## 3 System Optimization Model
### Redundancy Degree of Nodes  
In most ESN applications, the coverage rate is a key to demonstrate the capacity of monitoring the environment [49], and it also serves as a QoS standard for evaluating the effectiveness of data collection. Actually, small sensing holes do not affect the performance of the entire network. Therefore, partial coverage redundancy is used to measure the redundancy of nodes, which is more suitable for ESNs [50]. To judge whether the node may enter into the dormant state, it is necessary to determine the redundancy degree of the node. In some algorithms, we must know the geographic information of the node and the relative position of its neighbor nodes to calculate the degree of redundancy. Unlike these deterministic methods, the mathematical model was based on the theory of probability [51], and the expected value of a node's redundancy can be calculated in accordance with the number of valid neighbors and the degree of its neighbors' coverage. We consider a wireless sensor network of \(n\) arbitrarily distributed sensor nodes, whose location information is unknown. All nodes will no longer move after deployment. In addition, the node's perception model adopts the Boolean model based on circular region, where the coverage area of each node can be confined to a circle about its position with a sensing range \(r\). The sensing range of all nodes can be adjusted adaptively, and the sensor-to-sensor distance is obtained by using the signal strength indicator or time of arrival methods. Let \(R_{i}\) and \(C_{i}\) be the sensing range and communication range of node \(N_{i}\), respectively, and the relationship between them is defined as \(C_{i}=\lambda R\), \(1\leq\lambda\leq 2\). **Definition 2:**_The sufficient condition for the communication between the nodes is that the distance is less than or equal to the minimum communication radius, i.e., \(d(i,j)\leq\text{min}\{C_{i},C_{j}\}\)._  
**Definition 3:**_Only when the distance between the nodes \(N_{i}\) and \(N_{j}\) satisfy \(d(i,j)\leq R_{i}+R_{j}\), those nodes are neighbors and can communicate with each other directly._  
**Definition 4:**_If each point in a region of a node is covered by at least one of its neighbors, then this area can be regarded as the area of redundant coverage. The redundancy of a node is defined as the ratio of the redundant coverage area by its neighbors to the sensing area of the node._  
**Corollary 1:**_For any node \(i\), there exists the neighbor set with the sensing range \(R_{i}\). Assuming an arbitrary point \(x\) in the sensing area, if the probability of being covered by a neighbor node is denoted by Pr(x), there must be Pr(x) of sensing area covered by neighbor nodes of \(i\)._  
**Proof:** Assuming there are \(K\) perception points \(\{x_{1},x_{2},\,\ldots,x_{K}\}\) in the sensor field of radius \(R_{i}\) for node \(N_{i}\), the perception points' distributions are uniformed and the events that the points being covered by neighbor nodes is independent. Considering the probability that a sensing point is covered by its neighbor node is described by \(Pr(x)\), then, the possibility that there are \(k\) out of \(K\) nodes being covered by neighbor nodes based on the probabilistic model is given by:  
\[P\{X=k\}=C_{K}^{k}Pr(x)^{k}(1-Pr(x))^{K-k} \tag{16}\]  
\(\square\)  
From Equation (16), it should be pointed out that the event is subject to binomial distribution, namely \(X\sim\)\(B(Pr(x)K)\). By solving for the mathematical expression of \(X\), we have:  
\[E[X]=\sum_{k=0}^{K}nC_{K}^{k}Pr(x)^{k}(1-Pr(x))^{K-k}=KPr(x) \tag{17}\]  
So, in the sensing range of \(R_{i}\), there are about \(KPr(x)\) perception points covered by neighbor nodes. When \(K\) tends to infinity, the sensing area with the radius \(\pi{R_{i}}^{2}\) can be seen as an infinite number of sensing points.