## 4 The Design of PNLDC Approach  
### Research Motivation  
In wireless sensor networks, because each node has a transmission range, data packets usually need to pass through multiple hops on the transmission path. Thus, the farther the distance between the sink and the sender node, the more hops that the packet needs to pass through. At the same time, the network uses the duty-cycling method based on the asynchronous protocol, which means that the nodes make a periodic conversion between active modes and sleep modes. Therefore, packets can be sent and received only when corresponding nodes are in the active state. But for each hop on the transmission path, it does not guarantee that the receiver is in the active state. If the receiver is in the sleep state, the sender node must wait until it wakes up, and the worst case is that the waiting time is close to the whole cycle duration. As a result, the more hops on the transmission path, the larger the transmission delay. It can be obtained that the transmission delay of the nodes in the area far away from the sink is large, which will affect the communication delay of the entire network. Considering that the distribution of sensor nodes in the network is fixed, it is necessary to reasonably plan the working mode of nodes to reduce the transmission delay. There is good deal of research on the effective reduction of transmission delay between nodes and their candidate forwarding nodes in duty-cycled wireless sensor networks. Most of the proposed methods adopt an optimal value of the duty cycle, that is, the duty cycle of all nodes in the network are the same. In these methods, some other network performance often needs to be sacrificed to reduce the delay. This is because these methods usually increase the time of the nodes in the active state of a work cycle, that is, increasing the duty cycle of nodes. But after improving the duty cycle, since the energy consumed when a node is in the active state is more than one thousand times of the energy consumed when it sleeps, the energy consumption of the network will be improved. Considering that the maximum energy consumption has a great impact on network lifetime, reducing the delay on the premise of guaranteeing energy consumption that does not affect the network lifetime is a problem that should be solved urgently. Many studies show that the nodes in the region near the sink undertake a lot of data forwarding. Thus their energy consumption is high while the nodes far away from the sink maintain residual energy. Related studies indicate that the region far away from the sink held about eighty percent of the network energy. Selecting a proportion of nodes in the non-hotspots and setting their duty cycle to 1 can reduce the delay while using the residual energy. When there is at least one node with the duty cycle of 1 in the forwarding nodes set of the sender, the sender does not need to wait for the receiver to wake up before the transmission. This will effectively reduce delay. The comparison between the nodes with basic duty cycle and the nodes with the duty cycle of 1 is shown in Figures 1 and 2. The distribution of the nodes corresponding to Figure 1 is shown in Figure 3. The white nodes represent nodes that adopt a basic duty cycle. Figure 1: Timeline of nodes with basic duty cycle. Figure 2: Timeline of nodes with full duty cycle. According to Theorem 1 in Ref. [61], considering that the length of a duty period is \(\rho\) and in a duty period the length of the time that node in the active state is \(\mathcal{T}^{A}\), then the duty cycle of each node is \(\frac{\mathcal{T}^{A}}{\rho}\). And the number of nodes in the forwarding nodes set is \(p\). In this situation, the sender can forward the packet if there is a node in FNS in the active state, so the expected delay can be calculated by Equation (7).