# A Microscopic Traffic Model Considering Driver Reaction and Sensitivity
## 3 Stability Analysis
### Theoretical Stability  
The stability of both the ID and proposed models is analyzed over an infinite road. The driver response and vehicles are assumed identical and the equilibrium distance headway \(s_{e}\) is maintained between all vehicles [45]. Thus, drivers align to forward conditions with little acceleration, and only small changes in equilibrium velocity \(v_{e}(s_{e})\) occur. The changes in distance headway \(x\) are therefore small as are the changes in velocity \(y\). The distance headway during a change in velocity is then  
\[s=s_{e}+x, \tag{12}\]  
and the corresponding velocity is  
\[v=v_{e}(s_{e})+y. \tag{13}\]  
During traffic alignment over the distance headway, the temporal change in velocity [37] is  
\[x(t)=\frac{dx}{dt}=y_{l}-y_{F}, \tag{14}\]  
where the subscripts \(l\) and \(F\) denote the leading and following vehicles, respectively. The changes in headway are small as the changes in \(v_{e}(s_{e})\) are small, so \(y(t)\) during alignment can be expressed as [45]  
\[y(t)=\frac{dy}{dt}=f_{s}x_{F}+(f_{v}+f_{\Delta v})y_{F}-f_{\Delta v}y_{l}, \tag{15}\]  
where \(f_{\Delta v}\) is the partial derivative with respect to the change in velocity, \(f_{v}\) is the partial derivative with respect to velocity, and \(f_{s}\) is the partial derivative with respect to distance headway, and are given by  
\[f_{s}=\frac{\partial f}{\partial s},\;f_{v}=\frac{\partial f}{\partial v},\; \text{and}\;f_{\Delta v}=\frac{\partial f}{\partial\Delta v}.\]  
Equations (14) and (15) are solved using Fourier-Ansatz as  
\[y(t)=\hat{y}e^{\gamma t+i\hat{k}}, \tag{16}\]  
\[u(t)=\hat{u}e^{\gamma t+i\hat{k}}, \tag{17}\]  
so (16) and (17) take the form  
\[ \begin{pmatrix}x(t)\\ y(t)\end{pmatrix} = \begin{pmatrix}\hat{x}\\ \hat{y}\end{pmatrix} e^{\gamma t+i\hat{k}}, \tag{18}\]  
where \(\gamma=\alpha+i\omega\) is the change in traffic oscillations during alignment and \(i=\sqrt{-1}\). The real part \(\alpha\) denotes the change in amplitude, \(\omega=\frac{2\pi}{T}\) is the oscillation frequency, \(T\) is the oscillation period, \(k\) is the phase shift that corresponds to the delay experienced by a driver [45], and \(\hat{y}\) and \(\hat{x}\) are the changes in velocity and distance headway, respectively. Substituting (18) in (14) and (15) gives  
\[x(t)=y-ye^{ik}, \tag{19}\]  
\[y(t)=f_{s}xe^{ik}+(f_{v}+f_{\Delta v})ye^{ik}-f_{\Delta v}y. \tag{20}\]  
The model is stable if the real parts of the eigenvalues are negative. The eigenvalues are obtained from  
\[\left|f- \begin{pmatrix}\lambda&0\\ 0&\lambda\end{pmatrix} \right|=0, \tag{21}\]where \(J\) is the Jacobian matrix given by  
\[J= \begin{pmatrix}j_{11}&j_{12}\\ j_{21}&j_{22}\end{pmatrix} ,\]  
and \(j_{11}\) and \(j_{21}\) are the partial differentials of (19) and (20) w.r.t. \(x\) and \(j_{12}\) and \(j_{22}\) are the partial differentials of (19) and (20) w.r.t \(y\) so that  
\[J=e^{ik} \begin{pmatrix}0&e^{-ik}-1\\ f_{s}&(f_{v}+f_{\Delta v})-f_{\Delta v}e^{-ik}\end{pmatrix} . \tag{22}\]  
Then substituting (22) in (21) we have  
\[ \begin{vmatrix}\lambda&1-e^{-ik}\\ -f_{s}&\lambda-f_{v}-f_{\Delta v}+f_{\Delta v}e^{-ik}\end{vmatrix} =0, \tag{23}\]  
which gives  
\[\lambda^{2}+\left(-f_{v}-f_{\Delta v}+f_{\Delta v}e^{-ik}\right)\lambda+f_{s} \left(1-e^{-ik}\right)=0 \tag{24}\]  
Let \(X(k)=-f_{v}-f_{\Delta v}+f_{\Delta v}e^{-ik}\) and \(Y(k)=f_{s}\left(1-e^{-ik}\right)\) in (24) so that  
\[\lambda^{2}+X(k)\lambda+Y(k)=0 \tag{25}\]  
The eigenvalues from (25) are  
\[\lambda_{1,2}=-\frac{X(k)}{2}\Bigg{(}\,1\pm\sqrt{1-\frac{4Y(k)}{X^{2}(k)}} \Bigg{)}, \tag{26}\]  
The model is string stable [45] if the real parts of the eigenvalues are negative. In this case, vehicles maintain the distance headway and changes in the equilibrium velocity are small [46]. Then the oscillations in flow decay over time so it is stable (smooth). A model is unstable if the flow increases over time. An example is stop and go traffic, which occurs during congestion. In this case, acceleration is high, whereas in string stable traffic, acceleration is low [5].