## 2 System Model
### Omnidirectional and Probabilistic Perception Model  
The omnidirectional perception model in Figure 2a refers to a circular area on a plane, centered on the sensor node with a radius \(R_{\rm s}\). Within this area, the perception probability is set to 1, while outside this area, the sensor perception probability is 0. Here, \(R_{\rm s}\) is known as the perception radius and is determined by the physical characteristics of the sensor node. The omnidirectional perception model is commonly used in application scenarios requiring basic event detection or binary decision-making, such as intrusion detection, smoke detection, and door contact sensors [48]. The advantages of the omnidirectional perception model are its simplicity in calculations and low resource usage. However, its drawback is the inability to handle uncertainty in perception results.  
In practical application scenarios, the environments where wireless sensor networks are deployed often vary greatly and are affected by a range of environmental interferences, such as external temperature, noise, and humidity. Additionally, during signal transmission, sensor nodes also experience natural attenuation of wireless communication signals. Most sensor nodes have a disk model characteristic, where the sensing intensity continuously attenuates with sensing distance. In the paper [49], a corresponding probabilistic perception model has been proposed. In this model, when the distance between the sensor node and the target point is less than \(R_{\rm s}-R_{se}\), the sensor node perception probability for the target point is 1. When the distance between the sensor node and the target point is greater than \(R_{\rm s}+R_{se}\), the perception probability is 0. If the distance between the sensor node and the target point falls between \(R_{\rm s}-R_{se}\) and \(R_{\rm s}+R_{se}\), the perception probability follows an exponential distribution, decreasing from the inside out. The specific calculation formula is:  
Figure 1: Sensor deployment scheme in the IIoT.  
Figure 2: Diagram of sensor node perception model.  
\[\rho=\left\{\begin{array}{ll}0&if\;d_{ip}\geq R_{s}+R_{se}\\ e^{-\lambda a^{\beta}}&if\;R_{s}-R_{se}\leq d_{ip}\leq R_{s}+R_{se}\\ 1&if\;d_{ip}\leq R_{s}-R_{se}\end{array}\right. \tag{1}\]  
Here, \(R_{se}\) represents the error range of the sensor node perception radius, indicating the uncertainty in the sensor component perception performance. Parameters \(\alpha=d_{ip}-(R_{s}-R_{se})\), \(\lambda\), and \({}^{\beta}\) are adjustable parameters representing the attenuation degree of the sensor node perception between distances \(R_{s}-R_{se}\) and \(R_{s}+R_{se}\).  
The probabilistic perception model is a more complex sensing model in which the perception results of sensor nodes are probabilistic and may include errors or uncertainties. This model is suitable for applications that need to account for the uncertainty in perception results or require estimation of the target position and state, such as in localization and target tracking. It is often used for studying node deployment minimization and optimal deployment performance. The advantages of the probabilistic perception model provide probabilistic information on sensing results, reflecting the true state of perception and making it applicable to a broader range of scenarios, thereby enhancing system robustness. The disadvantages is the more complex computation.