## 4 Empirical Study and Result  
### Volume-Density Characteristics  
All three scenarios above separately calculated traffic parameters for the merge lane and open lanes. Traffic density and speed are computed through conventional methods for one frame. Density is calculated as the number of vehicles divided by the length of the merge lane. And for open lanes, we used the length of the total work zone instead:  
\[K_{m}=\frac{N_{m}}{L_{m}} \tag{6}\]  
\[K_{o}=\frac{N_{o}}{L} \tag{7}\]  
where \(m\) and \(o\) denote the merge lane and open lanes, respectively. \(L_{m}\) is the length of the merge lane, and \(L\) is the total length of the road segment in aerial videos. Speed is calculated as the harmonized mean of instantaneous speeds of vehicles:  
Figure 4: The representative trajectory plots of different sites in Scenarios 1, 2, and 3. Lane 1 means the merge lane; Lane 2 means the median lane.  
\[V_{m}=\frac{1}{\frac{1}{N_{m}}{\sum_{i=1}^{N_{m}}\frac{1}{v_{i}}}} \tag{8}\] \[V_{0}=\frac{1}{\frac{1}{N_{c}}{\sum_{j=1}^{N_{o}}\frac{1}{v_{j}}}} \tag{9}\]  
where vehicle \(i\) (\(1\leq i\leq N_{m}\)) is in the merge lane, and vehicle \(j\) (\(1\leq j\leq N_{o}\)) is in open lanes. However, due to the limitations of each video's size, the traffic volume cannot be counted directly over an aggregated time interval. Depending on the relationship between traffic volume, density, and speed, the volumes are defined as follows:  
\[Q_{m}=K_{m}\times V_{m} \tag{10}\]  
\[Q_{0}=K_{0}\times V_{0} \tag{11}\]  
The flow-density relationships of all sites from observed trajectory data are shown in Figure 5. In Scenario 1, sites A1, B1, B3, and B4 were under a free flow state, but site C1 was highly congested. And in both Scenario 2 and Scenario 3, the relationship between traffic flow and density was nearly linear. Highway G36 functions as both an urban expressway and a freeway, carrying a large amount of traffic and often experiencing severe congestion during the morning and evening rush hours. However, the traffic flow is not exceptionally high in the aerial videos taken for this study due to the location of the work zones and the timing of the construction. The traffic breakdown and capacity drop phenomenon can quickly occur, and traffic conditions can deteriorate rapidly with lane closures during peak hours. But in this study, Site C1 is an exception, which offers a chance to analyze vehicle behavior under heavy traffic.  
In Scenario 2 and Scenario 3, the through lanes cover a more extensive range of regions on flow-density diagrams than in Scenario 1. When the density is 30 veh/km/lane, the flow rate range in Scenario 1 is about 700 veh/h/lane, while under the same density, the fields in Scenario 2 and Scenario 3 are about 1700 veh/h/lane and 1200 veh/h/lane, respectively. Regarding the trajectory plots in Figure 3, under similar density levels, more oscillations occur in Scenario 2 and Scenario 3 than in Scenario 1. That may be due to the behavior of merging vehicles in the merging lane. With the density increase, the behavior of merging vehicles is more likely to interfere with the speed of vehicles in the through lane, causing deceleration of subsequent vehicles and traffic oscillation.  
These findings have significant implications for work zone traffic management. For instance, the influence of merging vehicles on flow stability suggests the need for more adaptive strategies, such as early lane merging or dynamic lane controls, to mitigate disruptions in high-density scenarios. By managing traffic flow more effectively, these measures can help maintain higher throughput and minimize the risk of capacity breakdown in complex work zone environments.