## 5 Performance Evaluation
### Example 1  
The time for both system simulations is 100 s. The maximum velocity is 0.964 m/s [54]; \(d_{t}\) is 8 m, and \(d_{s}=2\) m [65; 66; 67]. The velocity constant for the PW model is 25 m/s. The spatial road step for the proposed and PW systems is 5 m. For the proposed model, as per given condition (20), the maximum temporal time step is 4.99 s. Whereas per condition (22), the maximum temporal step for the PW model is 0.19 s. To ensure this CFL stability condition, the temporal step is chosen as 0.01 s, which is less than 4.99 s and 0.19 s, respectively, for the proposed and PW models [59]. That is, the speed of iteration is faster than the traffic speed in a temporal step. The traversed time is 18.08 s, which gives a driver sensitivity of 0.0031 s\({}^{-2}\) as shown in Table 2.  
The initial density over 2000 m at 0 s for the performance results in example 1 is  
\[\rho_{0}= \begin{cases}0.01&\text{for }x\leq 600\\ 0.3&\text{for }x\leq 1000\\ 1&\text{for }x\leq 1500\\ 0.5&\text{for }x\leq 2000.\end{cases}  \tag{24}\]  
The changes in density distribution \(\rho_{0}\) represent the traffic transition. The initial transitions are at 600 m, 1000 m, and 1500 m. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Time of simulation & 100 s \\ Length of the circular road & 2000 m \\ Maximum velocity & 0.964 m/s \\ Minimum velocity & 0 m/s \\ Velocity constant & \(A=25\) m/s \\ Equilibrium velocity & \(v(\rho)=\) Greenshields \\ Maximum normalized density & \(\rho_{m}=1\) \\ Minimum normalized density & \(\rho_{m}=0\) \\ Temporal step & 0.1 s \\ Spatial road step & 5 m \\ Relaxation time & \(\tau=2\) s \\ Safe distance headway & \(d_{s}=2\) m \\ Transition distance & \(d_{t}=8\) m \\ Traversed time & \(t=18.08\) s \\ Driver sensitivity & \(\eta=\frac{1}{2}=0.0031\) s\({}^{-2}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Traffic parameters. The proposed traffic system velocity over 2000 m at 1 s, 50 s, and 100 s is depicted in Figure 2. At 1 s, 0.57 m/s is at 0 m, which at 40 m grows to 0.968 m/s and continues up to 470 m. It drops to 0.7 m/s from 500 m/s, which continues up to 1000 m. A velocity of 0 m/s is at 1030 m, which continues up to 1500 m. At 50 s, it is 0.68 m/s, which, at 230 m, grows to 0.96 m/s. The speed at 700 m drops to 0.7 m/s and at 1080 m drops to 0 m/s. At 100 s, it is 0.65 m/s, which, at 270 m, grows to 0.96 and at 680 m drops to 0.7 m/s. The speed at 1200 m further drops to 0.014 m/s. It grows at 855 m to 0.51 m/s, which at 1994 m grows to 0.64. The proposed system traffic velocity behavior over 2000 m for 100 s is shown in Figure 3. The velocity behavior shows that the velocity remains within the bounds both spatially and temporally. Speed (velocity) behavior is correspondingly based on the density behavior, as shown in Figures 4 and 5. As the density grows, the velocity drops. Conversely, as the density decays, the velocity increases. That is, the proposed system behaves according to the behavior of the desired equilibrium velocity, which is realistic and adequate. Figure 3: Velocity with the proposed system on a 2000 m circular road with a driver sensitivity of 0.0031 s\({}^{-2}\). Figure 2: Velocity with the proposed system on a 2000 m circular road with a driver sensitivity of 0.0031 s\({}^{-2}\).