# 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
To characterize the influence of the travel-time reliability difference on the TTRN in the future, a correction coefficient \(\delta_{tra}\) is introduced:  
\[\delta_{tra}=1-\frac{\sigma_{tra}}{\mu_{tra}} \tag{6}\]  
where \(\sigma_{tra}\) and \(\mu_{tra}\) represent the standard deviation and the mean value of \(V\) for all of the routes. \(\sigma_{tra}/\mu_{tra}\) denotes the degree of discreteness of non-delay rate. When there are large gaps between the non-delay rate of different air routes, airlines would like to put more flights on the routes with a higher non-delay rate. Air traffic congestion may easily occur on these routes, and the overall reliability of the network will be reduced. Thus, the TTRN of the air traffic network can be calculated as follows:  
\[R_{tra}=\delta_{tra}\frac{1}{l}\sum_{l\in L}V_{l}=(1-\frac{\sigma_{tra}}{\mu_{ tra}})\times\mu_{tra}=\mu_{tra}-\sigma_{tra} \tag{7}\]  
where \(L\) represents the number of routes. (3) Modeling of the CARN  
The CARN can be described by the probability that the air traffic capacity is greater than demand at a certain service level. We quantify the unreliability of any airport by the following formula:  
\[f=\frac{\sum\limits_{i\in N}A_{i}}{\sum\limits_{i\in N}c_{i}},\ A_{i}=\left\{ \begin{array}{cc}0&(q_{i}\leq c_{i})\\ q_{i}-c_{i}&(q_{i}>c_{i})\end{array} \right. \tag{8}\]  
where \(f\) represents the capacity unreliability of one airport; \(A_{i}\) is the residual value of demand minus capacity at the \(i\)th time interval; and \(q_{i}\) and \(c_{i}\) are the airport demand and capacity at the \(i\)th time interval, respectively. There is no consideration of service level in the calculation of \(f\). The airport service level can be measured by the delay of departing flights. Departing flights with delays of more than 15 minutes are considered delayed flights. The capacity reliability of one airport at a certain service level can be calculated as follows:  
\[P=(1-f)(1-\frac{\sum\limits_{i\in N}\frac{d_{i}}{q_{i}}}{N}) \tag{9}\]  
where \(d_{i}\) and \(q_{i}\) are the number of delayed flights and air traffic demand at the \(i\)th time interval, respectively. To characterize the influence of the airport capacity reliability difference on the CARN in the future, a correction coefficient \(\delta_{cap}\) is introduced:  
\[\delta_{cap}=1-\frac{\sigma_{cap}}{\mu_{cap}} \tag{10}\]  
where \(\sigma_{cap}\) and \(\mu_{cap}\) represent the standard deviation and the mean value of \(P\), respectively. \(\sigma_{cap}/\mu_{cap}\) denotes the dispersion degree of \(P\). When there are large gaps between the capacity reliability of different airports, more flights will be distributed to the airports with higher capacity reliability. Air traffic congestion may occur in those airports and the CARN of the air traffic network will be reduced. Thus, the network CARN for a certain service level can be calculated as follows:  
\[R_{cap}=\delta_{cap}\frac{1}{m}\sum_{j\in M}P_{j}=(1-\frac{\sigma_{cap}}{\mu_ {cap}})\times\mu_{cap}=\mu_{cap}-\sigma_{cap} \tag{11}\]  
where \(m\) is the number of airports and \(P_{j}\) is the capacity reliability of airport \(j\). Comprehensive Evaluation of Air Traffic Network Reliability Based on the Uncertainty Transformation Model  
Transform the quantitative data of CORN, TTRN, and CARN into the qualitative language of network reliability is an important problem. The uncertainty transformation model is an effective tool in uncertainty transformation between qualitative concepts and quantitative expression. It reflects both uncertainty and fuzziness of human thinking.