# Reliability Analysis of an Air Traffic Network: From Network Structure to Transport Function
## 3 Reliability Measurement of the Air Traffic Network
### Measurement of the CORN, TTRN, and CARN
#### 3.3.1 Description of Uncertainty Transformation Model  
a. Concept of an uncertainty transformation model  
The uncertainty transformation model can realize the transformation between qualitative and quantitative values [40]. Suppose \(U\) is a quantitative numerical universe of discourse, and \(T\) is a linguistic term associated with \(U\). If \(x\in U\) is a random implementation of concept \(T\) and \(\mu(x)\in[0,1]\) stands for the certainty degree of \(x\) belonging to \(T\), then the distribution of \(x\) in the universe of discourse \(U\) is called a fuzzy set, and each \(x\) is called a fuzzy set drop. \[\mu:U\rightarrow[0,1]\ \ \forall x\in U\ \ x\rightarrow\mu(x)\]  
According to the above descriptions, the random implementation is in terms of probability. The certainty degree is the degree of membership in a fuzzy set according to the probability distribution. Therefore, the uncertainty transformation model shows the correlation of randomness and fuzziness. b. Characteristics of the uncertainty transformation model  
Expectation \(E_{x}\), entropy \(E_{n}\), and super-entropy \(H_{e}\) are the three digital characteristics describing fuzzy sets, as shown in Figure 3. \(Cr(x)\) is the membership degree. The three characteristics express the quantity feature of a qualitative concept. \(E_{x}\) is the center value of the concept in the theory field, and it is the most representative of the qualitative concept. \(E_{n}\) is the uncertainty measurement of the qualitative concept, which is determined by both the randomness and fuzziness of the concept. \(H_{e}\) is the uncertainty measurement of \(E_{n}\), which is determined by both the randomness and fuzziness of \(E_{n}\). Therefore, the fuzziness and randomness are completely combined by the three characteristics of the uncertainty transformation models, which constitute the mapping between quality and quantity as the foundation of knowledge expression. c. Algorithms in uncertainty transformation model  
The normal and converse generators are the most important algorithms in the uncertainty transformation model. The normal generator is used to convert qualitative language to a quantitative value, while the converse generator does the opposite [41]. Next, we will introduce the two algorithms. (1) The normal generator  
The fuzzy set can be generated by the set's digital characters \((E_{x},E_{n},H_{e})\), and its mathematic expectation curve (MEC) is:  
\[MEC_{A}(x)=\exp\left[-(x-E_{x})^{2}/\left(2E_{n}\right)^{2}\right] \tag{12}\]  
The generating algorithm of the fuzzy set is as follows:  
Input: digital character \((E_{x},E_{n},H_{e})\), the number of samples \(n\). Output: \(drop(x_{i},\mu_{i}),i=1,2,\cdots,n\). (1) Generate a normal random number \(E^{\prime}_{ni}=NORM(E_{n},H_{e}^{2})\) with expected value \(E_{n}\) and variance \(H_{e}^{2}\). (2) Generate a normal random number \(E^{\prime}_{ni}=NORM(E_{x},E^{\prime 2}_{ni})\) with expected value \(E_{x}\) and variance \(E^{\prime 2}_{ni}\). (3) Compute \(\mu_{i}=e^{-\frac{(x_{i}-E_{x})^{2}}{2E^{\prime 2}_{ni}}}\), where \((x_{i},\mu_{i})\) is a sample. (4) Repeat steps (1) to (3) until enough sample drops are generated. (2) The converse generator  
Input: sample points \(x_{i},i=1,2,\cdots,n\)  
Output: set characteristics \(E_{x}\), \(E_{n}\), \(H_{e}\)  
(1) Calculate the expectation of the samples, \(\overline{x}=\frac{1}{N}\sum\limits_{i=1}^{N}x_{i}\), center distance of the samples, \(B=\frac{1}{N}\sum\limits_{i=1}^{N}\left|x_{i}-\overline{x}\right|\), and sample variance, \(S^{2}=\frac{1}{N-1}\sum\limits_{i=1}^{N}\left(x_{i}-\overline{x}\right)^{2}\). (2) Let \(E_{\widetilde{x}}=\overline{x}\)  
(3) Let \(E_{\widetilde{n}}=\left(\frac{\pi}{2}\right)^{\frac{1}{2}}\times B\)  
(4) Let \(\widetilde{He}=\left(S^{2}-E_{\widetilde{n}}^{2}\right)^{\frac{1}{2}}\). Suppose there are two sets \(C1(E_{x1},E_{n1},H_{e1})\) and \(C2(E_{x2},E_{n2},H_{e2})\); the rules used by the algorithm for these sets is shown in Table 1.  
Figure 3: Three digital characteristics of a fuzzy set.