## 3 System Model and Problem Statements
### System Parameters  
Sensor nodes are mainly composed of two units: a communication unit and sensing unit. Among them, the sensing unit is in charge of the perception of network events and invoking the communication unit for packet transmission. Because in the network sensor nodes can both perceive and forward packets, the energy consumption of nodes is serious. To reduce the energy consumption in the system, as mentioned above, this network uses a method called duty-cycling so that the sensor nodes can make a periodic conversion between active modes and sleep modes according to the specified wake-up interval. Therefore, the time is divided into some frames of the same length in the working cycle, and each frame is composed of an active state and a sleep state. In the active state, nodes can both send and receive packets. In contrast, nodes cannot complete these operations in the sleep state, which saves energy. Based on the above description, in a unit period, the duty cycle of each node can be defined as the ratio of the time of the node in the active mode and the time of the entire unit cycle.  
In this paper, for sensor nodes, there are three main periods: communication, sensing, and self learning. In the self learning period, nodes remain in an active state and perceive the state of their candidate forwarding nodes. After this period, the sender nodes start to communicate. The communication duty cycle is indicated by \(\delta_{COM}\), and the sensing duty cycle is denoted as \(\delta_{SEN}\), \(\delta_{COM}\) and \(\delta_{SEN}\) can be expressed by Equations (1) and (2).  
\[\delta_{COM}=\mathcal{T}^{A}_{COM}/\left(\mathcal{T}^{A}_{COM}+\mathcal{T}^{S} _{COM}\right) \tag{1}\]  
\[\delta_{SEN}=\mathcal{T}^{A}_{SEN}/\mathcal{T}_{SEN}=\mathcal{T}^{A}_{SEN}/ \left(\mathcal{T}^{A}_{SEN}+\mathcal{T}^{S}_{SEN}\right) \tag{2}\]  
where \(\mathcal{T}^{A}_{COM}\) is the time that nodes in the active mode during the communication period and \(\mathcal{T}^{S}_{COM}\) is the time that nodes in the sleep mode during the communication period. In the sensing duration (denoted as \(\mathcal{T}_{SEN}\)), \(\mathcal{T}^{A}_{SEN}\) is the duration when nodes in active state and \(\mathcal{T}^{S}_{SEN}\) is the duration when nodes in sleep state.  
In most of the previous research, the duty cycle of all sensor nodes is the same. But in this paper, we will select some nodes in the non-hot areas on the basis of self learning, and set the duty cycle of them to 1, so as to reduce delay, improve the quality of monitoring packets and effectively use energy. Thus, the power consumption of nodes mainly includes four parts: (1) the power required to send or receive packets, denoted as \(\mu_{T}\) and \(\mu_{R}\). (2) the power used in low power listening (_LPL_), denoted as \(\mu_{LPL}\). (3) the power consumption of sensing when nodes in active state, denoted as \(\mu_{SEN}\). (4) the power used when nodes in sleep state, denoted as \(\mu_{S}\).  
In this system model, the main parameters are listed in the Table 1[42; 61]. The other parameters are explained in the later text.  
\begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Description** & **Value** \\ \hline \(E_{ini}\) & Initial energy & 0.5 J \\ \(\mathcal{T}_{COM}\) & Communication duration & 100 ms \\ \(\mu_{T}\) & Power consumed by transmission & 0.0511 W \\ \(\mu_{R}\) & Power consumed by reception & 0.0588 W \\ \(\mu_{S}\) & Power consumed by sleeping & 2.4 \(\times\) 10\({}^{-7}\) W \\ \(\mathcal{T}_{B}\) & Preamble duration & 0.26 ms \\ \(\mathcal{T}_{ACK}\) & Knowledge window duration & 0.26 ms \\ \(\mathcal{T}_{P}\) & Packet duration & 0.93 ms \\ \(\mathcal{T}_{SEN}\) & Sensing duration & 15 s \\ \hline \end{tabular}
\end{table}
Table 1: System Parameters.