# 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  
(1) Modeling of the CORN  
In this paper, network efficiency is introduced to measure the CORN of an air traffic network. The network efficiency is the average value of the connectivity efficiency between all pairs of airports, which is expressed as Equation (1):  
\[E=\frac{1}{N(N-1)}\sum_{i\
eq j\in G}e_{ij} \tag{1}\]  
where \(N\) represents the total number of airports. The network is completely connected when \(E=1\), and the connectivity of the network is the worst when \(E=0\). Therefore, the value of \(E\) is \(0\leq E\leq 1\). \(\varepsilon_{ij}\) is the connectivity efficiency between airports \(i\) and \(j\). It can be calculated by the shortest path \(d_{ij}\) between \(i\) and \(j\). \(d_{ij}\) can be calculated by the Floyd algorithm [39]. The algorithm starts from any one-sided path. The distance between any two points is the weight of the edge. When \(i\) and \(j\) are disconnected, \(d_{ij}\) tends toward infinity and \(\varepsilon_{ij}=0\). \[\varepsilon_{ij}=\frac{1}{d_{ij}} \tag{2}\]  
Though the network efficiency represents the network connectivity, it does not include the impact of connectivity differences on the CORN evolution. To characterize the influences of connectivity difference on the CORN in the future, we introduce a correction coefficient \(\delta_{con}\):  
\[\delta_{con}=1-\frac{\sigma_{con}}{\mu_{con}} \tag{3}\]  
where \(\sigma_{con}\) and \(\mu_{con}\) represent the standard deviation and the mean value of \(\varepsilon_{ij}\), respectively. \(\sigma_{con}/\mu_{con}\) denotes the degree of discreteness of connectivity reliability. When the connectivity difference is very large, there are some routes with much higher connectivity efficiency than other routes, and airlines will like to put more flights on these routes. Air traffic congestion may easily occur on the routes, and the network reliability will be reduced quickly. Thus, the CORN of the air traffic network can be calculated as follows:  
\[R_{con}=\delta_{con}E=(1-\frac{\sigma_{con}}{\mu_{con}})\times\mu_{con}=\mu_{ con}-\sigma_{con}=E-\sigma_{con} \tag{4}\]  
(2) Modeling of the TTRN  
In air traffic operation, the flight time \(T\) of an individual route always changes in the external environment. The less the flight delay, the higher the travel-time reliability. A large number of historical data show that flight time obeys a normal distribution, as shown in Figure 2.  
Figure 2: Morphology map of the travel-time distribution. \(t_{0}\) represents the expected flight time. The average flight time of curve 1 is less than \(t_{0}\), indicating that the flight time reliability is higher than the expected level. In contrast, the flight time reliability of curve 3 is lower than the expected level. The travel-time reliability \(V\) of an air route can be measured by the flight delay, as follows:  
\[V=\frac{1}{n}\sum_{i\in N}c_{i},c_{i}=\left\{ \begin{array}{cc}1&(T_{i}\leq t _{i})\\ 1-k\frac{T_{i}-t_{i}}{t_{i}}&(t_{i}<T_{i}\leq\frac{1+k}{k}t_{i})\\ 0&(T_{i}>\frac{1+k}{k}t_{i})\end{array} \right. \tag{5}\]  
where \(n\) represents the flight volume, \(c_{i}\) is the non-delay rate of flight \(i\), \(T_{i}\) is the average actual flight time of flight \(i(i\in[0,N])\), \(t_{i}\) is the estimated flight time of flight \(i(i\in[0,N])\), and \(k\) is the correction coefficient, which is used to ensure that the reliability is in \([0,1]\). When flight \(i\) is not delayed, \(c_{i}\) equals \(1\). Generally, when the actual flight delay is greater than half of the estimated flight time, the actual flight delay is not acceptable to potential passengers. Thus, the value of \(k\) in this paper is 2; that is, when \(T_{i}>\frac{k+1}{k}t_{i}=\frac{3}{2}t_{i}\), the value of \(c_{i}\) is 0.