# A Microscopic Traffic Model Considering Driver Reaction and Sensitivity
## 3 Stability Analysis
### Numerical Stability  
Both models are evaluated over a 1000 m single-lane circular road for 180 s with a platoon of 21 vehicles. The time headway with the ID model is 2 s. For the proposed model, \(\tau=1\) s, which corresponds to an aggressive driver, \(\tau=2.5\) s, which corresponds to a sluggish driver, and \(\tau=2\) s, which corresponds to a typical driver, are considered. The value of \(h=\frac{s}{D}\) ranges between 0 and 1, so here 0.3, 0.5, and 1 are used. The value of \(\delta\) varies between 1 and \(\infty\), and is usually 4 [17]. Hence, the values of \(\delta\) considered are 1, 4, and 20. The initial equilibrium velocity is 4 m/s and the number of vehicles is 21. A perturbation is induced at 30 s based on the minimum acceleration. The stability analysis parameters for the ID and proposed models are given in Table 2.  
Figure 2 presents the velocity trajectories for the ID and proposed models. The trajectory of the 1 st vehicle is represented by the red line, while the black lines indicate the trajectories of the following 20 vehicles. For the ID model, when \(\delta=1\), the velocity of the 1 st vehicle decreases to 3.1 m/s and then increases to 4.3 m/s at 38.0 s. At 47.5 s, the velocity oscillates between 3.8 and 4.2 m/s and increases over time. At 179.5 s, the velocity fluctuates between 2.5 and 5.1 m/s as shown in Figure 2a. When \(\delta=4\), the velocity of the 1 st vehicle decreases to 3.1 m/s and then increases to 4.3 m/s at 37.0 s. At 47.5 s, the velocity oscillates between 3.7 and 4.3 m/s, and at 179.5 s, it fluctuates between 0.03 and 7.4 m/s as shown in Figure 2b. When \(\delta=20\), the velocity of the 1 st vehicle decreases to 3.1 m/s and then increases to 4.3 m/s at 37.0 s. At 45.5 s, the velocity oscillates between 3.7 and 4.3 m/s, and at 179.5 s, it fluctuates between 0.03 and 7.4 m/s as shown in Figure 2c. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Proposed model acceleration, \(a\) & \(1.5\) m/s\({}^{2}\) \\ Time headway for the ID model, \(\tau\) & 2 s \\ Time headway for the aggressive driver, \(\tau\) & 1 s \\ Time headway for the sluggish driver, \(\tau\) & 2.5 s \\ Time headway for the typical driver, \(\tau\) & 2 s \\ Ratio of actual to desired distance headway, \(h=\frac{s}{D}\) & \(0.3,\ 0.5,\) and 1 \\ Jam spacing, \(s_{j}\) & 5 m \\ Maximum acceleration, \(a\) & 0.73 m/s\({}^{2}\) \\ Minimum acceleration, \(b\) & 1.67 m/s\({}^{2}\) \\ Vehicle length, \(l\) & 5 m \\ Time step, \(\Delta t\) & 0.5 s \\ Initial equilibrium velocity, \(v\) & 4 m/s \\ Acceleration exponent, \(\delta\) & 1, 4, and 20 \\ Safe time headway, \(\tau_{s}\) & 2 s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters for stability analysis. For the proposed model with \(\frac{\tau}{\tau_{k}}<1\) (aggressive driver), when \(h=0.3\), the velocity of the first vehicle decreases to 3.2 m/s and then increases to 4.3 m/s at 43.5 s. At 63.0 s, it oscillates between 3.5 and 4.5 m/s. These oscillations increase over time, and at 178.0 s, the velocity fluctuates between 0.03 and 7.3 m/s as shown in Figure 2d. When \(h=0.5\), the velocity of the first vehicle decreases to 3.1 m/s and then increases to 4.3 m/s at 40.0 s. At 50.5 s, the velocity oscillates between 3.6 and 4.4 m/s, and at 179.5 s, it fluctuates between 0.07 and 8.5 m/s as shown in Figure 2e. When \(h=1.0\), the velocity of the 1 st vehicle decreases to 3.1 m/s and then increases to 4.3 m/s at 37.5 s. At 69.5 s, the velocity oscillates between 3.1 and 5.1 m/s, and at 179.0 s, it fluctuates between 0.16 and 11.1 m/s as shown in Figure 2f. With \(\frac{\tau}{\tau_{k}}>1\) (sluggish driver), at \(h=0.3\) the velocity of the 1 st vehicle decreases to 3.1 m/s and then increases to 4.2 m/s at 38.0 s. At 52.0 s, the velocity oscillates between 3.8 and 4.2 m/s, and at 179.5 s, it fluctuates between 3.6 and 4.3 m/s as shown in Figure 2g.