# A Framework to Determine the Utilization of Vacant Taxis on HOV Lanes with the Optimal Deployment
## 1 Introduction
[19] confirmed that the Route 55 HOV facility encouraged more travelers to carpool during peak hours. [PERSON] et al. [20] revealed the success of the HOV lanes in terms of maximizing people throughout, managing congestion, offering travel time saving, and reliability. On the other hand, some studies demonstrated that HOV lanes failed to reduce traffic congestion. [PERSON] and [PERSON] [21] conducted an investigation on the performances of the HOV lanes in California. The results showed the ineffectiveness of the HOV lanes on mitigating congestion, namely, reducing total travel time. The most likely reason is that insufficient travelers are willing to choose carsharing mode, leading to lower HOV lane usage. [PERSON] [22] also stated that HOV lanes are not always better than general purpose (GP) lanes in alleviating congestion. From the perspective of the environment, HOV lanes are partially effective [23; 24]. Therefore, several approaches, involving converting HOV lanes to HOT lanes, optimal deployment design of lanes, and allowing other types of vehicles like plug-in electric vehicles to use HOV lanes, were proposed to remedy the drawbacks of HOV lanes [25; 26; 27; 28; 29; 30]. Most of the studies above proposed models based on theoretical analysis. Note that variational inequality is a useful tool to analyze traffic networks. It can be used to describe travelers' mode and route choice. [PERSON] first introduced VI into transportation network analysis [31]. [PERSON] finished a series of work on the application to transportation networks involving models and algorithms [32]. VI has been widely used in the field of transportation research, with various cases such as ridesharing [33; 34]; the ride-sourcing market [35]; equilibrium, including autonomous vehicle [36]; etc. Therefore, VI is an appropriate choice for ridesharing equilibrium, taxi market equilibrium, and analysis. Nevertheless, the impacts of HOV lanes on networks with the presence of taxis are unclear. In other words, few studies have investigated the effects of the network when allowing/forbidding vacant taxis from using HOV lanes. We propose a framework includingthe variational inequality (VI) model and bi-level programming to explore the performance of the network and the optimal deployment of HOV lanes, respectively, in which VI is used to describe the travelers' mode and route choice when vacant taxis are allowed to use/forbidden from using HOV lanes, respectively, and bi-level programming aiming at maximal social welfare based on the two situations. We explain the procedures to solve VI and bi-level programming and use numerical examples to examine the performances of the network to demonstrate the feasibility of our models and algorithms. The rests of this paper is organized as follows: Section 2 introduces the preliminaries of this paper. Section 3 demonstrates the formulations, including cost functions, principles, constrains, and the VI model. Section 4 explores the performances of the network in a given network with/without the scheme allowing/forbidding vacant taxis from using HOV lanes as a numerical example. Bi-level programming for the optimization of the HOV lane deployment is provided in Section 5 and the algorithm of the bi-level model is explained in Section 6, both of which are applied as a simple network and a Sioux Falls network in Section 7 to detect the performances. The conclusions and further research directions are outlined in Section 8.