# An Emergency Message Routing Protocol for Improved Congestion Management in Hybrid RF/VLC VANETs
## 4 Algorithm Description
### Congestion Management Algorithm
[PERSON] (LWR) partial differential equation introduced in [39; 40], which describes the continuum relationship between the traffic density \(\rho\) and the flow \(F\), i.e.,  
\[\frac{\partial\rho}{\partial t}+\frac{\partial F(\rho)}{\partial x}=0 \tag{10}\]  
where, \(\rho\) is the vehicle density and \(F(\rho)\) is the flow-density relationship (for instance, the Greenshields model). This equation allows for the characterization of how the traffic conditions propagate through a road network. To do so, a proper calibration of the flow-density relationship \(F(\rho)\) to the local road characteristics is necessary for achieving accurate predictions. **Clearance time simulation:** In our proposed model, the RSU simulates the clearance times under different strategies (i.e., reduced speed vs. rerouting). Indeed, this step is meant to estimate and compare how long it can take for different congestion mitigation strategies to clear the congestion and return the traffic flow to normal conditions. Specifically, a real-time traffic flow model is utilized to simulate the densities of the vehicles over the time and location that will result from each potential strategy, such as reduced speed limits, rerouting of vehicles to alternate routes, and changes in signal timings. Then, the predicted clearance time for each strategy is calculated by evaluating the integral of the density from the initial congested state to the density at the outflow point of the affected area. This helps quantify a key factor in evaluating the effectiveness of the strategy, i.e., how long will it take for this approach to resolve the congestion if implemented? Comparing the clearance times (\(T_{clear}\)) between the strategies allows our algorithm to objectively identify which options are expected to clear the roadway bottleneck in the least amount of time based on the traffic modeling simulations. Here, the purpose is to assign scores to the strategies based on their potential to shorten the duration of the vehicles' experience of congestion delays by de-congesting the network as rapidly as possible. Typically, for a reduced speed strategy vs. a strategy with normal speeds, we obtain the following:  
\[T_{\text{clear,reduced}}=\int_{\rho_{\text{reduced}}(x,t)}d\tau/\rho_{\text{ outflow}}\ T_{\text{clear,normal}}=\int_{\rho_{\text{normal}}(x,t)}d\tau/\rho_{\text{ outflow}} \tag{11}\]  
where, \(\rho_{\text{reduced/normal}}(x,t)\) are the densities over the time/location from the LWR model. Here, it should be noted that the real-time traffic, calendar events, and weather, are factored into the strategy selection, and the strategies are divided into tiers based on the congestion severity. These strategies are scored based on the total travel time, the emissions from the idling/stop-and-go traffic, and the road user satisfaction/stress levels. **Travel time impact scoring:** This step is to provide a metric for evaluating and comparing how much each potential congestion mitigation strategy considered by the RSU can reduce travel times. Specifically, it helps in scoring and ranking the potential strategies based on their estimated ability to bring the travel times closer to the ideal free-flow condition. The strategies that achieve the larger travel time savings will receive a higher score. As the vehicles comply with the messages, the densities in the modeling equation should decrease over time and location. This indicates a de-congestion as the bottleneck is cleared. To calculate the score of a strategy, the following equation is utilized:  
\[\text{Score}_{TT}=\Sigma(T_{\text{current}}-T_{\text{strategy}}) \tag{12}\]  
where \(T_{current}\) is the actual travel time and \(T_{strategy}\) is for each strategy from the LWR. **Emissions scoring:** This step is to evaluate and rank the potential congestion mitigation strategies based on their estimated impact on the vehicle's emissions levels during the clearance process.