# A New Driver Model Based on Driver Response
## 3 Stability Analysis
Moreover, traffic models become unstable as \(k\to 0\), i.e., when there is no delay between traffic waves which occur in congestion. In this case, the wavelengths and periods increase, and thus so do the distance headway and velocity [41]. The coefficients of (27) can be expressed as Taylor series. For small \(k\), \(A(k)\) can be approximated as [41]  
\[A(k)\approx-f_{v}-f_{\Delta v}+f_{\Delta v}e^{-i(0)}-\frac{if_{\Delta v}e^{-i(0)} }{1}k\]  
\[=-f_{v}-if_{\Delta v}k, \tag{29}\]  
and \(B(k)\) can be approximated as [41]  
\[B(k)\approx f_{s}\Big{(}1-e^{-i(0)}\Big{)}+\frac{f_{s}ie^{-i(0)}}{1}k+\frac{f_ {s}e^{-i(0)}}{2}k^{2}\]  
\[=if_{s}k+\frac{f_{s}}{2}k^{2}.\]  
According to the steady-state (equilibrium) condition, the interdependence of \(f_{s}\) and \(f_{v}\)[41] is  
\[f_{s}=-v_{e}^{\prime}(D)f_{v}. \tag{30}\]  
where \(v_{e}^{\prime}(D)\) is the change in velocity for a given distance headway at equilibrium. Substituting \(f_{s}\) from (30) in \(B(k)\) gives  
\[B(k)=-iv_{e}^{\prime}(D)f_{v}-\frac{v_{e}^{\prime}(D)}{2}f_{v} \tag{31}\]  
Let  
\[A(k)=a_{0}+a_{1}k,B(k)=b_{1}k+b_{2}k^{2} \tag{32}\]  
With  
\[\lambda^{2}+A(k)\lambda+B(k)=0. \tag{27}\]0  
Expressing the square root in (28) as a Taylor series gives the approximation  
\[\lambda^{2}+A(k)\lambda+B(k)=0. \tag{27}\]1  
Substituting (34) in (28) and considering the negative sign for the square root results in  
\[\lambda^{2}+A(k)\lambda+B(k)=0. \tag{27}\]2  
Substituting \(A(k)\) and \(B(k)\) from (32) gives  
\[\lambda^{2}+A(k)\lambda+B(k)=0. \tag{27}\]3  
and then substituting (33) in (35) results in  
\[\lambda^{2}+A(k)\lambda+B(k)=0. \tag{27}\]4  
The real part of (36) describes the wave growth and the traffic flow is string stable if this term is negative. As  
\[\lambda^{2}+A(k)\lambda+B(k)=0. \tag{27}\]5the term \(\left[\frac{-2f_{\Delta v}-f_{v}}{2}-v^{\prime}_{e}(D)\right]\) gives the following string stability criteria [27]  
\[\lambda^{2}+A(k)\lambda+B(k)=0. \tag{27}\]6  
From (37) and (38), the product of \(\left[-\frac{2f_{\Delta v}-f_{v}}{2}-v^{\prime}_{e}(D)\right]\) and \(\frac{v^{\prime}_{e}(D)}{f_{v}}\) results in a negative real part of \(\lambda_{1}\) in (38). The criterion in (38) indicates that the model is very stable for a driver following large changes in traffic. That is, the traffic flow is smooth and changes in velocity are small. Further, traffic is stable when a driver tracks changes in velocity. Drivers not doing this may produce stop and go traffic. In this case, the model is not string stable. The partial derivatives \(f_{v}\) and \(f_{\Delta v}\) of (1) at steady state with traffic moving at the equilibrium velocity \(v_{e}(D)\) can be expressed as  
\[\lambda^{2}+A(k)\lambda+B(k)=0. \tag{27}\]7  
\[\lambda^{2}+A(k)\lambda+B(k)=0. \tag{27}\]8  
Substituting (39) and (40) in (38) gives the string stability criteria  
\[\lambda^{2}+A(k)\lambda+B(k)=0. \tag{27}\]9  
Changes in velocity for the ID model are based on a constant exponent \(\delta\), and thus so are changes in the distance headway. To ensure string stability, a large value of \(\delta\). is employed in practice which results in large changes in velocity. Then, vehicles will leave congestion quickly while maintaining the desired distance headway. Using a large value of \(\delta\) is not based on traffic physics but rather is a means of ensuring stability. Thus, it results in an unrealistic and inadequate characterization. In reality, changes in velocity occur based on the rearward and forward driver responses. Therefore, these responses must be considered, and this should ensure traffic string stability. Thus, for the proposed model, replacing \(\delta\) in (41) with (7) gives  
\[\lambda_{1,2}=-\frac{A(k)}{2}\Bigg{(}1\pm\sqrt{1-\frac{4B(k)}{A^{2}(k)}}\Bigg{)}, \tag{28}\]0  
With the proposed model, changes in velocity are based on the forward and rearward traffic conditions, i.e., traffic stability is affected by the forward and rearward time and distance headways. With a long time headway, vehicles move slowly and stay longer in stop and go traffic.