## 3 System Optimization Model
### Redundancy Degree of Nodes
Therefore, the probability of the perception points being covered by the neighbor nodes can be regarded as the ratio of the total covered area, i.e.,  
\[\lim_{K\rightarrow\infty}\frac{KPr(x)}{K}=Pr(x) \tag{18}\]  
Let \(Sx=\{X_{1},X_{2},\ldots,X_{i},\ldots\}\) be the set of perceptual points in the sensing field of node \(N_{i}\) and \(x\) be the distance from the point to the node. In terms of the Corollary 1, if the sensing set can be covered by its neighbor nodes, the node \(N_{i}\) can be covered by its neighbor nodes completely. Assuming that \(Y\) is a random variable used to represent the Euclidean distance between the node and a perceptual point \(x\) within its range, the probability density function of \(x\) can be calculated as:  
\[f_{Y}(x)=2x/{\lambda_{i}}^{2}R_{i}^{2} \tag{19}\]As shown in Figure 4, the sensing area \(\rho(N_{i},R_{i})\) is the range of the node's perception. For the point \(X\) within the range of node \(N_{i}\), the probability that the point is covered by the neighbor node \(N_{j}\) is denoted \(Pr_{i,j}(x)\). If \(X\in\rho(N_{i},R_{i})\) and the point is covered by another neighbor's node \(N_{j}\), we can determine that the distance from point \(X\) to node \(N_{j}\) is less than the sensing range of \(N_{j}\), meanwhile, these two nodes are able to communicate with each other. Thus, \(Pr_{i,j}(x)\) is equal to the ratio of the shaded part within the sensing range \(R_{j}\) to the communication range of node \(N_{j}\). The probability \(Pr_{i,j}(x)\) of perception point \(X\) within the sensing range of \(N_{i}\) being covered by the neighbor node \(N_{j}\) can be expressed as follows:  
\[ \begin{split}& Pr_{i,j}(x)=\frac{\rho(X,R_{j})\cap\rho(N_{i},C_{i}) \cap\rho(N_{j},C_{j})}{\rho(N_{i},C_{i})\cap\rho(N_{j},C_{j})}\times\frac{f_{Y} (x,N_{i})}{f_{Y}(x,N_{j})}\\ &=\frac{\rho(X,R_{j})\cap\rho(i,\min\left\{C_{i},C_{j}\right\})} {\rho(N_{i},\min\left\{C_{i},C_{j}\right\})}\times\frac{\lambda_{j}^{2}R_{j}^{ 2}}{\lambda_{i}^{2}R_{i}^{2}}\end{split}  \tag{20}\]  
According to the previous definition, the sensing range of a sensor node is less than the communication range. Then, we can discuss the problem in two scenarios as shown in Figure 5a,b, respectively. If \(0<x<\min\{C_{i},C_{j}\}-R_{j}\), the maximum distance between point \(X\) and \(N_{j}\) is about \(\min\{C_{i},C_{j}\}\). But if \(\min\{C_{i},C_{j}\}-R_{j}\leq x\leq R_{j}\), the distance between them can be obtained as \([\min\{C_{i},C_{j}\}-R_{j},R_{i}]\). Therefore, \(Pr_{i,j}(x)\) can be expressed as:  
Figure 4: The point within the sensing range of \(N_{i}\) being covered by its neighbor. \[Pr_{t_{ij}}(x)=\left\{ \begin{array}{c}\dfrac{\rho(X,R_{j})}{\rho(i,\min\{C_{i},C _{j}\})}\times\dfrac{\lambda_{j}^{2}R_{j}^{2}}{\lambda_{i}^{2}R_{i}^{2}},0<x< \min\{C_{i},C_{j}\}-R_{j}\\ \dfrac{\rho(X,R_{j})\cap\rho(s_{i},\min\{C_{i},C_{j}\})}{\rho(i,\min\{C_{i},C_{ j}\})}\times\dfrac{\lambda_{j}^{2}R_{j}^{2}}{\lambda_{i}^{2}R_{i}^{2}},\min\{C_{i},C_{ j}\}-R_{j}\leq x\leq R_{j}\end{array} \right. \tag{21}\]  
Suppose the point \(X\) can be covered at least one node with the same category of \(N_{j}\), then the probability is:  
\[\widetilde{\Pr}_{t_{ij}}(x)=1-(1-\Pr_{t_{ij}}(x))^{n_{j}(i)} \tag{22}\]  
where \(n_{j}(i)\) is the number of sensor nodes with the same category of \(N_{j}\). Let \(\psi(X)\) be the random event that point \(X\) within the scope of \(N_{i}\) being covered by at least one sensor node of a different category.