## 3 Methodology
### Car-Following Models for AVs and MVs  
This paper only considers longitudinal car-following models for two kinds of vehicles (AVs and MVs) on the approaching lanes. The movements of AVs follow the optimal control inputs designed by this study, and the Krauss model in [29] is adopted to describe the driving behavior of MVs.  
The longitudinal movement dynamics of AVs can be modeled by the following discrete-time model in [30]:  
\[\begin{cases}p_{i}(t+1)=p_{i}(t)+\delta v_{i}(t)+\frac{\delta^{2}}{2}u_{i}(t) \,\ \ \ i\in\Omega_{A}(t)\\ v_{i}(t+1)=v_{i}(t)+\delta u_{i}(t)\end{cases} \tag{1}\]  
where \(\delta\) is the discrete time interval, and \(p_{i}(t)\) and \(v_{i}(t)\) denote the position and speed of AV \(i\) at time step \(t\), respectively. \(u_{i}(t)\) is the control input to be designed at the control sample point \(t\), and \(\Omega_{A}(t)\) represents the time-varying set of AVs in the CZ at time step \(t\).  
The movements of MVs are not under control, and the Krauss model is introduced to estimate the longitudinal behavior of MVs. The Krauss model is a conflict-free model based on the safe distance with immediately preceding vehicles. Algorithm 1 is given to describe the updating strategy of the speed and position of MVs according to [29].  
```
0: The speed of the immediately preceding vehicle \(v_{p}\), the position and speed of ego vehicle \(p_{m}\) and \(v_{m}\) (\(m\in\Omega_{M}(t)\)), the vehicle gap with its immediately preceding vehicle \(g\);
0: Updated position and speed of the ego vehicle at the next step.
1: Repeat
2: Acquire \(v_{p}(t)\), \(p_{m}(t)\), \(v_{m}(t)\), and \(g\).
3:\(v_{safe}\gets v_{p}(t)+\frac{g-v_{m}(t)\tau}{(v_{p}(t)+v_{m}(t))/2v_{max}+\tau}\)
4:\(\gamma_{0}\leftarrow\min\left[v_{m}(t)+a_{max},V_{max},v_{safe}\right]\)
5:\(\gamma_{1}\leftarrow\gamma_{0}-\epsilon\{\gamma_{0}-[v_{m}(t)-a_{max}]\}\)
6:\(v_{m}(t+1)\gets v_{ran(\gamma_{0},\gamma_{1})}\)
7:\(p_{m}(t+1)\gets p_{m}(t)+\delta v_{m}(t+1)\)
8:Return \(p_{m}(t+1)\) and \(v_{m}(t+1)\);
9:\(t\gets t+1\)
10:Until The ego vehicle departs from the CZ.
```  
**Algorithm 1** Trajectory-Updating Algorithm for MVs  
In Algorithm 1, the position and speed of the ego vehicle at the next time step are estimated according to the speed of its immediately preceding vehicle, the position and speed of the ego vehicle, and the vehicle gap between the two vehicles at the current time step. \(\Omega_{M}(t)\) represents the dynamic set of MVs in the CZ at time step \(t\delta\). \(v_{safe}\) denotes the maximum safe speed of the ego vehicle without collision with its immediately preceding vehicle at the next time step, which is computed by step 3 in Algorithm 1. \(\tau\) is the reaction time of drivers, and \(\epsilon\) is a constant. \(a_{max}\) and \(b_{max}\) are the maximum acceleration and braking deceleration, respectively. Steps 4-7 in Algorithm 1 show the updating rules of the Krauss model, where \(\gamma_{0}\) denotes the desired speed, and \(\gamma_{1}\) represents the speed with the maximum difference from the desired speed due to imperfect driving behaviors of drivers. \(v_{ran(\gamma_{0},\gamma_{1})}\) denotes a random value for the speed picked from \(\gamma_{0}\) and \(\gamma_{1}\). Note that although this randomly selected updating strategy leads to different driving behaviors, the stronginteractions between successive vehicles will decrease the freedom of individual drivers to choose their velocity.