## 3 Multimetric Map-Aware Routing Protocol (MMMR)
### Metrics Evaluation
Figure 1:0 Figure 1:1  
where \(\langle\cdot,\cdot\rangle\) refers to the scalar product. Notice that comparing Equation (10) to the [PERSON] approximation in Equation (9), where \(d(0)^{\alpha}=d(\vec{x},\vec{x}_{D})^{\alpha}=\parallel\vec{x}-\vec{x}_{D} \parallel^{\alpha}\) (see Equation (2)), we obtain:  
Figure 1:2  
Applying the chain rule in \(\frac{d}{dt}(d(t)^{2})^{\frac{\alpha}{2}}\) yields:  
Figure 1:3At \(t=0\), we have \(d(0)=d\) and \(\left.\frac{d}{dt}d(t)^{2}\right|_{t=0}=2\cdot\langle\vec{x}-\vec{x}_{D},\vec{v}\rangle\). Hence,  
Figure 1:4  
Let \(\theta\) be the angle between vectors \(\vec{x}-\vec{x}_{D}\) and \(\vec{v}\cdot t\); see Figure 2a. On account of the fact that \(\langle\vec{x}-\vec{x}_{D},\vec{v}\rangle=d\cdot\|\vec{v}\|\cdot\cos\theta\), we have:  
Figure 1:5 Figure 1:6  
Replacing Equation (14) in Equation (9), we obtain:  
Figure 1:7  
Next, comparing Equation (15) to Equation (7), we achieve an expression for \(\Delta_{d}(t)^{\alpha}\), _i.e._, our proposed measure of the variation of the distance in a time, \(t\), to the power of \(\alpha\) with respect to the current distance in \(t=0\) to the power of \(\alpha\). Figure 1:8  
Equation (15) can be divided into two parts: the current distance, \(d\), to the power of \(\alpha\), and our measure of the future position, \(\Delta_{d}(t)^{\alpha}\). Finally, replacing Equation (16) in Equation (8), we obtain an equation for the metric of the trajectory of vehicles, where we set \(t=1\) s as the interval of time to evaluate the trajectory of the vehicle, \(i\):  
Figure 1:9  
Due to the exponential behavior of metrics \(u_{1,i}(d_{i})\) and \(u_{2,i}(d(t),d(0))\), we decided to use exponential functions also to define the metrics of density \(u_{3,i}\) and available bandwidth \(u_{4,i}\). This way, we avoid having an excessive influence of metrics \(u_{1,i}\) and \(u_{2,i}\) in the final value of the multimetric score, because these metrics could obtain high values that might affect the final score. **Density**: This is computed as the number of vehicles in the list of neighbors (\(N_{neigh}\)) of each node at the moment, \(t\), of sending the current hello message, divided by the transmission range (\(Tx\)) of the node. The list of neighbors is composed of vehicles in transmission range that send packets with enough power to be considered stable neighbors. Each node, \(i\), computes its current density of nodes \(\rho_{i}(t)\) and includes it in the next hello message. \[u_{1,i}(d_{i})=e^{-\left(\frac{d_{i}}{d_{ref}}\right)^{\alpha}} \tag{3}\]0  
According to Equation (19), the algorithm gives a higher score (bounded by one) to this metric when the node has a higher value of the density of nodes, \(\rho_{i}\). Nodes with a denser area in the transmission range will have more possibilities to forward the packet to a better next node. This is true until reaching a maximum traffic density, above which, the too high number of vehicles in the surrounding area of each node increases the frequency of collisions. This maximum traffic density will depend on the specific scenario and configuration parameters and will be determined in each simulation. We evaluate our proposal in two scenarios with different vehicle densities, to be able to analyze the performance of the next forwarding selection used by our protocol. Our approach to calculate the density metric uses a simple algorithm that gives a higher score to those nodes with higher vehicle densities. Moreover, we leave to the available bandwidth metric the task of dealing with the data congestion problem. For instance, a node with a high vehicle density metric that has a congestion problem (_i.e._, a low available bandwidth) will be highly rated by the density metric, but will also be penalized by the available bandwidth estimator (ABE) metric. In this way, we face the trade-off between density and congestion.