## 4 Example Application
### Sensitivity Matrix  
After getting a set of equilibrium solutions, the sensitivity matrix can quickly predict the new equilibrium solution after the OD link demand changes. According to the previous model and algorithm design, the reserve capacity can be easily obtained according to the sensitivity matrix.  
Taking the capacity state of the first section (25, 25, 15, 15, 15, 15, 15) as an example, the equilibrium solution can be obtained according to MSA, as shown in Table 3. According to the equilibrium solution of the road section, the Hessian matrix \(\
abla^{2}L\) can be obtained, and then the sensitivity matrix related to the input flow \(q\) can be obtained by calculating the sensitivity equation according to the \(M\) and \(N\) components.  
\[\
abla^{2}L=\left[\begin{array}{cccccc}0.02862&0&0&0&0&0\\ 0&1.21165&-1.20252&0&0&0&0\\ 0&-1.20252&1.20483&0&0&0&0\\ 0&0&0&3.98755&0&-2&0\\ 0&0&0&0&2.65285&0&-1.33333\\ 0&0&0&-2&0&1.10764&0\\ 0&0&0&0&-1.33333&0&0.71905\\ \end{array}\right]\]  
\[\left[\begin{array}{cccccc}\frac{\partial\tau_{12}}{\partial\tau_{14}},\frac {\partial\tau_{13}}{\partial\tau_{14}},\frac{\partial\tau_{21}}{\partial\tau_{ 14}},\frac{\partial\tau_{34}}{\partial\tau_{14}},\frac{\partial\tau_{25}}{ \partial\tau_{14}},\frac{\partial\tau_{34}}{\partial\tau_{14}},\frac{\partial \tau_{35}}{\partial\tau_{14}},\frac{\partial\tau_{34}}{\partial\tau_{14}}, \frac{\partial\tau_{35}}{\partial\tau_{14}}\right]=\left[0.6630,0.3370,0.3300,0.3343,-0.0013,0.6657,0.0013\right]\]  
\[\left[\frac{\partial\tau_{12}}{\partial\tau_{15}},\frac{\partial\tau_{13}}{ \partial\tau_{15}},\frac{\partial\tau_{21}}{\partial\tau_{15}},\frac{\partial \tau_{21}}{\partial\tau_{15}},\frac{\partial\tau_{24}}{\partial\tau_{15}}, \frac{\partial\tau_{25}}{\partial\tau_{15}},\frac{\partial\tau_{34}}{\partial \tau_{15}},\frac{\partial\tau_{35}}{\partial\tau_{15}}\right]=\left[0.6654,0.3 346,0.3276,-0.0008,0.3386,0.0008,0.6614\right]\]  
At state 7, the equilibrium solution is shown in Table 4, and the road Section 3 reaches the capacity limit, the sensitivity matrix between the link flow and the OD demand can be obtained.  
It can be found that according to the sensitivity matrix, when OD demand increases, the traffic flow of Section 3 will not increase. In the analysis of reserve capacity, it can be considered that the increase of traffic demand will not bring about the increase of saturation of this section, and this section can be eliminated in the analysis.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**States** & \(P_{S}\) & **States** & \(P_{\mathbf{S}}\) \\ \hline (25,25,15,15,15,15,15) & 0.3024 & (25,25,7,5,15,15,7,5,15) & 0.0224 \\ (25,25,15,15,15,15,7,5,15) & 0.2016 & (25,12,5,15,7,5,15,7,5,15) & 0.0216 \\ (25,25,15,7,5,15,15,15) & 0.1296 & (25,25,7,5,7,5,15,15,15) & 0.0144 \\ (25,25,15,7,5,15,7,5,15) & 0.0864 & (25,25,7,5,7,5,15,7,5,15) & 0.0096 \\ (25,12,5,15,15,15,15,15) & 0.0756 & (25,12,5,7,5,15,15,15,15) & 0.0084 \\ (25,12,5,15,15,15,15,7,5,15) & 0.0504 & (25,12,5,7,5,15,15,15,7,5,15) & 0.0056 \\ (25,25,7,5,15,15,15,15) & 0.0336 & (25,12,5,7,5,7,5,15,15,15) & 0.0036 \\ (25,12,5,15,7,5,15,15,15) & 0.0324 & (25,12,5,7,5,7,5,15,7,5,15) & 0.0024 \\ \hline \hline \end{tabular}
\end{table}
Table 2: State probability of different road network capacity.