# Modeling the Accuracy of Estimating a Neighbor's Evolving Position in VANET
## 1 Introduction  
Vehicular ad-hoc networks (VANETs) are the driving force for realizing intelligent transport systems such as vehicular safety systems and autonomous driving systems. In these systems, accurate positioning of surrounding vehicles (i.e., neighbors) is essential to securing safety of a driver. As the technology for connected vehicles has become more widespread, periodic beaconing has become fundamental for the accurate positioning. Specifically, each vehicle periodically broadcasts a beacon message including its kinematic data (e.g., position, velocity, and acceleration). Upon receiving this message, vehicles can estimate the evolving position of the sender of the message. Several mathematical models were proposed when periodic beaconing was adopted in VANET [1; 2; 3; 4; 5; 6; 7; 8; 9]. In [1; 2], the authors proposed packet delivery ratio (PDR) and delay models in which multi-channel operation was considered. The authors in [3] focused on the possible effects of a hidden terminal problem in establishing a PDR model. In [4], the authors established an analytic model that incorporated the hidden terminal problem and general service time distribution. In the PDR model proposed in [5], a statistical channel model was considered. In [6], the authors developed a model for delay distribution on periodic beaconing. In [7; 8], an analytic model focused on 802.11p EDCAperformance was established based on Markov chain models. In [9], the authors developed an analytic model for the delay distribution of a safety message's dissemination using a white TV spectrum. The limitation of the previous models involves focusing only on network performances (e.g., PDR and delay) that fail to evaluate the accuracy of estimating a neighbor's evolving position in intelligent transport systems. Specifically, the accuracy can be quantified with an estimation error of a neighbor's evolving position, but the network performances cannot be directly translated into the estimation error. For example, as shown in Figure 1, the estimation error of a neighbor's evolving position could be different even with the same PDR. Even if PDR in Figure 1a is the same as that in Figure 1b (0.5), the error in estimating the position of VEH 2 by VEH 1 is 0.75 m in Figure 1a, whereas the estimation error is 0.5 m in Figure 1b. In [10], the authors proposed a model for tracking uncertainty when periodic beaconing was adopted. However, the tracking uncertainty is an upper bound of a tracking error rather than a specific value for the tracking error. In [11], the authors proposed a congestion control algorithm depending on estimating tracking error and channel load. However, the authors did not provide a mathematical model for tracking error but the algorithm for getting the approximate tracking error in real time. To address this limitation of previous models, in this paper, we propose a new analytic model for an estimation error of a neighbor's evolving position in VANET (The estimated position is the absolute position of the neighbor). To derive the estimation error, we consider that a vehicle estimates a neighbor's evolving position using a constant speed and position estimator (CSPE) and kinematic data of the beacon received from the neighbor. In the CSPE, the current position of a neighbor is estimated using a position and a velocity included in a received beacon. To establish a mathematical model, we first establish a model for the estimation error using a simple equation, which is associated with a probability of successful reception. Then, we obtain the probability of successful reception that will be used in the error model. To the best of our knowledge, this is the first attempt to establish an  
Figure 1: Estimation error of a neighbor’s position can be different even with the same packet delivery ratio (PDR). analytic model for evaluating the accuracy of estimating a neighbor's evolving position in VANET. To validate the proposed model, we compare numerical results of our model with those of the NS-2 simulation [12].