## 3 Strategy and Model
### Basic Time Slice Division Strategy  
Before dividing the time slices for space slices, it is first necessary to determine whether the cars (virtual lane-borrowing vehicles) have the right to use the bus lane at the current moment, where the key is to compare the travel time between the virtual lane-borrowing vehicles and the approaching bus to the end of the space slice at the current moment. The right-of-way priority of the virtual lane-borrowing vehicles is defined as \(p\) and assume that they are located at the beginning position of the space slice (the position of point \(a\) in Figure 4) and try to drive on the bus lane. The right-of-way priority of the approaching bus is defined as \(q\) (\(q>p\)) and its position determines whether the virtual lane-borrowing vehicles can use the bus lane of this space slice, where the approaching bus can be identified by floating vehicle data. If the virtual lane-borrowing vehicles can pass through the space slice before the approaching bus reaches the end position of the space slice (the position of point \(b\) in Figure 4), the virtual lane-borrowing vehicles are allowed to use (or borrow) the bus lane of this space slice, i.e., the lane-borrowing assumption is tenable under the assumption that there is no influence of the intersection signal; otherwise, the virtual lane-borrowing vehicles are forbidden to travel in the bus lane. This relationship can be formulated as follows:  
\[\left\{ \begin{array}{ll}\text{Vehicles with priority $p$ are allowed to borrow the bus lane,}&\text{when}T_{\text{travel}}^{q}>T_{\text{travel}}^{p}+T_{\text{min}}+h_{ \text{t}}\\ \text{Vehicles with priority $p$ are forbidden to borrow the bus lane,}&\text{when}T_{\text{travel}}^{q}\leq T_{\text{travel}}^{p}+T_{ \text{min}}+h_{\text{t}}\end{array} \right. \tag{1}\]  
where \(T_{\text{travel}}^{q}\) and \(T_{\text{travel}}^{p}\) are the predicted travel time of the approaching bus and the virtual lane-borrowing vehicles traveling to the end position of the space slice at the current moment, respectively, including the vehicle lane travel time, lane-changing time, delay time at bus stops, and intersections, etc. A segmented travel time prediction method can be used here, which is described in [19], and the specific details are no longer detailed. the minimum time slice, which is equivalent to the minimum green time of intersection signals. The purpose of setting is to prevent the fragmentation of the lane control signal of the space slice caused by too small time slice division. The parameter design details of the minimum time slice are described in the reference [35], which is set to 11 s here. \(h_{t}\) is the minimum safe headway when vehicles change lanes, which is set to 2 s according to the relevant literature [36]. It is obvious that the prediction accuracy of vehicle travel time is related to the accuracy of lane system control. Generally, a longer space slice will increase the fluctuation range of the vehicle travel time in the space slice, which will lead to a larger prediction error of the vehicle travel time and a decrease in the effectiveness of lane utilization. Therefore, an appropriate space slice length should be set within the acceptable range of prediction error, which is discussed in Section 3.3. The division method of dynamic time slices (i.e., the duration of time allowed to enter the bus lane) for vehicles with priority \(p\) is as follows. 1. Identify the position of the high-priority vehicle (i.e., the approaching bus) in the upstream area at the current moment that is closest to the beginning position of the space slice. 2. Predict the travel time of the approaching bus and the virtual lane-borrowing vehicles traveling to the end position of the space slice at the current moment and substitute the prediction result into Equation (1) to determine whether the virtual lane-borrowing have the right to use the bus lane of the space slice at the current moment. 3.