## 5 Performance Evaluation
### Example 2  
The proposed system is analyzed with traversed time during transition as 1 s and 20 s, which, respectively, gives the driver sensitivity as 1 s\({}^{-2}\) and 0.0025 s\({}^{-2}\) as shown in Table 3. The temporal step is 0.2 s, with the spatial road step chosen as 14 m to satisfy the CFL conditions [59]. The value of \(d_{t}\) is 18 m. The maximum and minimum velocity, respectively, are 25 m/s and 0 m/s [17]. The total simulation time for example 2 is 200 s. The initial density distribution \(\rho_{0}\) over 2000 m at time \(t=0\) for the performance results in example 2 is  
\[\rho_{0}= \begin{cases}0.15&\text{for }x\leq 300\\ 0.8&\text{for }x\leq 600\\ 0.3&\text{for }x\leq 1000\\ 0.8&\text{for }x\leq 1500\\ 0.2&\text{for }x\leq 2000\end{cases}  \tag{25}\]  
The changes in density distribution (25) presents the traffic transition. This distribution has transitions at 300 m, 600 m, 1000 m, and 1500 m.  
Figure 10 depicts the density of the proposed system on a 2000 m circular road with 0.0025 s\({}^{-2}\) and 1 s\({}^{-2}\) driver sensitivity at 0.2 s, 12 s, 100 s, and 200 s. With 0.0025 s\({}^{-2}\), at 0.2 s, the density is 0.15 from 0 m to 300 m, whereas from 301 m to 600 m, it is 0.8. From 601 m to 1000 m, it is 0.3, whereas the density from 1001 m to 1500 m is 0.8. At 12 s, the density bounds between 0.17 and 0.92. At 0 m, it is 0.2, and it is 0.17 at 285 m. The density grows to 0.89 at 390 m, and it drops to 0.29 at 800 m. This grows to 0.92 at 1040 m, which then drops to 0.8 at 1200 m and stays contant up to 1300 m. The density drops to 0.18 at 1800 m. At 100 s, the density bounds between 0.29 and 0.81. At 0 m, it is 0.4, and it is 0.29 at 635 m. The density grows to 0.81 at 785 m and drops to 0.5 at 1000 m. This grows to 0.6 at 1190 m and then drops to 0.5 at 1500 m. At 200 s, it is between 0.47 and 0.6. At 0 m, it is 0.45, and it is 0.4 at 900 m. The density grows to 0.56 at 1050 m and then grows to 0.6 at 1280 m, but then drops to 0.47 at 1500 m. With 1 s\({}^{-2}\), at 0.2 s, it is shown in Figure 10 that the density behavior with the proposed system is the same as with the driver sensitivity 0.0025 s\({}^{-2}\). At 12 s, the density is between 0.19 and 0.77. At 0 m it is 0.2, and it is 0.19 at 150 m. The density grows to 0.6 at 410 m, whereas it drops to 0.37 at 800 m. This grows to 0.77 at 1200 m and then drops to 0.2 at 2000 m. At 100 s, the density bounds between 0.38 and 0.51. At 0 m, it is 0.42, and it is 0.38 at 500 m. The density grows to 0.51 at 1000 m, whereas it drops to 0.4 at 1500 m. This further drops to 0.4 at 2000 m. At 200 s, the density bounds between \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Simulation time & 200 s \\ Length of the circular road & 2000 m \\ Maximum velocity & 25 m/s \\ Minimum velocity & 0 m/s \\ Equilibrium velocity & \(v(\rho)=\) Greenshields \\ Maximum normalized density & \(\rho_{m}=1\) \\ Minimum normalized density & \(\rho_{m}=0\) \\ Temporal step & 0.2 s \\ Spatial road step & 14 m \\ Relaxation time & \(\tau=2\) s \\ Safe distance headway & \(d_{s}=2\) m \\ Transition distance & \(d_{t}=18\) m \\ Traversed time & \(t=20\) s \\ Driver sensitivity \(\eta\) & \(\frac{1}{t^{2}}=0.0025\) s\({}^{-2}\) \\ Traversed time & \(t=1\) s \\ Driver sensitivity \(\eta\) & \(\frac{1}{t^{2}}=1\) s\({}^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Traffic parameters. 0.43 and 0.47. At 0 m, it is 0.45, and it is 0.43 at 600 m. The density grows to 0.47 at 1500 m, whereas it drops to 0.46 at 2000 m. Figure 10 demonstrates that the variation in density with 0.0025 s\({}^{-2}\) is larger than with 1 s\({}^{-2}\) and shows appropriate evolution. For 1 s\({}^{-2}\), at 0.2 s, Figure 11 shows that the velocity behavior with the proposed system is the same as with the driver sensitivity of 0.0025 s\({}^{-2}\). At 12 s, it is bounded from 5.7 m/s to 22.3 m/s.