## 3 System Optimization Model
### One-Dimension Normal Cloud Model
\tag{5}\]  
Since \(En^{\prime}\)\(\sim\)\(N(Ex,He^{2})\), it can be given that \(X\) should be a standard normal random variable with mathematical expectation \(EX=Ex+\sqrt{-\pi lny}En\). a standard deviation \(\sigma=\sqrt{-\pi lny}He\). Therefore, the expected curve of the normal cloud can be obtained as:  
\[h(x)=\exp\{-\frac{(x-Ex)^{2}}{\pi En^{2}}\} \tag{6}\]  
The degree of dispersion of cloud drops is proportional to \(He\), and is inversely proportional to \(y\). In other words, \(He\) is directly proportional to the dispersion of cloud drop, and conversely to the value of \(y\). As can be seen from Figure 2, the significant geometric characteristics of the normal cloud can be well reflected by the expected curve method. Besides, all cloud drops fluctuate randomly around the expected curve of normal cloud. According to Equation (6) and the expression of the desired curve, the expected curves of the two cloud models are:  
\[\left\{ \begin{array}{l}h_{1}(x)=\exp\{-\frac{(x-Ex)^{2}}{\pi En_{1} ^{2}}\}\\ \\ h_{2}(x)=\exp\{-\frac{(x-Ex)^{2}}{\pi En_{2}^{2}}\}\end{array} \right. \tag{7}\]  
Typically, under the premise that the two expected curves are definitive, the area of the overlapping region can be calculated by an integral method. As shown in Figure 3, if the horizontal coordinate of the intersection point of the two desired curve is \(x_{0}\), then:  
\[\Delta s=\int_{-\infty}^{x_{0}}h_{2}(x)dx+\int_{x_{0}}^{+\infty}h_{1}(x)dx \tag{8}\]  
where \(h_{1}(x)\) and \(h_{2}(x)\) are the expectation curve equation for the normal cloud models \(C_{1}\) and \(C_{2}\), respectively. This function is a non-integral function, so it can only be approximated by a numerical approximation method. However, the approximate solution method is quite time consuming, which is not feasible to the similarity comparison between plenty of cloud models. Figure 3: The similarity of two normal cloud models. Figure 2: The expected curves of the normal cloud mode. From the expected curve, it is intrinsic that the expression may be approximated by a probability density function of the normal distribution. Then, the expression can be transformed as follows:  
\[h(x)=\sqrt{2\pi}En\times f(x) \tag{9}\]  
where \(f(x)\) is a probability density function of normal distribution. Next, the relative properties of the normal distribution are used to solve the overlapping area of the two expected curves. Thus, we have:  
\[\Delta s=\sqrt{2\pi}\int_{-\infty}^{z_{0}}En_{2}f(x)dx+\sqrt{2\pi}\int_{x_{0}} ^{+\infty}En_{1}f(x)dx \tag{10}\]  
Suppose \(\theta_{1}=(x_{0}-Ex_{1})\) / \(En_{1}\) and \(\theta_{2}=(x_{0}-Ex_{2})\)/ \(En_{2}\), then Equation (10) can be expressed as:  
\[\Delta s=\sqrt{2\pi}En_{2}\int_{-\infty}^{\theta_{2}}\phi(x)dx+\sqrt{2\pi}En_ {1}\int_{-\infty}^{\theta_{1}}\phi(x)dx \tag{11}\]  
Compared to the conventional model, the cloud model involves the transformation of a qualitative notion and its quantitative instances, and can systematically consider the randomness and fuzziness of eutrophication assessment. The normal cloud model is the most rudimentary model for its adoption of normal distribution and membership function, which should be satisfied by the node characteristics of wireless sensor networks.