## 4 The Design of PNLDC Approach
### The PNLDC Approach Design
: _For the region \(x\) away from the sink, the proportion of nodes with full duty cycle (denoted as \(\theta\)) can be calculated by the following equation._  
\[\theta=(\epsilon_{R}^{x}-\epsilon_{R}^{\prime})\cdot\xi_{R}/\partial \tag{10}\] \[\partial=(\mu_{R}-\mu_{S})(\delta_{COM}-1)\mathcal{T}_{COM}+(\mu_ {SEN}-\mu_{S})(\delta_{SEN}-1)\mathcal{T}_{SEN}+(\epsilon_{R}^{x}+\epsilon_{T}^ {x}-\epsilon_{R}^{\prime}-\epsilon_{T}^{\prime})\cdot\mathcal{T}_{COM}+\delta _{T}^{\prime}\epsilon_{T}^{\prime}-\delta_{T}^{x}\epsilon_{T}^{x}\]  
Proof. : Before the adjustment of the duty cycle of partial nodes in this phase, all nodes in the region \(x\) away from the sink have the same duty cycle. As mentioned earlier in this paper, it takes a lot of energy to change the nodes from the original duty cycle to the full duty cycle (that is, the duty cycle is 1). Considering that the regions far away from the sink have available remaining energy, this part of energy can be used to complete the operation in this phase. After the selection of the nodes with full duty cycle and the optimization of their duty cycle, the energy consumption of the nodes in the region \(x\) away from the sink will be equal to the energy consumption of nodes very close to the sink, which can be expressed by the formula:  
\[\big{(}\mathbb{E}_{T}^{\prime}+\mathbb{E}_{SEN}^{\prime}+\mathbb{E}_{LPL}^{ \prime}-\mathbb{E}_{T}^{x}-\mathbb{E}_{SEN}^{x}-\mathbb{E}_{LPL}^{x}\big{)} \theta=\mathbb{E}_{R}^{x}-\mathbb{E}_{R}^{\prime} \tag{11}\]  
The above formula can also be expanded as follows:  
\[[(\mu_{R}-\mu_{S})(\delta_{COM}-\delta_{COM}^{x})\mathcal{T}_{COM}+(\mu_{SEN}- \mu_{S})(\delta_{SEN}^{\prime}-\delta_{SEN}^{x})\mathcal{T}_{SEN}+(\epsilon_{ R}^{x}+\epsilon_{T}^{x}-\epsilon_{T}^{\prime})\mathcal{T}_{COM}+\delta_{T}^{ \prime}\epsilon_{T}^{\prime}-\delta_{T}^{x}\epsilon_{T}^{x}]\theta=(\epsilon _{R}^{x}-\epsilon_{R}^{x})\xi_{R} \tag{12}\]  
Because the duty cycle of nodes with full duty cycle is 1, then  
\[[(\mu_{R}-\mu_{S})(\delta_{COM}-1)\mathcal{T}_{COM}+(\mu_{SEN}-\mu_{S})( \delta_{SEN}-1)\mathcal{T}_{SEN}+(\epsilon_{R}^{x}+\epsilon_{T}^{\prime}- \epsilon_{R}^{\prime}-\epsilon_{T}^{\prime})\mathcal{T}_{COM}+\delta_{T}^{ \prime}\epsilon_{T}^{\prime}-\delta_{T}^{x}\epsilon_{T}^{x}]\theta=(\epsilon _{R}^{x}-\epsilon_{R}^{\prime})\xi_{R} \tag{13}\]  
Based on the Formula (13), the proportion of nodes with full duty cycle can be calculated by the following equation. Figure 10:0  
Figure 10:1  
\(\square\)  
According to the above theorems, the greater the distance between the area and the sink, the higher the proportion of the selected nodes with full duty cycle in this area, which can be seen from Figure 10:2  
Based on Equation (15), the relationship between the proportion of nodes with full duty cycle and the distance to the sink is more intuitively shown in Figure 10:3 The proportion of nodes with full duty cycle is 1.2  
Figure 10:4 The distribution of two kinds of nodes relative to the sink. duty cycle increases with the increase of distance. This is because the residual energy is much more when the distance between the node and the sink is larger. Combined with the changing trend of the proportion under a different duty cycle, the larger the basic duty cycle is, the greater the number of nodes with a full duty cycle.