# A Macroscopic Traffic Model based on Driver Reaction and Traffic Stimuli
## 5 Performance Results  
The performance of the proposed and PW models is evaluated in this section. The boundary conditions employed are periodic, which denote a circular road. These boundary conditions were implemented in the simulations such that the density and flow at \(x=300\) m move to \(x=0\) m in the next time step. The simulation parameters are given in Table 2. The stability of the models can be guaranteed by employing the Courant, Friedrich and Lewy (CFL) stability conditions [51]. The road and time steps for the proposed model were then 1 m and 0.1 s and for the PW model were 5 m and 0.1 s. The total simulation time in both cases was 60 s and the maximum velocity was \(v_{m}=10\) m/s. The maximum normalized density was 1, which means that the road was 100% occupied. Typical values of the relaxation time range from \(\tau=0.5\) s to \(\tau=3\) s. The relaxation time considered was 2.5 s and the headway was 20 m [52; 53]. The initial density \(\rho_{0}\) at time \(t=0\) for free flow traffic was  
\[\rho_{0}= \begin{cases}0.01,&\text{for }x<100\\ 0.2,&\text{for }x\geq 100,\end{cases}  \tag{73}\]  
whereas, for congestion, the initial density was  
\[\rho_{0}= \begin{cases}0.15,&\text{for }x\leq 130\\ 0.8,&\text{for }130<x<180\\ 0.10,&\text{for }x\geq 180.\end{cases}  \tag{74}\]  
The density between 130 m and 180 m was \(0.8\rho_{m}\), which was well above the critical density \(0.5\rho_{m}\). The speed constant for the PW model varies between 2.4 m/s to 57 m/s in the literature, thus here \(C_{0}=10\) m/s was used. The density with the proposed model over the 300 m road at 1 s, 20 s, 40 s and 60 s is shown in Figure 1 and given in Table 3. Comparing the results from 1 s to 60 s, the density becomes smoother over time. At 1 s, the density is 0.19 at 1 m, and from 10 m to 104 m it is 0.01. It increases to 0.2 at 111 m and stays at this level to 300 m. At 20 s, the density is 0.20 at 1 m, and decreases to 0.19 at 107 m and 0.008 at 197 m. Between 197 m and 200 m, it increases from 0.008 to 0.01 and then it is 0.20 at 267 m. At 40 s, the density is 0.12 at 1 m, decreases to 0.01 at 93 m, and is 0.19 at 227 m and 0.12 at 300 m. At 60 s, the density is 0.20 at 1 m and decreases to 0.02 at 268 m. From 268 m and 283 m, the density varies between 0.02 and 0.19 and is 0.20 at 300 m. \begin{table}
\begin{tabular}{c c} \hline \hline
**Description** & **Value** \\ \hline Simulation time for both models with different initial densities (Equations (73) and (74)) & 60 s \\ Length of the circular road & 300 m \\ Maximum velocity & 10 m/s \\ Time step for both models & 0.1 s \\ Road step for the proposed model & 1 m \\ Road step for the PW model & 5 m \\ Relaxation time & \(\tau=2.5\) s \\ Equilibrium velocity distribution, \(v_{e}(\rho)\) & Greenshields \\ Maximum normalized density & \(\rho_{m}=1\) \\ Speed constant, \(C_{0}\) & 10 m/s \\ Resolution test time step for the proposed model & 0.01 s \\ Resolution test road step for the proposed model & 2 m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. The velocity with the proposed model over the 300 m road at 1 s, 20 s, 40 s and 60 s is shown in Figure 2 and given in Table 3. At 1 s the velocity is 8.0 m/s at 1 m and increases to 9.9 m/s at 10 m. The velocity is a constant 8.0 m/s between 111 m and 300 m. At 20 s, the velocity is 8.0 m/s at 1 m and increases to 9.9 m/s at 200 m. Between 267 m and 300 m, it is a constant 8.0 m/s. At 40 s, the velocity increases from 8.7 m/s at 1 m to 9.9 m/s at 96 m. The velocity is 9.9 m/s at 109 m, decreases to 8.0 m/s at 227 m and then increases to 8.7 m/s at 300 m. At 60 s, the velocity is 8.0 m/s at 1 m and smoothly increases to 9.7 m/s at 268 m. The velocity is 8.0 m/s between 282 m and 300 m. The density and velocity behavior of the proposed model is realistic and becomes smooth over time. When there is a change in density, the velocity is as expected.