## 1 Introduction  
Networked signals from advanced communication devices and other detection devices can sense intersection traffic demand and make intelligent control decisions [1, 2, 3]. Over time, the rapid development of vehicle-to-everything (V2X) and connected and automated vehicle (CAV) technologies will provide more real-time information about vehicles at signalized intersections and optimize vehicle trajectories [4]. A common belief among policy makers and scholars is that heterogeneous traffic flows with HV and CAV coexistence will be prevalent in the next 20-30 years [5, 6, 7]. The construction of China's urban roads and increasing car ownership have made peak traffic flows at urban road intersections saturated or oversaturated [8, 9, 10]. Collaborative control of heterogeneous traffic flows based on dedicated CAV lanes has been explored to improve intersection capacity and reduce vehicle delays while avoiding uncertainty in HV driving behaviour [11, 12, 13, 14, 15]. Based on these advanced technologies and realistic situations, we need to study a new oversaturated heterogeneous traffic flow signal control model with a variable virtual waiting zone based on dedicated CAV lanes to utilize the advantages of CAV network linkages and exploit the spatiotemporal resources of intersections. This new signal control model consists of signal cycle control, dedicated lane variable virtual waiting zone time division, and vehicle trajectory control. However, the challenges of the new signal control model include the following three aspects. First, the signal control period and phase duration are directly related to intersection traffic efficiency and vehicle delay duration, and reasonable regulation of the signal control period needs to fully consider the minimum vehicle delay at the entrance. Second, to fully exploit the space-time resources inside an intersection, the length and holding time of the variable virtual waiting zone need to be set reasonably. Third, the trajectory planning and grouping of CAVs in dedicated lanes also becomes complicated under oversaturated heterogeneous traffic demand. In this paper, our objectives are to solve the above problems. An oversaturated heterogeneous traffic flow signal cooperative control model based on a variable virtual waiting zone for dedicated CAV lanes is proposed. The contributions of this study include signal cycle duration optimization, variable virtual waiting zone maintenance time optimization, and CAV trajectory optimization. (1) An oversaturated heterogeneous traffic-flow cooperative signal-control model based on a variable virtual waiting zone in a dedicated CAV lane is proposed for the first time. The three-level structure of this model has coupling constraints and nested correlations. The variable virtual waiting zone can provide terminal constraints for a CAV to optimize its trajectory within the dedicated lane. The period and phase duration of the signal are based on the delay of the heterogeneous traffic flow, and the determination of the delay depends on the maintenance time of the variable virtual waiting zone and the trajectory planning of the CAV. Therefore, these three components are considered in this paper in an interrelated and comprehensive manner. (2) Different from existing dedicated CAV lanes, to improve intersection capacity and reduce vehicle delays strategy we propose a variable virtual waiting zone concept based on dedicated CAV lanes combined with signal phasing. The variable virtual waiting zone fully exploits the spatial and temporal value of intersections and organically integrates the signal and CAV network characteristics to provide redundant space for CAV trajectory planning. The remainder of this essay is structured as follows. Section 2 examines related research. The issue is discussed in Section 3 together with introduction of the variable virtual waiting zone. Section 4 presents the variable virtual waiting zone oversaturated heterogeneous traffic flow traffic signal and vehicle trajectory control model based on dedicated CAV lanes.