## 5 Performance Analysis and Simulation Results
### Delay of Forwarding Data  
In the process of data transmission, besides sending and receiving data, there are also the sending and receiving of the preamble and the ACK. We analyze the delay for forwarding data in this section. **Theorem 3**. : _In the PNLDC approach, if the sender node is \(x\) away from the sink and the proportion of nodes with full duty cycle is \(\theta\), then the average delay for forwarding data per hop can be calculated by Equation (27)._  
\[\mathcal{Y}_{hop}=\Bigg{[}\mathcal{T}_{P}+\mathcal{T}_{B}+\mathcal{T}_{ACK}+ \frac{\mathcal{T}_{COM}(1-\delta_{COM})^{2}}{2}\Bigg{]}\bigg{(}1-\theta asin \frac{R_{S}}{x}\bigg{)} \tag{27}\]  
_where \(\delta_{COM}\) is the communication duty cycle, \(\mathcal{T}_{COM}\) represents the communication duration, and \(\mathcal{T}_{P}\), \(\mathcal{T}_{B}\), \(\mathcal{T}_{ACK}\) represent the duration of packet, preamble and ACK message, respectively._  
**Proof. ** Considering that the state of the receiver node, the probability that this receiver is in the active state when the sender node starts sending a packet is \(\delta_{COM}\). In such a situation, the transmission delay is \(\mathcal{T}_{P}+\mathcal{T}_{B}+\mathcal{T}_{ACK}\). In contrast to this case, the receiver is in the sleep state, the corresponding probability is \(1-\delta_{COM}\). To calculate the average transmission delay for each hop, the best and worst situations needs to be taken into account. In the best situation, the sender node starts sending a packet when the corresponding receiver starts the low power listening. Therefore, the packet will be successfully transferred to the receiver after \(\mathcal{T}_{P}+\mathcal{T}_{B}+\mathcal{T}_{ACK}\). In the worst situation, the sender node needs to wait for the whole duration of sleep state. Because the receiver should send the ACK back to the sender after receiving the entire preamble, this may require two transmissions of preambles before the packet transmission in the worst case situation. Thus, the transmission delay is \(\mathcal{T}_{P}+\mathcal{T}_{B}+\mathcal{T}_{ACK}+\mathcal{T}_{COM}(1-\delta _{COM})\). Based on the above analysis, according to [65], the average delay per hop in the transmission path can be expressed as:  
\[\mathcal{Y}_{basic}=\mathcal{T}_{P}+\mathcal{T}_{B}+\mathcal{T}_{ACK}+\frac{ \mathcal{T}_{COM}(1-\delta_{COM})^{2}}{2} \tag{28}\]  
This equation is suitable for the case that all nodes take the same duty cycle. Figure 17: The network lifetime under different transmission range. In the PNLDC approach, because some nodes are selected as nodes with full duty cycle, if the packet meets these nodes on the transmission path, the delay after detection will be decreased. According to [66], the probability of nodes with a full duty cycle on the transmission path is \(\theta a\sin\frac{R_{S}}{x}\). Thus, the delay for forwarding data per hop can be calculated by Equation (28). \(\square\)  
The proportion of nodes with a full duty cycle in the region \(x\) away from the sink is calculated by Equations (14) and (15) in Section 4.2. **Corollary 1**. : _The total delay for forwarding data can be calculated as:_  
\[\mathcal{Y}_{x,hop}=\vartheta\Bigg{[}\mathcal{T}_{P}+\mathcal{T}_{B}+ \mathcal{T}_{ACK}+\frac{\mathcal{T}_{COM}(1-\delta_{COM})^{2}}{2}\Bigg{]} \bigg{(}1-\thetaasin\frac{R_{S}}{x}\bigg{)} \tag{29}\]  
_where is \(\vartheta=\left\lceil\frac{x}{r}\right\rceil+1\) and \(0<x\leq R\)._  
**Proof. ** Assuming that the sender node is \(x\) away from the sink, and the packet sent by it is required to pass through \(\vartheta\) hops, this means that there are \(\vartheta\) nodes on the transmission path of a packet. Therefore, the total delay for forwarding data can be expressed as the product of the delay of each hop and the number of hops. \(\square\)  
Figures 18 and 19 show the relationship between the delay for forwarding data and the distance between nodes and the sink.