# 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
### Derivation of the Probability of Successful Reception  
A vehicle successfully receives a beacon when two events occur simultaneously: (1) a direct collision does not occur and (2) a hidden collision does not happen. Thus, a probability of successful reception can be calculated by [5]  
\[p_{s}=P(X_{D})\cdot P(X_{H}) \tag{6}\]  
where \(X_{D}\) is an event that a signal from a sender does not interfere with signals from any direct collisioners (DCs); \(X_{H}\) is an event in which a signal from a sender does not interfere with signals from any hidden collisioners (HCs). Here, DC refers to a vehicle that could induce direct collisions at a receiver and HC refers to a hidden vehicle that might cause hidden collisions at a receiver.  
The event \(\overline{X_{D}}\) (the complement of an event \(X_{D}\)) happens when two events occur simultaneously: (1) a sender broadcasts a beacon and (2) at least one DC transmits a signal. First, a sender can broadcast its beacon only if it has a non-empty buffer (\(\rho\)) and the channel is idle (\(1-P_{busy}\)). Second,the complement of the second event is that no DC transmits a signal (\(((1-\rho\tau)^{N_{D}})\). Hence, \(P(\overline{X_{D}})=1-P(X_{D})\) can be expressed by  
\[\begin{split}& P(\overline{X_{D}})\\ &=\rho(1-P_{busy})(1-(1-\rho\tau)^{N_{D}})\end{split} \tag{7}\]  
where \(\rho\) is the probability of a non-empty queue; \(N_{D}\) is the number of DCs and can be expressed with \(2\beta R\); \(\beta\) is a vehicle density; \(R\) is a transmission range; \(\tau\) is a transmission attempt probability and can be obtained by \(\frac{2}{\overline{CW_{min}+1}}\)[15]; and \(P_{busy}\) is the probability that a channel is busy. According to a queuing theory [16], \(\rho\) can be expressed as \(\lambda\cdot E[S]\) where \(\lambda\) and \(E[S]\) are an averaged message arrival rate and expected service time, respectively. \(E[S]\) can be obtained by  
\[\begin{split} E[S]&=T_{data}+(1-(1-\rho)(1-P_{busy }))\cdot\overline{CW}\cdot E[slot]\\ \overline{CW}&=\frac{CW_{min}+1}{2}\end{split} \tag{8}\]  
where \(CW_{min}\) is the minimum contention window size. \(E[slot]\) is an average slot time and can be obtained by  
\[E[slot]=\sigma\cdot(1-P_{busy})+T_{data}\cdot P_{busy} \tag{9}\]  
where \(\sigma\) is the duration of an empty slot time and \(T_{data}\) is the transmission duration of a beacon. \(P_{busy}\) is obtained by  
\[P_{busy}=1-(1-\rho\tau)^{N_{D}} \tag{10}\]  
The event \(X_{H}\) occurs when the following two events happen simultaneously. The first event (\(X_{H1}\)) is that all HCs must not transmit any packets when a sender starts to broadcast its beacon. The second event (\(X_{H2}\)) is that all HCs must not start their transmissions while a sender broadcasts a beacon. Hence, \(P(X_{H})\) can be calculated by  
\[P(X_{H})=P(X_{H1})\cdot P(X_{H2}) \tag{11}\]  
where \(P(X_{H1})\) is obtained by  
\[P(X_{H1})=(1-\rho\tau)^{N_{H}} \tag{12}\]  
where \(N_{H}\) is the number of HCs and can be expressed by \(2\beta R\).  
For event \(X_{H2}\), all HCs must not start their transmissions while a sender broadcasts a beacon. No HCs start their transmissions when the following two conditions are satisfied. First, no HCs have a beacon to send at the starting time of the sender's transmission. Second, no beacon arrives while a sender broadcasts a beacon. Hence, \(P(X_{H2})\) can be expressed by  
\[P(X_{H2})=(1-\rho)^{N_{H}}\cdot e^{-\lambda\cdot N_{H}\cdot T_{data}}. \tag{13}\]  
Equations (6)-(13) describe a non-linear system with unknowns \(\tau,\rho\), and \(P_{busy}\). We can solve this nonlinear system using numerical techniques such as the Newton method.