# Optimal Lane Allocation Strategy in Toll Stations for Mixed Human-Driven and Autonomous Vehicles
## 3 Methodology
### Optimization Model
#### 3.1.2 Multi-Class User Equilibrium  
Due to the traffic conditions inherent in the toll plaza queue system, ETC-HVs do not experience temporal or spatial invisibility. Consequently, they are already aware of the traffic situation at the toll plaza before selecting a lane. Moreover, based on their experience, ETC-HVs can identify the lane that minimizes their time spent at the toll plaza. Therefore, we assume that all users are rational and have a clear understanding of the traffic conditions within the toll plaza queue system. Under a given toll lane allocation strategy, we determine the lane choice of ETC-HVs according to the user equilibrium principle. It is important to note that when the lane allocation strategy is fixed, vehicles entering and exiting the toll plaza do not compete with each other, as users can only access lanes corresponding to their direction of travel and compete solely with vehicles moving in the same direction. Furthermore, even when considering dynamic demand, vehicles during different time intervals do not compete with one another. Thus, we can treat the lane choices of users in various directions and time intervals as independent and analyze them separately. If we regard the highway toll plaza queue system as an equivalent traffic network, then for an ETC-HV user traveling in one direction, there are only two available paths: the ETC lane and the MTC lane. In this scenario, we need not adhere strictly to the conventional objective function forms of traffic network user equilibrium models. For an individual ETC-HV, the lane selection is guided by the principle of minimizing travel time. Specifically, if the travel time on the MTC lane is less than that on the ETC lane, the vehicle will choose the MTC lane; otherwise, it will opt for the ETC lane. This relationship can be expressed mathematically as follows:  
Figure 4: Bi-level programming model for lane allocation at highway toll stations. \[choice= \begin{cases}ETC,&\text{if }Ws_{d1}\leqslantWs_{d2},\\ MTC,&\text{if }Ws_{d1}>Ws_{d2}.\end{cases}  \tag{12}\]  
According to Equation (1), the average travel time \(Ws_{i}\) for lane type \(i\) increases as the user arrival rate \(\lambda_{i}\) for that lane type rises. Consequently, when an individual ETC-HV opts for a lane with a shorter travel time, it results in an increase in the average travel time for that lane type. Therefore, the lane selection behavior of ETC-HVs tends to equalize the average travel times of both the MTC and ETC lanes. This relationship can be represented mathematically as follows:  
\[\min_{\lambda_{di}}|Ws_{d1}-Ws_{d2}| \tag{13}\]  
subject to  
\[\lambda_{di}<\mu_{di}n_{di},\quad\forall d\in\mathbf{D},i\in\mathbf{I},n_{di}\
eq 0, \tag{14}\]  
\[\lambda_{di}\geqslant p_{i}\lambda_{d},\quad\forall d\in\mathbf{D},i\in\mathbf{I}, \tag{15}\]  
\[\sum_{i\in\mathbf{I}}\lambda_{di}=\lambda_{d},\quad\forall d\in\mathbf{D}, \tag{16}\]  
where Equation (14) is derived from Equation (2); Equations (15) and (16) are derived from Equations (3)-(6). However, the objective function in Equation (13) contains absolute values, and the expression for \(Ws_{di}\) is a fractional structure. Given that \(Ws_{di}>0\), for computational and analytical convenience, we can reframe the model using \(\frac{n_{di}n_{d2}}{Ws_{di}}\). This indicates that an individual ETC-HV user will select the lane type that minimizes \(\frac{n_{d2}n_{d2}}{Ws_{di}}\). Thus, the selection behavior of a single ETC vehicle can be expressed as follows:  
\[choice= \begin{cases}ETC,&\text{if }\frac{n_{d1}n_{d2}}{Ws_{d2}}\leqslant\frac{n_{ d1}n_{d2}}{Ws_{d1}},\\ MTC,&\text{if }\frac{n_{d1}n_{d2}}{Ws_{d2}}>\frac{n_{d1}n_{d2}}{Ws_{d1}}.\end{cases}  \tag{17}\]  
Let \(Z_{low}\) denote the objective function for the lower-level planning.