# Modeling the Accuracy of Estimating a Neighbor's Evolving Position in VANET
## 3 Model Validation  
To validate our model, we compare the numerical results of the proposed model with those of the NS-2 simulation [12]. In the NS-2 simulation, every vehicle broadcast its beacon periodically by following Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) from IEEE 802.11p [17]; every vehicle moved along a highway with a random acceleration; the average of accelerationwas set to be 1 m/s\({}^{2}\). We set the average of acceleration to be 1 m/s\({}^{2}\) based on experimental study in [18]. Moreover, we validated our model in other acceleration settings, which were less than 1 m/s\({}^{2}\). This is because experimental study [18] showed that the average acceleration is less than 1 m/s\({}^{2}\) when vehicle speed is higher than 10 km/h, and the speed is highly likely to be higher than 10 km/h on a highway. In both the simulation and an analytic model, we used 3 Mbps for a transmission rate, which is normally used for periodic beaconing due to its high decoding reliability [4, 5]. We adopted a channel model in [19], which considered shadowing loss caused by other cars' obstructions. Specifically, the model employs different parameters according to the existence of obstruction in the propagation path, and we validated its accuracy by comparing the numerical results of the model with the experimental measurements. Every vehicle periodically estimated positions of all its neighbors at every beacon interval (\(T_{BI}=100\) ms). The default parameter settings are summarized in Table 2. We note that we did not derive the numerical results of previous works because there was no previous work deriving the estimation error of a neighbor's evolving position. Specifically, [1, 2, 3, 4, 5, 6, 7, 8, 9] provided only network performances (e.g., PDR and delay) and [10] derived the upper bound of tracking error rather than the specific value for the tracking error. Figure 4 depicts an estimation error according to a vehicle density, with different curves that are characterized by a beacon interval. Here, a beacon interval is associated with a period for estimating a neighbor's position and a packet generation interval. In this figure, we observe that the numerical results of our model are consistent with those of the NS-2 simulation. Specifically, in all curves, the results of the model are located within 95% confidence intervals of the simulation results. We can find two interesting points in Figure 4. First, an estimation error increases with vehicle density. This is because network congestion rises as the density grows. Second, an estimation error rises with the beacon interval. This trend comes from the fact that the deviation from the actual position rises with the duration of missing a beacon from its neighbor; said duration tends to increase with the beacon interval. We can further analyze the relationship between the beacon interval and the estimation error from Equation (5), which shows that \(e^{est}\) is proportional to \(T_{BI}^{2}\). However, Figure 4 demonstrates that \begin{table}
\begin{tabular}{c c} \hline \hline
**Data Rate** & **3 Mbps** \\ \hline Duration of an empty slot time (\(\sigma\)) & 16 \(\mu\)s \\ Minimum Contention Window (\(CW_{min}\)) & 15 \\ Transmission range (\(R\)) & 450 m \\ Average acceleration (\(a\)) & 1 m/s\({}^{2}\) \\ Number of iterations & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Default parameter settings. Figure 4: Comparison between numerical results of our model and those of the NS-2 simulation as a vehicle density varies with various a beacon interval settings: (1) 100 ms and (2) 200 ms, and (3) 300 ms.  
\(e^{est}\) with \(T_{BI}=200\) ms is smaller than four times \(e^{est}\) with \(T_{BI}=100\) ms, and \(e^{est}\) with \(T_{BI}=300\) ms is smaller than nine times \(e^{est}\) with \(T_{BI}=100\) ms. This is because network congestion becomes smaller as \(T_{BI}\) rises. Hence, an ego vehicle is more likely to receive beacons from its neighbors as \(T_{BI}\) increases.