## 5 Performance Analysis and Simulation Results  
### Energy Consumption and Network Lifetime  
In the PNLDC approach, nodes mainly have four different states: low power listening, sensing, sending packets, and receiving packets. Because in the network the amount of data packets is large and the reception and transmission of packets are the main energy-consuming operations, the energy consumption is serious. The energy consumption of the node \(x\) away from the sink can be calculated by the following equation. \[\xi_{TOT}^{x}=\xi_{T}^{x}\epsilon_{T}^{x}+\xi_{R}^{x}\epsilon_{R}^{x}+\xi_{ LPL}^{x}+\xi_{SEN}^{x} \tag{16}\]  
where \(\xi_{TOT}^{x}\) represents the total energy consumption of the node, \(\xi_{T}^{x}\) and \(\xi_{R}^{x}\) are the energy consumed in the transmission and reception respectively, and \(\xi_{LPL}^{x}\) is the energy consumption of low power listening and \(\xi_{SEN}^{x}\) represents the energy consumption in sensing. \(\epsilon_{T}^{x}\) is the number of packets in transmission and \(\epsilon_{R}^{x}\) represents the number of packets in reception. According to Ref. [61], assuming that the distance between the node and sink is \(x\), \(N\) packets are sent by this node, then  
\[N=\frac{(\tau+1)\cdot\tau r}{2x}+\tau+1 \tag{17}\]  
where \(\tau\) is an integer so that \(\tau r+x\) is less than the radius of this network. When the probability of event production is \(\beta\), the amount of packets in reception is  
\[\epsilon_{T}^{x}=\left(\frac{(\tau+1)\cdot\tau r}{2x}+\tau+1\right)\beta \tag{18}\]  
The amount of packets sent by the node is equal to the amount of packets received by it plus the packets generated by itself. \[\epsilon_{R}^{x}=\epsilon_{T}^{x}+\beta \tag{19}\]  
Figure 14: The proportion under different distance. For the nodes with a distance of \(x\) from the sink, the energy consumed in the transmission can be presented as the following equation. \[\xi_{T}^{x}=(\mathcal{T}_{B}\mu_{T}+\mathcal{T}_{ACK}\mu_{R})\frac{\mathcal{T}_{ COM}(1-\delta_{COM})}{2(\mathcal{T}_{B}+\mathcal{T}_{ACK})}+\mathcal{T}_{P}\mu_{T} \tag{20}\]  
where \(\mathcal{T}_{B}\mu_{T}\) denotes the energy consumption in the preamble duration, \(\mathcal{T}_{ACK}\mu_{R}\) represents the energy consumed by the transmission of ACK message and \(\mathcal{T}_{P}\mu_{T}\) represents the energy consumption of the packet transmission. The energy consumption in the reception of a packet can be expressed as:  
\[\xi_{R}^{x}=\mathcal{T}_{B}\mu_{R}+\mathcal{T}_{ACK}\mu_{T}+\mathcal{T}_{P}\mu _{R} \tag{21}\]  
The energy consumption of sensing can be expressed by:  
\[\xi_{SEN}^{x}=(1-\delta_{SEN})\mu_{S}+\mu_{SEN}\delta_{SEN} \tag{22}\]  
where \(\delta_{SEN}\) is the sensing duty cycle, \(\mu_{SEN}\) is the power consumption of sensing when nodes are in the active state and \(\mu_{S}\) is the power used when nodes are in the sleep state. For the nodes \(x\) away from the sink, the energy consumption of low power listening can be calculated as:  
\[\xi_{LPL}^{x}=\delta_{COM}\mu_{R}+(1-\delta_{COM})\mu_{S}-\xi_{T}^{x}-\xi_{R}^ {x} \tag{23}\]  
where the first part of Equation (24) represents the energy used by listening and the second part represents the energy consumption of sleeping. According to Ref. [61], \(\xi_{T}^{x}\) and \(\xi_{R}^{x}\) can be calculated by the following equations. \[\xi_{T}^{x}=\frac{\varepsilon_{T}^{x}}{\mathcal{T}_{COM}}\bigg{\{}\mathcal{T}_ {B}\mu_{R}+\bigg{[}\mathcal{T}_{B}+\mathcal{T}_{ACK}+\frac{\mathcal{T}_{COM} (1-\delta_{COM})}{2}\bigg{]}\mu_{S}\bigg{\}} \tag{24}\]  
\[N=\frac{(\tau+1)\cdot\tau r}{2x}+\tau+1 \tag{17}\]0  
The comparison of energy consumption of the fixed duty cycle strategy, dynamic duty cycle strategy, and PNLDC approach is shown in Figures 15 and 16, when the basic duty cycle is 0.2. It can be obtained from the change curve of energy consumption in various regions that energy consumption decreases with the increase of distance to the sink in the fixed duty cycle strategy.