## 3 Strategy and Model
### Intersection Signal Coordination Model Based on Time Slice Division  
For the intersection space slice, the prerequisite for allowing the virtual lane-borrowing vehicles to use the bus lane is \(T^{p}_{\text{basic}}>0\):  
\[T^{p}_{\text{basic}}=T^{q}_{\text{travel}}-\left(T^{p}_{\text{travel}}+h_{ \text{t}}\right)>0 \tag{3}\]  
When \(T^{p}_{\text{basic}}>0\), the virtual lane-borrowing vehicles in the space slice will not interfere with the normal running of the approaching bus, and the bus approach of the intersection space slice has the possibility to be open to the virtual lane-borrowing vehicles. On the premise that the above prerequisite is met, when a virtual lane-borrowing vehicle reaches the intersection, if the signal phase of the intersection is green, or the signal phase of the intersection is red and the virtual lane-borrowing vehicle can pass through the intersection before the approaching bus reaches the intersection, it is allowed to use the bus lane (bus approach) of the intersection space slice; otherwise, it is forbidden to travel in the bus lane. According to the above logic analysis, on the basis of obtaining the phase, phase remaining time and cycle of the intersection signal at the current moment, the dynamic time slice divisions of the intersection space slice are modeled for different intersection signal states. When the current signal phase is red,  
\[\text{mod}(T_{\text{travel}}^{p}-T_{\text{remain}}^{\text{red}},T_{\text{cycle}})<T _{\text{green}}? \tag{4}\]  
where mod( ) is the remainder function; \(T_{\text{cycle}}\) is the cycle of the intersection signal; \(T_{\text{remain}}^{\text{red}}\) is the remaining time of the current red phase; \(T_{\text{green}}\) is the green period duration of the intersection signal. If mod(\(T_{\text{travel}}^{p}-T_{\text{remain}}^{\text{red}},T_{\text{cycle}})<T_{ \text{green}}\), the signal phase of the intersection is green when the virtual lane-borrowing vehicles reach the intersection; thus, virtual lane-borrowing vehicles are allowed to enter the bus approach of the intersection space slice. The time slice of vehicles with priority \(p\) allowed to enter the bus approach at the current moment \(T_{\text{allow}}^{p}\) is calculated as follows:  
\[T_{\text{allow}}^{p}=\min\Bigl{\{}T_{\text{basic}}^{p}\,T_{\text{green}}- \text{mod}(T_{\text{travel}}^{p}-T_{\text{remain}}^{\text{red}},T_{\text{ cycle}})\Bigr{\}} \tag{5}\]  
If mod(\(T_{\text{travel}}^{p}-T_{\text{remain}}^{\text{red}},T_{\text{cycle}})\geq T_{ \text{green}}\), the signal phase of the intersection is red when the virtual lane-borrowing vehicles reach the intersection. \[T_{\text{travel}}^{q}-T_{\text{travel}}^{p}>T_{\text{cycle}}-\text{mod}(T_{ \text{travel}}^{p}-T_{\text{remain}}^{\text{red}},T_{\text{cycle}})+T_{\text{ queue}}? \tag{6}\]  
where \(T_{\text{queue}}\) is the queue dissipation time of the intersection approach during the red signal phase, which can be estimated as follows [29]:  
\[T_{\text{queue}}=\psi*T_{\text{red}}/(s-\psi) \tag{7}\]  
where \(\psi\) is the arrival rate of vehicles in the traffic flow direction of the intersection approach; \(s\) is the saturated flow rate of all approaches in this traffic flow direction; \(T_{\text{red}}\) is the red period duration of the intersection signal, \(T_{\text{red}}=T_{\text{cycle}}-T_{\text{green}}\). If \(T_{\text{travel}}^{q}-T_{\text{travel}}^{p}>T_{\text{cycle}}-\text{mod}(T_{ \text{travel}}^{p}-T_{\text{remain}}^{\text{red}},T_{\text{cycle}})+T_{\text{ queue}}\), the virtual lane-borrowing vehicles can pass through the intersection before the approaching bus reaches the intersection.