## 3 Problem Statement
### Downstream Queue Area of the Calibrated CTM Method  
Considering the limitations of traditional CTMs, we modify the three sub-cells \(\lambda_{b}\), \(\lambda_{c}\), and \(\lambda_{d}\) into independent cells \(\lambda-2\), \(\lambda-1\), and \(\lambda\), respectively. Figure 3 illustrates the modified downstream queues of link \(a\). Moreover, the maximum number of vehicles that can occupy cell \(i\) of link \(a\) is specified as \(N_{a}^{i}(t)=20\), where \(i\in\{1,2,\ldots,\lambda\}\). However, we notice that the upstream cell has two lanes, whereas the downstream cell has three lanes, but the capacity of the lanes to accommodate vehicles remains unchanged. The definition of the traditional CTM method does not appear to consider the number of different lanes. Therefore, we try to modify \(\mathrm{N_{a}^{i}(t)=20,i\in\{1,2,\ldots,\lambda\}}\) into \(\mathrm{N_{a}^{i}(t)=20,\ i\in\{1,2,\ldots,\lambda-3\}}\) and \(\mathrm{N_{a}^{i}(t)=10,\ i\in\{\lambda-2,\lambda-1,\lambda\}}\). Three cases of jammed cells are discussed below. Specifically, with \(\lambda=11\), we assign \(\lambda-2,\lambda-1\), and \(\lambda\) to the left-turn lane, straight lane, and right-turn lane, respectively. Figure 3: The calibrated downstream queues area of link \(a\). Figure 2: The process of jam growth caused by traffic jammed on Link \(n\). Case I: Left-Turn Lane Jammed  
Figure 4a shows that once downstream cell 9 is jammed, we stop upstream cell 8 from allocating vehicles to enter downstream cell 9. Based on the proportion of vehicles turning left, turning right, and going straight, we allow 50% of the upstream vehicles to be allocated to the downstream straight lane and 30% to the right-turn lane. Note that this allocation rule is different from the strategy used in the traditional CTM method. Case II: Straight Lane Jammed  
Figure 4b shows that once downstream cell 10 is jammed, we stop upstream cell 8 from allocating vehicles to enter downstream cell 10. Then, we allow 20% of the upstream vehicles to be allocated to downstream cell 9 (i.e., the left-turn lane) and 30% to downstream cell 11 (i.e., the right-turn lane). Case III: Right-Turn Lane Jammed  
Figure 4c shows that once downstream cell 11 is jammed, we stop upstream cell 8 from allocating vehicles to enter downstream cell 11. Then, we allow 50% of the upstream vehicles to be allocated to the downstream straight lane (i.e., cell 10) and 20% to the left-turn lane (i.e., cell 9). Based on the traditional CTM method, if congestion occurs in any of the three lanes (i.e., left-turn, right-turn, or straight lane) when vehicles are moving forward downstream, the method restricts the upstream vehicles from entering downstream cell 9. This highlights an important relationship between the upstream and downstream cells. When vehicles move from the upstream cell (\(\lambda-3\)) to the downstream cells (\(\lambda-2\), \(\lambda-1\), or \(\lambda\)), two key factors must be considered: the vehicle density in each downstream cell and the number of vehicles in the upstream cell. To better reflect real-world road conditions, we appropriately calibrate the parameter \(\varphi_{ax}\). Figure 2 illustrates how congestion in any downstream lane can lead to congestion in the entire upstream lane. However, unlike the traditional CTM method, our improved approach allows vehicles to move into uncongested lanes. Even if two of the three lanes are jammed, vehicles can still enter the remaining open lane. Notably, the traditional CTM method would restrict movement into any downstream lane, even if it remains uncongested. To illustrate how our new CTM method operates, we designed a pseudocode to explain the updated vehicle allocation rules, as shown in Algorithm 2. This pseudocode shows that if the capacity of any downstream cell is not full, i.e., \(n_{a}^{\lambda}\), \(\lambda\in[9,\,10,\,11]\), then the upstream cell (cell 8) will allocate vehicles accordingly. Notably, this new CTM method differs significantly from the traditional approach.