# Coverage, Rate, and Last Hop Selection in Multi-Hop Communications in Highway Scenarios
## 4 Numerical Results
### Different Antenna Locations  
We start with Figure 4 illustrating the data rate and mean bridge length as a function of the density of vehicles for different antenna locations. The other parameters such as the number of unidirectional adjacent lanes \(M\) and the maximum allowed number of hops \(N_{max}\) are set to \(M=2\) and \(N_{max}=\infty\), while the applied last hop selection strategy is FO. Here, we see that there is an inherent trade-off between the considered metrics--as the data rate increases with \(\lambda\), the mean bridge length decreases. This trend is observed for all the considered locations with the notable exception of windshield location, where the data rate plateaus. This behavior is explained by the fact that, at large inter-vehicle distances, the propagation through the windshield becomes impossible while the under-vehicle propagation path is not available for this location option. Furthermore, observe that the difference between bumper and engine locations in terms of the data rate is rather insignificant while there is a notable difference between these schemes from the bridge length point of view. Similar results are observed for other last hop strategies. Thus, considering both metrics altogether, we observe that the bumper antenna location is preferable. We continue our discussion with Figure 5a, where, contrary to Figure 4, we show conditional data rates and bridge length as a function of the vehicle density on the road for three considered antenna locations. The other parameters are set as \(M=2\), \(N_{max}=\infty\), while the applied last hop selection strategy is FO. Here, we see that logically the better bridge length and data rates are observed for bumper location. These effects are explained by the availability of propagation paths with better propagation conditions, specifically, the under-vehicle propagation path in case of bumper location. For example, for the windshield antenna location, the through-the-window propagation path that can be considered an alternative to the under-vehicle path for bumper location is char  
Figure 4: Trade-off between the bridge length and data rate. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Notation** & **Value** & **Description** \\ \hline \(G_{A}\), \(G_{U}\) & 17.58 dBi & Transmit and receive side gains [29] \\ \(\gamma\) & 2.1 & Path loss exponent [23] \\ \(\phi\) & 23,000 & Propagation coefficient at 0.4 m [23] \\ \(\psi\) & 3.4 & Propagation coefficient at 0.4 m [23] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._attenuation as shown in [23]. However, for larger values of the vehicle density and the windshield location, the bridge length approaches the values of the bumper location. The reason is that, for these densities, the propagation paths are dominated by direct LoS and side-reflections paths, and they have similar attenuations for windshield and bumper locations. However, as shown in Section 2, the engine location is characterized by much higher attenuation associated with the side-reflection propagation path, thus resulting in worse coverage and data rates. The number of hops is one of the most important parameters for multi-hop communications. In fact, in practical implementation, the higher the number of hops, the more complex the control signaling. As a result, the number of hops might be limited. The behavior of the data rate is further explained in Figure 5b showing the data rate and the mean number of hops as a function of the vehicle density for all the considered antenna locations with the parameters set as \(M=2\), \(N_{max}=\infty\), and last hop selection strategy is FO. As one may observe, the engine location is characterized by the least number of hops, thus requiring the least signaling overhead. Although this also translates to a better data rate, the bumper antenna locations are not far behind while showing the smaller number of hops. These trends are again related to the availability and \"quality\" of the propagation paths for different antenna locations as we discussed above.