# Queue Spillover Management in a Connected Vehicle Environment
## 3 Intersection Signal Control Modeling
### Intersection Queue Overflow Judgment  
The maximum phase control distance was used as the standard to determine whether the current phase vehicle queue will overflow. The initial queue length was compared with the maximum control distance between the phase, to determine whether the current phase allowed vehicles to pass through the intersection. The following two situations were considered. 1. \(L_{0}<L_{c}\)  
That is, the initial queue length is less than the phase maximum control distance. This situation indicates that the current phase allows the vehicle to pass through the intersection, and the green time may still be in surplus after the queuing vehicles pass the intersection. That is, (6) satisfies \(\raisebox{-1.0 pt}{\text{\textcircled{1}}}\)\(S_{n}(0)=-L_{0}\)\(\raisebox{-1.0 pt}{\text{\textcircled{2}}}\)\(S_{n}(t_{0})=0\). Then the time of initial queue length through intersection can be calculated. \[t_{0}=\left\{ \begin{array}{ll}\sqrt{\frac{2l_{0}}{a}}+t_{n}&t\leq t_{n}^{ \text{max}}\\ \frac{\left(a\cdot t_{n}-v_{n}^{\text{max}}\right)+\sqrt{\frac{v_{n}^{\text{ max}}-2}{a}\cdot{a}\cdot{v}\frac{v_{n}^{\text{max}}}{a}+2\cdot{a}\cdot{l}_{0}+2 \cdot{v}\frac{v_{n}^{\text{max}}}{a}}}{a}&t>t_{n}^{\text{max}}\end{array} \right. \tag{9}\]  
The remaining time \(t_{s}\) of the green light is as follows:  
\[t_{s}=t_{g}-t_{0}\quad(t_{s}\geq 0) \tag{10}\]  
In the remaining time, to ensure that as many vehicles as possible go through the intersection, the vehicle in the rear will be able to implement the vehicle speed guidance. When the vehicle speed is guided, it is necessary to judge whether the vehicle can pass the intersection at the current vehicle speed, and the speed of the approaching \(i\)-th car is \(v_{i}\). There are,  
If \(v_{i}\cdot t_{s}\geq L_{i}\), the current speed at a constant speed can be used to cross the intersection;  
If \(v_{i}\cdot t_{s}<L_{i}\), the current speed cannot meet the requirements of the vehicle through the intersection, the vehicle should be accelerated to \(v_{G}\), to ensure that it can safely pass the intersection. The setting of guide speed needs to be satisfied:  
\[\frac{v_{G}^{2}-v_{i}^{2}}{2a}+v_{G}\cdot t_{s}-\frac{v_{G}-v_{i}}{a}\cdot v_{ a}\geq L_{i} \tag{11}\]  
From (11), the speed \(v_{G}\) can be obtained as follows:  
\[v_{G}\geq\frac{\left(2\cdot a\cdot t_{s}+2\cdot v_{i}\right)+\sqrt{\left(2 \cdot a\cdot t_{s}+2\cdot v_{i}\right)^{2}-4\left(v_{i}^{2}+L_{i}\right)}}{2} \tag{12}\]  
It is important to note that the guide speed limited condition is \(v_{G}<v_{n}^{\text{max}}\). 2. \(L_{0}>L_{c}\)  
That is, the initial queue length is more than the phase maximum control distance. When this situation occurs, the current phase is not sufficient to adjust the vehicle through the intersection. At this point, it is necessary to judge whether the queue length is overflowing. Queuing vehicles through the intersection of computing time with (9), the calculated time is represented as \(t_{o}^{\prime}\). When the number of queuing vehicles is greater than the largest phase control vehicles, signal controller starts overflow decision function. Then calculate the vehicle's biggest difference between queue length and phase control of each cycle. If the difference value is negative, regard as zero, and save the difference analysis on vehicle queue change trend. The specific steps are as follows:  
* The maximum control distance of intersection phase is assumed as \(L_{c}\). The measured vehicle queue length of each cycle was \(L_{r1}\), \(L_{r2}\), \(L_{r3}\),... \(L_{ri}\). The queue length of the vehicle can be calculated by the position coordinates of the vehicle at the end of the queue; then get the vehicle queue length and maximum control distance differential record, as shown in Table 1.
* In order to prevent the overflow of road queues, we need to find an optimal overflow control threshold value, which needs to be obtained according to the actual traffic conditions of the intersection roads.