# Optimal Lane Allocation Strategy in Toll Stations for Mixed Human-Driven and Autonomous Vehicles
## 5 Conclusions and Future Research  
This study investigates the TSLAP under heterogeneous traffic flow, considering the lane selection rights of different vehicle types. Based on the user equilibrium principle and queueing theory, the lane selection behavior of ETC-HVs is modeled. The TSLAP is formulated as a mixed-integer nonlinear bilevel optimization problem, aiming to minimize the total passage time of the toll station by optimizing the number of ETC lanes at the station's entrances and exits. Since both the upper and lower-level problems are convex, the KKT conditions of the lower-level problem are used to transform the bilevel problem into an equivalent single-level optimization. The problem is solved numerically using the commercial solver Gurobi, and a relaxed analytical solution is derived using a parameterized iterative method. The results show that the proposed approach significantly improves the performance of toll station queuing systems under heterogeneous traffic flow, reducing the total passage time by 90.44% compared to the current system. Impact analysis and comparative experiments reveal that the proportion of MTC-HVs has the greatest impact on the optimal total passage time while increasing the market penetration of CAVs or ETC-HVs helps manage high traffic demand. Additionally, dynamically adjusting lane allocation in response to changes in vehicle arrival rates proves to be a more effective strategy than maintaining a static allocation. The developed model and obtained results can serve as a basis for planning and managing toll station lane allocation in heterogeneous traffic flow environments. However, this study has several limitations. First, the study assumes that the toll station queuing system follows an M/M/1 model. On one hand, each lane is treated as an independent single-server queuing system, which is a reasonable assumption for the Liulin toll station, as its lanes are equipped with barriers and extended vehicle separation guidelines that prevent vehicles from switching lanes freely during the queuing process. However, other toll stations may allow vehicles to switch lanes. For toll stations that allow lane switching, a multi-server queueing system could potentially be used. However, considering the practical situation, it is quite difficult for vehicles to change lanes continuously before the toll plaza--such as from the outermost lane to the innermost lane--and during congestion, the spacing between vehicles does not allow lane changes. This may not align with the characteristics of a multi-server queueing system, which typically allows free switching between servers. Additionally, lane switching would disrupt the toll plaza system, potentially affecting the speed of vehicles behind. We suggest that for toll stations where lane switching is allowed, simulation techniques may be a better approach for modeling such systems. On the other hand, this study focuses on the lane allocation problem at toll stations and assumes a simple M/M/1 queuing system, characterized by Poisson arrivals and exponential service times. In practice, the situation is more complex and may change over time, requiring more sophisticated models. Second, while numerical experiments in this study demonstrate that dynamic lane allocation can effectively manage traffic fluctuations, it is important to recognize that real-world applications involve more complex operational processes. With the increasing sophistication of traffic management systems, we believe dynamic adjustment holds significant application potential. Third, this study assumes that ETC-HV users are fully rational in their lane selection, always choosing the lane with the shortest waiting time. However, real-world behavior is likely more complex, with factors such as driver preferences, experience, and external influences affecting lane choice.