## 2 Methodology  
### Length of the Transition Area  
As shown in Figure 1, a transition area should guide vehicles to change lanes and merge at the speed limit, enabling them to safely cross through the work zone. In most previous studies, only the speed limit and width of offset have been considered to calculate the transition area length [31]:  
\[ \begin{array}{l}L=\frac{Wv^{2}}{60}\ \ (v\leq 60\text{km}/\text{h})\\ L=Wv\ \ (v>60\text{km}/\text{h})\end{array}  \tag{1}\]  
where \(L\) is the transition area length, m, \(W\) is the width of offset, m, and \(v\) is the speed limit, km/h. However, this calculation model does not consider traffic flow and drivers' psychophysiological and driving behavior characteristics in the work zone. Vehicles located in the closed lane need to find suitable gaps (same as the time headway, \(t\)) to merge into the adjacent lane before reaching the end of the transition area. When the traffic volume is low, drivers can easily choose larger merging gaps. The probability of critical gap appearance decreases when the traffic volume is higher and the transition area length is limited. When approaching the end of the transition area, drivers will choose smaller gaps to complete merging as soon as possible, and at the end of the transition  
Figure 1: Schematic diagram for calculating the length of the transition area. area, drivers must stop and forcibly merge. According to observations, when the vehicle running speed is 40-80 km/h and the lane width is 3.25-3.75 m, the acceptable critical gap for a vehicle in the closed lane is 1.6-4.3 s. The time headway of the adjacent lane follows a continuous distribution: when the time headway is greater than the critical gap, the vehicle merges. In order to determine the time headway distribution of the adjacent lane, a survey was conducted using an unmanned aerial vehicle in a work zone of a highway in China, as shown in Figure 2. The time headway of the adjacent lane in the closed lane of the transition area under different traffic volumes is shown in Table 1.  
The adjacent lane is a single-lane traffic flow that cannot overtake, and the minimum time headway is 0.8 s, which follows to the shifted negative exponential distribution function, as shown in Equation (2):  
\[F(t)=1-\mathrm{e}^{-\lambda(t-\tau)},\ t\geq\tau \tag{2}\]  
where \(\lambda\) is parameter and \(\tau\) is the minimum time headway, s. If it follows to the shifted negative exponential distribution, then the expected time headway: \(E(t)=1/\lambda+\tau\), and variance: \(Var(t)=1/\lambda^{2}\). We used the \(X^{2}\) test to test its distribution: \(X^{2}=16.541\)-31.684, and when the number of groups was 30 and the significance level was 0.05, the critical value \(X^{2}_{0.05}=40.113>16.541\)-31.684. Therefore, we accepted the original assumption and followed to the shifted negative exponential distribution. When the traffic volume was 500 pcu/h, 1000 pcu/h, and 1500 pcu/h, its distribution function was as shown in Equations (3)-(5):  
\[F(t)=1-\mathrm{e}^{-0.181(t-0.8)},\ t\geq 0.8 \tag{3}\]  
\[F(t)=1-\mathrm{e}^{-0.358(t-0.8)},\ t\geq 0.8 \tag{4}\]  
\[F(t)=1-\mathrm{e}^{-0.513(t-0.8)},\ t\geq 0.8 \tag{5}\]  
The vehicle recognizes the need for merging in the transition area before the transition area. In the transition area, it needs to travel a certain distance (\(l_{1}\)) to wait for the critical \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Survey Time** & **Traffic Volume** & **Maximum Time** & **Minimum Time** & **Mean Time** & **SD** \\  & **(pcu/h)** & **Headway (s)** & **Headway (s)** & **Headway (s)** & **SD** \\ \hline A & 537 & 20.61 & 0.83 & 6.32 & 5.53 \\ B & \(1011\) & 15.37 & 0.79 & 3.59 & 2.80 \\ C & 1468 & 11.88 & 0.76 & 2.75 & 1.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The time headway of the adjacent lane in the closed lane of the transition area under different traffic volumes. Figure 2: Survey of time headway in the work zone. gap, and then make decisions and change lanes (\(l_{2}\)).