frac{d}{\gamma}>\mathcal{T}_{\mathit{SEN}}(1-\delta_{\mathit{SEN}})\\ \int_{0}^{R}\int_{0}^{2\pi}x\cdot\mathcal{Y}_{DET}dxd\varphi,\ else\end{array} \right. \tag{37}\]  
where the sensor node is \(x\) away from the sink and the network radius is \(R\). Figure 22 shows the variation trend of delay for detection under different radius of network, when the transmission range of the node is ninety meters and the basic duty cycle is 0.2. As can be seen in this figure, with the increase of the network radius, the delay for detection also increases. Compared with the dynamic duty cycle strategy, the overall performance of the PNLDC approach in terms of delay for detection is no worse than that of the dynamic duty cycle strategy. It can be obtained from the contrast between the grey and black columns that the PNLDC approach is actually better able to reduce the delay for detection. Compared with the fixed duty cycle strategy, the delay for detection is reduced by 2.1-24.6% in the PNLDC approach. Therefore, in terms of delay for detection, the PNLDC approach outperforms the fixed duty cycle strategy. Figure 21: The \(\mathcal{P}_{\mathit{misp}}\) under different duty cycle.