## Appendix A
### Stability Analysis of the Proposed Model
\begin{array}{l}y^{\prime}_{n}(t+2t_{d})-y^{\prime}_{n}(t+t_{d})=\\ t_{d}f^{s}_{n}(y_{n-1}(t)-y_{n}(t))+f^{\circ n}_{n}(y_{n}(t+t_{d})-y_{n}(t))+ \\ f^{\Delta v}_{n}(y_{n}(t+t_{d})-y_{n}(t)-y_{n-1}(t+t_{d})+y_{n-1}(t))+\\ f^{\prime n Substituting \(y_{n}(t)=ce^{\omega a_{k}n+zt}\) and \(y^{\prime}_{n}(t)=zce^{\omega a_{k}n+zt}\) into (A6) and simplifying, we obtain:  
\[\begin{array}{c}\left(e^{zt_{d}}-1\right)\left[ze^{zt_{d}}-f_{n}^{v_{n}}-f_{n}^ {\Delta v}(1-e^{-\omega a_{k}})-\right.\\ \left.\left(zf_{n}^{a_{n-i}}+f_{n}^{v_{n-i}}\right)e^{-i\cdot\omega a_{k}}- \left(zf_{n}^{a_{n+i}}+f_{n}^{v_{n+i}}\right)e^{j\cdot\omega a_{k}}\right]=\\ t_{d}\Big{[}f_{n}^{s}(e^{-\omega a_{k}}-1)+f_{n}^{\Delta x_{i}}\big{(}e^{-i\cdot \omega a_{k}}-1\big{)}+f_{n}^{\Delta x_{j}}\big{(}e^{j\cdot\omega a_{k}}-1 \big{)}\Big{]},\end{array} \tag{10}\]  
We then expand the power series for the parameters in (10).  
\[z=z_{1}(\omega a_{k})+z_{2}(\omega a_{k})^{2}+\cdots, \tag{11}\]  
\[e^{zt_{d}}=1+t_{d}z+\frac{t_{d}^{2}z^{2}}{2}+\cdots, \tag{12}\]  
\[e^{-\omega a_{k}}=1-\omega a_{k}+\frac{(\omega a_{k})^{2}}{2}-\cdots, \tag{13}\]  
Expanding this gives the first- and second-order expressions for z:  
\[\begin{array}{c}\left(z_{1}(\omega a_{k})+z_{2}(\omega a_{k})^{2}+\frac{t_ {d}}{2}\Big{(}z_{1}(\omega a_{k})+z_{2}(\omega a_{k})^{2}\Big{)}^{2}\right) \times\\ \left(z_{1}(\omega a_{k})+z_{2}(\omega a_{k})^{2}+t_{d}\Big{(}z_{1}(\omega a_{ k})+z_{2}(\omega a_{k})^{2}\Big{)}^{2}+\right.\\ \left.\frac{t_{d}^{2}}{2}\Big{(}z_{1}(\omega a_{k})+z_{2}(\omega a_{k})^{2} \Big{)}^{3}-f_{n}^{v_{n}}-f_{n}^{\Delta v}\bigg{(}\omega a_{k}-\frac{(\omega a _{k})^{2}}{2}\Big{)}-\\ \left.\Big{(}\left(z_{1}(\omega a_{k})+z_{2}(\omega a_{k})^{2}\right)f_{n}^{a_ {n-i}}+f_{n}^{v_{n-i}}\Big{)}\Big{(}1-i\cdot(\omega a_{k})+\frac{\rho_{i}^{2} }{2}(\omega a_{k})^{2}\Big{)}-\\ \left.\Big{(}\left(z_{1}(\omega a_{k})+z_{2}(\omega a_{k})^{2}\right)f_{n}^{a_ {n+i}}+f_{n}^{v_{n+i}}\Big{)}\Big{(}1+j\cdot(\omega a_{k})+\frac{\rho_{i}^{2} }{2}(\omega a_{k})^{2}\Big{)}\right)=\\ \left.\Big{(}-f_{n}^{s}-if_{n}^{\Delta x_{i}}+jf_{n}^{\Delta x_{j}}\Big{)}( \omega a_{k})+\Big{(}\frac{1}{2}f_{n}^{s}+\frac{\rho_{i}^{2}}{2}f_{n}^{\Delta x _{i}}+\frac{\rho_{i}^{2}}{2}f_{n}^{\Delta x_{j}}\Big{)}(\omega a_{k})^{2}, \end{array} \tag{14}\]  
and thus yields:  
\[z_{1}=\frac{f_{n}^{s}+if_{n}^{\Delta x_{i}}-jf_{n}^{\Delta x_{j}}}{f_{n}^{v_{n }}+f_{n}^{v_{n-i}}+f_{n}^{v_{n+i}}}, \tag{15}\]  
\[z_{2}=\left(z_{1}{}^{2}\Big{(}1-f_{n}^{a_{n-i}}-f_{n}^{a_{n+i}}-\frac{t_{d}}{2 }f_{n}^{v_{n}}-\frac{t_{d}}{2}f_{n-i}^{v_{n-i}}-\frac{t_{d}}{2}f_{n}^{v_{n+i}} \Big{)}-\right.\\ z_{1}\Big{(}f_{n}^{\Delta v}-if_{n}^{v_{n-i}}+jf_{n}^{v_{n+i}}\Big{)}-\left( \frac{1}{2}f_{n}^{s}+\frac{\rho_{i}^{2}}{2}f_{n}^{\Delta x_{i}}+\frac{\rho_{i}^ {2}}{2}f_{n}^{\Delta x_{j}}\Big{)}\right)\times\\ \left.\Big{(}\frac{1}{f_{n}^{\Delta v}+f_{n}^{v_{n-i}}+f_{n}^{v_{n+i}}+f_{n}^{ v_{n+i}}}\Big{)},\end{array}  \tag{16}\]  
Traffic flows become unstable when \(z_{2}<0\) in response to small perturbations and stabilise when \(z_{2}>0\). The stability conditions are  
\[ \begin{array}{c}\left(\left(\frac{f_{n}^{s}+if_{n}^{\Delta x_{i}}-jf_{n}^{ \Delta x_{j}}}{f_{n}^{v_{n}}+f_{n}^{v_{n+i}}+f_{n}^{v_{n+i}}}\right)^{2}\Big{(}1 -f_{n}^{a_{n-i}}-f_{n}^{a_{n+i}}-\frac{t_{d}}{2}f_{n}^{v_{n}}-\frac{t_{d}}{2}f_ {n}^{v_{n-i}}-\frac{t_{d}}{2}f_{n}^{v_{n+i}}\Big{)}-\\ \left(\frac{f_{n}^{s}+if_{n}^{\Delta x_{i}}-jf_{n}^{\Delta x_{j}}}{f_{n}^{v_{n} }+f_{n}^{v_{n-i}}+f_{n}^{v_{n+i}}}\right)\Big{(}f_{n}^{\Delta v}-if_{n}^{v_{n-i }}+jf_{n}^{v_{n+i}}\Big{)}-\\ \left(\frac{1}{2}f_{n}^{s}+\frac{\rho_{i}^{2}}{2}f_{n}^{\Delta x_{i}}+\frac{\rho _{i}^{2}}{2}f_{n}^{\Delta x_{j}}\Big{)}\right)<0,\end{array}  \tag{17}\]Simplifying and rearranging the terms yields:  
\[