# Modeling the Accuracy of Estimating a Neighbor's Evolving Position in VANET
## 2 Analytic Model for the Estimation Error of a Neighbor's Evolving Position
### System Model  
We make some assumptions for establishing an analytic model. First, it is assumed that vehicles are deployed along a one-dimensional line topology. This assumption is reasonable because the width of a lane is negligible compared to the length of a road, particularly in highway areas. Second, we assume that the arrival of a message follows the Poisson process, similarly to several previous models on periodic beaconing [3; 5; 13]. In our model, each vehicle estimates evolving positions of vehicles within transmission range. For ease of explanation, we call a vehicle within a transmission range a \"neighbor\" and a vehicle estimating an evolving position of a neighbor an \"ego vehicle.\" Similarly to [10; 11], we adopt a constant speed position estimator (CSPE) to estimate an evolving position of a neighborSpecifically, an ego vehicle calculates an evolving position of its neighbor at every beacon interval (BI) using kinematic information in the beacon and CSPE. Figure 2 illustrates how an ego vehicle estimates an evolving position of its neighbor. When receiving a beacon from N-VEH, an ego vehicle exploits position information in the beacon (e.g., the first estimation attempt in Figure 2). Failing to receive such a beacon message from N-VEH, an ego vehicle relies on a linear equation for CSPE with those most recently received (e.g., the second and the third estimation attempts in Figure 2). In our model, an ego vehicle fails to receive beacons from its neighbors when packet collision happens. We describe vehicle mobility with two random processes: (1) the number of vehicles passing a tagged point in a road (previous work on traffic flow revealed that the number of vehicle arrivals could be accurately modeled with the Poisson arrival process [14]) and (2) an acceleration from among those vehicles. For analytical tractability, we take a snapshot to gather statistics on periodic beaconing and kinematic data of vehicles at certain times, and then derive an estimation error of neighbors' evolving positions at those moments. At each snapshot, the acceleration of a vehicle follows a random variable with the mean \(a_{m}\). (In a practical scenario, we can obtain the average acceleration by averaging the acceleration information contained in the beacons received from neighbors). Here, we can use any random variable for modeling an acceleration. The benefit of this approach is that we can describe various mobility patterns by using a random variable corresponding to the mobility pattern. Using the mobility model with constant mean acceleration does not reduce the accuracy of a model because an acceleration rarely changes within a short interval in the real-world (i.e., piecewise constant). Specifically, the duration for gathering statistics of our model is short (e.g., order of a few seconds) and the abrupt changes in acceleration within this duration rarely occur in the real world. Thus, we can insist that the acceleration is constant for that short duration. To justify the mobility model with constant mean acceleration, we captured the mean acceleration over time using two mobility models (Krauss car-following model with and without lane-changing models) in SUMO. Then, we investigated the time-varying trend of the accelerations captured every 10 s. Figure 3 demonstrates that the mean acceleration is almost constant within 10 s window (black dotted box) except for some time points (red circle). However, as the proportion of those time points is very small, we can conclude that our mobility model with constant acceleration is valid.