## 4 Simulation of Traffic Jam Propagation and Dispersion
_Example 1_[34]:  
This example illustrates the differences between traditional vehicle allocation rules 6 and our new allocation rules 6. We analyze traffic jam formation and dispersion using three strategies, i.e., no strategy, \(Q\)-strategy, and \(H\)-strategy, under both the traditional and calibrated CTM methods. In this case, an accident occurs between the 301 st and 1500 th time intervals. Here, \(d_{A}\) = 1.0 with \(d_{B}\) = 0.6 and \(d_{C}\) = 0.8 are for the _Q-strategy_, and \(d_{A}\) = 1 and \(d_{B}\) = \(d_{C}\) = 0.8 are for the _H-strategy_. Figure 6a shows traffic jam propagation and dispersion using the traditional CTM method with \(t_{d}\) = 0.75, based on parameters from [PERSON] et al. [17]. For comparison, Figure 6b presents the simulation results for both the traditional and calibrated CTM methods, also with \(t_{d}\) = 0.75. A key observation is the significant difference between the blue dashed and solid lines. The dashed line shows that the traffic jam starts at the 343 rd time interval and peaks at 421 jammed cells by the 1513 th interval. In contrast, the solid line shows a jam initiation at the same time but with a peak of only 48 jammed cells at the 1486 th interval. For the Q1-strategy, Q2-strategy, and H-strategy, traffic jams start at the 341 st, 338 th, and 337 th time intervals, respectively, with peak jammed cells of 37 (1460 th interval),  
Figure 6: (**a**) Propagation and dispersion of traffic jams using the traditional CTM method (\(t_{d}\) = 0.75); (**b**) Propagation and dispersion of traffic jams using both the traditional and calibrated CTM methods. (**a**) [34]. (**b**) [34]. 36 (1453 rd interval), and 19 (1386 th interval). The results indicate that the H-strategy (\(d_{A}=1\) and \(d_{B}=d_{C}=0.8\)) outperforms the other two Q-strategy conditions (\(d_{A}=1.0\) with \(d_{B}=0.6\); 0.8). Overall, any controlled strategy significantly reduces the congestion duration compared to having no strategy. Additionally, Table 2 confirms that the H-strategy is more effective than both the Q1- and Q2-strategies. _Example 2_:  
The two vehicle allocation rules, along with \(t_{d}=0.8\), are simulated to analyze traffic jam formation and dispersion under identical accident conditions. Figure 7 presents four simulation results, where the traffic jam starts at the 339 th time interval with no strategy and peaks at 481 jammed cells by the 1515 th interval. Notably, it takes over 4000 time intervals for dispersion. Similarly, both CTM methods with \(t_{d}=0.8\) are simulated together, with the results shown in Figure 7.  
The blue dashed line in Figure 7 shows that the traffic jam starts at the 339 th interval and peaks at 481 jammed cells at the 1515 th interval. In contrast, the blue solid line indicates a later onset at the 342 nd interval, with a significantly lower peak of 73 jammed cells at the 1516 th interval. For the Q1-strategy, Q2-strategy, and H-strategy, traffic jams begin at the 338 th, 333 rd, and 332 nd intervals, respectively, with peak jammed cells of 49 (1479 th interval), 47 (1476 th interval), and 33 (1461 st interval). The results confirm that the H-strategy outperforms the Q1- and Q2-strategies in traffic jam dispersion. Overall, any controlled strategy significantly reduces the congestion duration compared to having no strategy. Additionally, Table 3 indicates that the H-strategy consistently delivers the best performance, regardless of whether \(t_{d}=0.75\) or \(t_{d}=0.8\).