## 5 Simulation and Results
### Scenario Design and Network Specifications  
The simulation scenario is based on hypothetical variations to test the robustness of our hierarchical Stackelberg model under different traffic conditions. These scenarios include normal traffic conditions, emergency vehicle priority situations, and peak congestion times. Normal condition scenario reflects the typical weekday traffic patterns during morning peak hours, while emergency priority scenarios simulate the network traffic flow of emergency vehicles. Peak congestion scenarios model high-density traffic with queues to evaluate system performance under pressure. The simulated urban subnet covers an area of 4 km\({}^{2}\) and represents mixed-use areas for commercial and residential activities. The network consists of 12 signal crossings and 24 bidirectional road segments. This configuration enables us to study the interaction between emergency vehicles and regular vehicles in a controlled but representative urban environment. The network includes three functional road classes: arterial, collector, and local access roads. Arterial roads make up 60% of the network, with a speed limit of 60 km/h and three routes in each direction. Collector roads account for 30%, 40 km/h speed limit, and two lanes per direction. Local access roads constitute the remaining 10% with a speed limit of 30 km/h and a lane per direction. The intersections are primarily signaled (12 of which are in total) using adaptive time-controlled active signals, while four unsigned intersections are modeled with priority rules. Traffic demand is synthesized to reflect real urban conditions. Origin Destination matrices (ODs) are constructed to represent baseline traffic flows, with peak time demand reduced to 120% of baseline to simulate congestion. Emergency vehicles are introduced at 5-10 vehicles per hour, representing 2% of total traffic flow, while regular vehicles are 1800-2400 vehicles per hour and vary by scenario. Turn ratios at intersections are set at 25% left, 50% straight, and 25% right to mirror typical urban traffic behavior.  
Traffic Flow Modeling Assumptions  
The speed-flow relationship used in our model is based on the Macroscopic Traffic-Flow Formula of Greenshields, which assumes a linear relationship between the vehicle speed (\(v\)) and the traffic density (\(k\)). This relationship is mathematically expressed as:\[v=v_{f}\left(1-\frac{k}{k_{j}}\right) \tag{11}\]  
where \(v_{f}\) represents the free-flow speed (i.e., the maximum speed possible under low traffic density), \(k_{j}\) represents the jam density (critical traffic density), and \(k\) represents the current vehicle density per unit length.  
The formula allows a direct estimate of traffic flow (\(q\)) as the product of density and speed:  
\[q=k\cdot v_{f}\left(1-\frac{k}{k_{j}}\right) \tag{12}\]  
With increasing density, the speed gradually decreases, eventually reducing the flow rate and simulating the beginning of congestion. The Greenshields model is chosen for its empirical relevance and analytical utility, especially in urban traffic systems, and is a standard reference in traffic engineering literature.  
In terms of road capacity and free-flow speed, identifying the peak of the speed curve determines the maximum theoretical capacity of a road segment (\(q_{\max}\)), which is given by:  
\[q_{\max}=\frac{v_{f}\cdot k_{j}}{4} \tag{13}\]  
The capacity values are determined according to established standards such as the _Highway Capacity Manual (HCM)_, which estimates about 1900 vehicles per hour per lane for urban arterials. These values were then refined by empirical adjustments, considering contextual factors such as traffic signal timing and pedestrian interference, based on local traffic surveys conducted at key intersections.