## 4 Example Application
### Reliability Analysis by Traditional Method  
Equation (4) is the traditional model to obtain the reserve capacity, its lower-level model has no capacity constraints, so at the state of low road capacity, the distribution flow may exceed the capacity of the road section, the reserve capacity is 1. Taking state of 7 as an example, in the case of no capacity constraint, if the results are not optimized by the Lagrange multiplier method, the results are shown in Table 8. At this time, the flow of the road Section 3 is greater than the capacity: \(x_{23}=8.2742>7.5\), so we get the \({\mu_{r}}^{n+1}<1\) by upper-level model, so the reserve capacity is \(\mu_{r}=\max({\mu_{r}}^{n+1}{\mu_{r}}^{n},{\mu_{r}}^{n})=1\). But in the actual situation, the distribution flow will not be higher than the link capacity, so this analysis result has some problems. The traditional upper-level model is simple, there is no limit to the acceptability of the link formation time, so the result will be different from that of the improved model, such as the state of 2, it is shown in Table 9. Then, through the approximation form of upper bound and lower bound function, the reliability of the example road network is 0.54, as shown in Table 10 and Figure 4. The reliability of the road network is far lower than that result of the improved model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Road Section** & **Capacity** & **Capacity Increased by 10\%** & **Reliability** & **Reliability Improvement Range** \\ \hline \(1\) & 25 & 27.5 & 0.6224 & 0.00\% \\ \hline \multirow{2}{*}{2} & 25 & 27.5 & \multirow{2}{*}{0.6224} & \multirow{2}{*}{0.00\%} \\  & 12.5 & 13.75 & & \\ \hline \multirow{2}{*}{3} & 15 & 16.5 & \multirow{2}{*}{0.6332} & \multirow{2}{*}{1.74\%} \\  & 7.5 & 8.25 & & \\ \hline \multirow{2}{*}{4} & 15 & 16.5 & \multirow{2}{*}{0.6344} & \multirow{2}{*}{1.93\%} \\  & 7.5 & 8.25 & & \\ \hline \multirow{2}{*}{5} & 15 & 16.5 & \multirow{2}{*}{0.6224} & \multirow{2}{*}{0.00\%} \\  & 7.5 & 8.25 & & \\ \hline \multirow{2}{*}{6} & 15 & 16.5 & \multirow{2}{*}{0.9944} & \multirow{2}{*}{59.77\%} \\  & 7.5 & 8.25 & & \\ \hline \multirow{2}{*}{7} & 15 & 16.5 & \multirow{2}{*}{0.6224} & \multirow{2}{*}{0.00\%} \\  & & & & \\ \end{tabular}
\end{table} 
Table 7: Analysis table of contribution degree of road network reliability improvement. Figure 3: Reliability upper and lower bounds analysis based on road network capacity. Because the results of traffic flow assignment of traditional model do not conform to the law of reality, and the travel time acceptability is not considered, so the improved model allocation results are more reliable. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**States**} & \multicolumn{2}{c}{\(P_{\text{s}}\)} & \multirow{2}{*}{**States**} & \multicolumn{2}{c}{\(P_{\text{s}}\)} \\ \cline{2-3} \cline{5-6}  & **Upper Bound** & **Lower Bound** & & **Upper Bound** & **Lower Bound** \\ \hline (25,25,15,15,15,15,15) & 1 & 0.3024 & (25,25,7.5,15,15,7.5,15) & 0.6056 & 0.54 \\ (25,25,15,15,15,7.5,15) & 0.7984 & 0.3024 & (25,12.5,15,7.5,15,7.5,15) & 0.584 & 0.54 \\ (25,25,15,7.5,15,15,15) & 0.7984 & 0.432 & (25,25,7.5,7.5,15,15,15) & 0.5696 & 0.54 \\ (25,25,15,7.5,15,7.5,15) & 0.712 & 0.432 & (25,25,7.5,7.5,15,7.5,15,7.5,15) & 0.56 & 0.54 \\ (25,12.5,15,15,15,15,15) & 0.712 & 0.5076 & (25,12.5,7.5,15,15,15,15) & 0.5516 & 0.54 \\ (25,12.5,15,15,15,7.5,15) & 0.6616 & 0.5076 & (25,12.5,7.5,15,15,7.5,15) & 0.546 & 0.54 \\ (25,25,7.5,15,15,15,15) & 0.628 & 0.5076 & (25,12.5,7.5,7.5,15,15,15) & 0.5424 & 0.54 \\ (25,12.5,15,7.5,15,15,15) & 0.628 & 0.54 & (25,12.5,7.5,7.5,15,7.5,15) & 0.54 & 0.54 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Reliability upper and lower bounds analysis based on road network capacity. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Road Section** & **Flow(pcu/h)** & **Road Section** & **Flow(pcu/h)** \\ \hline
1 & 16.6713 & 5 & 5.0440 \\
2 & 8.3287 & 6 & 6.6469 \\
3 & 8.2742 & 7 & 9.9560 \\
4 & 3.3531 & & \\ \hline \hline \end{tabular}
\end{table}