## 3 Problem Statement
### Downstream Queue Area of the Traditional CTM Method  
In previous studies [10; 14; 27; 28], a specific cell (\(\lambda=9\)) was divided into three sub-cells (\(\lambda_{b}\), \(\lambda_{c}\), and \(\lambda_{d}\)) to accommodate vehicles going straight, turning left, and turning right, respectively. A pseudocode illustrating the vehicle allocation rule of the traditional CTM method is shown in Algorithm 1. ```
1:\(N_{i}(t+1)=N_{i}(t)+\text{inflow}_{i-1}(t)-\text{outflow}_{i}(t)\)
2:\(N_{i}(t)=N_{i}(t)+\text{inflow}_{i-1}(t)-\text{outflow}_{i}(t)\)
3:for\(i=1,\ldots,N_{i}(t)\)do
4:\(N_{i}(t)=N_{i}(t)+\text{inflow}_{i-1}(t)-\text{outflow}_{i}(t)\)
5:for\(i=1,\ldots,N_{i}(t)\)do
6:\(N_{i}(t)=N_{i}(t)+\text{inflow}_{i-1}(t)-\text{outflow}_{i}(t)\)
7:endfor
8:\(N_{i}(t)=N_{i}(t)+\text{inflow}_{i-1}(t)-\text{outflow}_{i}(t)\)
```  
**Algorithm 1** Downstream Queue Area of the Traditional CTM Method  
This pseudocode indicates that if any of the three sub-cells reaches full capacity, the upstream cell (\(\lambda=8\)) will cease allocating vehicles to any of the three sub-cells. For convenience, we use the notation \(\varphi_{\lambda\in\{\lambda\},\lambda_{c},\,\lambda_{d}\}\) to replace \(\varphi_{ab}\), \(\varphi_{ac}\), and \(\varphi_{ad}\). In this context, \(N\varphi_{\lambda}\) denotes the highest number of vehicles that can be held in cell \(\lambda\) during the current time interval. For example, i.e., \(N=20\) and \(\varphi_{\lambda c}=0.2\), \(\varphi_{\lambda b}=0.5\), and \(\varphi_{\lambda d}=0.3\), if \(n_{a}^{\lambda c}=N\times\varphi_{\lambda c}\), \(n_{a}^{\lambda b}=N\times\varphi_{\lambda b}\), or \(n_{a}^{\lambda a}=\)\(N\times\varphi_{\lambda d}\), then \(y_{a}^{\lambda}=0\). This means that the upstream cell will stop allocating vehicles to the downstream cell (\(\lambda=9\)). In other words, if \(n_{a}^{\lambda c}<N\times\varphi_{\lambda c}\), \(n_{a}^{\lambda b}<N\times\varphi_{\lambda b}\), and \(n_{a}^{\lambda a}<\)\(N\times\varphi_{\lambda d}\), then \(y_{a}^{g}\
eq 0\). This implies that the upstream cell will continue allocating vehicles to the downstream cell. This suggests that the left-turn lane has not yet reached a congested state, allowing vehicles to enter freely. ```
for\(\lambda\) in {\(\lambda\_b\), \(\lambda\_c\), \(\lambda\_d\)}: if n_a^\(\lambda<\) N_\(\varphi\_\lambda\): total_ratio = \(\varphi\_\)ac + \(\varphi\_\)ab + \(\varphi\_\)ad if total_ratio == 1.0: allocate_vehicles(n_a^\(\lambda\)* \(\varphi\_\)ac, n_a^\(\lambda\)* \(\varphi\_\)ab, n_a^\(\lambda\)* \(\varphi\_\)ad) elif total_ratio == 0.0: trigger_congestion_control() else: raise InvalidRatioError else: restrict_flow(\(\lambda\))
```  
**Algorithm 1** The vehicle allocation rule of the traditional CTM. According to the vehicle allocation rules of the traditional CTM method, once link \(c\) becomes congested, the left-turn lane (\(\lambda_{c}\)) will quickly reach its capacity. As a result, vehicles in the upstream cell will no longer be able to enter the left-turn lane, leading to congestion across the entire link \(a\). Similarly, if link \(b\) or \(d\) is congested, it will cause \(\lambda_{b}\) or \(\lambda_{d}\) to reach full capacity, ultimately resulting in congestion of the entire link \(a\). This indicates that under the traditional CTM method, congestion in any one of the three sub-cells can propagate and block the entire link \(a\), which does not align with actual traffic conditions. Therefore, we believe that the traditional CTM method needs to be calibrated to better reflect real-world road traffic dynamics. To better explain the challenges associated with the traditional CTM method, we will first outline the reasons why calibration is necessary. Using Figure 2 as an example, we illustrate the need for calibration. Suppose a traffic road (link \(n\)) is blocked due to an accident occurring in the fifth cell of link \(n\). We then calculate the number of jammed cells with parameters \(\varphi_{ac}=0.2\), \(\varphi_{ab}=0.5\), and \(\varphi_{ad}=0.3\).