## 4 The Design of PNLDC Approach
### Research Motivation
\[\mathcal{Y}_{FFSC}=\sum_{i=0}^{\frac{p}{\mathcal{T}^{A}}-2}i\Bigg{[}1-\bigg{(}1 -\frac{\mathcal{T}^{A}}{\rho}\bigg{)}^{2}\Bigg{]}\bigg{(}1-\frac{\mathcal{T}^ {A}}{\rho}\bigg{)}^{pi}+\Big{(}\frac{\rho}{\mathcal{T}^{A}}-1\Big{)}\bigg{(}1- \frac{\mathcal{T}^{A}}{\rho}\bigg{)}^{p(\frac{\rho}{\mathcal{T}^{A}}-1)} \tag{7}\]  
Based on Equation (7), Figure 4 shows the relationship of the delay, and the number of nodes in the FNS and the duty cycle. With the increase of the duty cycle, the expected delay decreases. This is because a longer duty cycle means that the nodes are in the active state for a longer period of time. Therefore, the waiting time before the receiver wakes up is reduced. It can also be seen that as the number of nodes in FNS increases, the expected delay decreases. This is because each node in this network has an independent duty cycle, with the increase of the number of nodes in the FNS, the probability that a node in the FNS is exactly in the active state when the sender has a packet to send is greater. The distribution of the nodes corresponding to Figure 2 is shown in Figure 5. The black nodes represent nodes that adopt a full duty cycle, that is, \(\delta=1\). And the white nodes represent nodes that adopt a basic duty cycle. In this situation, selecting some nodes and setting their duty cycle to 1  
Figure 4: The expected delay decreases as the duty cycle increases. Figure 3: The distribution of all nodes with a basic duty cycle. can reduce delay. This is because as long as a node in the forwarding nodes set is in the active state, data transmission can be carried out. As shown in Figure 6, the larger the duty cycle is, the more energy the nodes consume. This is because the energy consumed by nodes in the active state is more than during sleep, and the larger the duty cycle is, the longer the time that the node in the active state is. Due to the imbalance of energy consumption in the network (see Figure 7), it is feasible to set up a part of the nodes with a duty cycle of 1 by using the residual energy of the non-hotspots. Figure 5: The distribution of two kinds of nodes. Figure 6: The energy consumption increases as the duty cycle increases. Figure 7: The energy consumption in different regions of the network. The residual energy in different regions can make the proportion of nodes with duty cycle of 1 different in the regions far or close to the sink, as can be seen in Figure 1:0  
The theoretical calculation (see corollary in Section 5.2) shows that the higher the proportion of nodes with a duty cycle of 1 is, the smaller the delay is, which can be seen in Figure 1:1 No matter the value of the base duty cycle, the delay for forwarding data increases with the increase of the distance to the sink. Combined with Figures 8 and 9, we can see that the larger the base duty cycle is, the more the number of nodes with full duty cycle is, and the smaller the delay is. On the basis of the above analysis, this paper proposes a better approach called the PNLDC approach, which can effectively reduce the delay. Figures 10 and 11 compare the different performances of the fixed duty cycle strategy and the PNLDC approach in two aspects, delay for detection and delay  
Figure 1:2 The proportion under different distance. Figure 1:3 The delay for forwarding data under different distance. for forwarding, respectively. It can be observed that the PNLDC approach performs better than the fixed duty cycle strategy in reducing delay.