# Vehicle Speed Estimation and Forecasting Methods Based on Cellular Floating Vehicle Data
## 3 Traffic Information Estimation and Forecasting Methods
### Traffic Information Estimation Methods
#### 3.1.2 Traffic Density Estimation  
For traffic density estimation, this study considers the amount of CAs (\(a_{i}\)) and the amount of periodic location updates (PLUs) (\(p_{i}\)) to estimate traffic density (\(K_{i}\)) on the road segment covered by Cell\({}_{i}\). Traffic Density Estimation by Using CA events  
In cellular networks, a cell is supplied with radio service by vicinal BTSs or Node Bs. The BSC in GSM and RNC in UMTS are responsible for the network control. When a call arrives within the coverage area of a cell, the BSC or the RNC provides a free channel to the MS, and the MS will be connected with the corresponding base station if there is a free channel. Figure 3a depicts the scenario diagram for vehicle movement and call arrivals on the road. Figure 4b shows the timing diagram for call arrivals on the road segment covered by a specific Cell\({}_{i}\). The MS in a car moving along the road performs the first call at time \(t_{0}\) (in Figure 3a,b) and enters the coverage area of Cell\({}_{i}\) at time \(t_{1}\) (in Figure 3a,b). The MS performs another call at time \(t_{2}\) (in Figure 3a,b)  
Figure 2: The scenario diagram of location update events. LA, location area. before leaving Cell\({}_{i}\) (time \(t_{3}\) in Figure 3a,b), and the call performed by the MS is called a call arrival in Cell\({}_{i}\). This study assumes that the call inter-arrival time (\(\tau\)) is exponentially distributed with the mean \(1/\lambda\) to generate call arrivals [25]. The length of road segment covered by Cell\(i\) is \(l_{i}\), and the average speed of cars is \(U_{i}\). Therefore, \(l_{i}/U_{i}\) denotes the time difference between \(t_{1}\) and \(t_{3}\). This study analyzes the amount of call arrivals (\(a_{i}\)) that occur between \(t_{1}\) and \(t_{3}\) on the road segment covered by Cell\({}_{i}\). The amount of call arrivals can be expressed as Equation (5), and Equation (6) indicates that this study can derive estimated traffic density (\(k_{i,a}\)) from the amount of call arrivals. \[ \begin{array}{ll}a_{i}&=Q_{i}\times\Pr\left[t_{1}<t_{2}<t_{3}\right]\\ &=Q_{i}\times\int_{x=0}^{\infty}\Pr\left[x<\tau<x+\frac{l_{i}}{U_{i}}\right] dx\\ &=Q_{i}\times\int_{x=0}^{\infty}\sum_{\tau=x}^{x+\frac{l_{i}}{U_{i}}} \lambda e^{-\lambda\tau}d\tau dx\\ &=Q_{i}\times\frac{1-e^{-\frac{\lambda l_{i}}{U_{i}}}}{\lambda} \Rightarrow& Q_{i}\left[1-\left(1-\frac{\lambda a_{i}}{Q_{i}}\right)\frac{1}{ \lambda l_{i}}\right]\approx\frac{Q_{i}}{U_{i}}=K_{i},\text{ where }\lim_{U_{i}\to\infty}1-e^{-\frac{1}{U_{i}}}= \frac{1}{U_{i}}\end{array}  \tag{5}\]  
Figure 3: (**a**) The scenario diagram for vehicle movement and call arrivals on the road; (**b**) the timing diagram for call arrivals on the road segment covered by Cell\(i\). \[k_{i,a}=q_{i}\left[1\text{-}\left(1\text{-}\frac{\lambda a_{i}}{q_{i}}\right) \frac{1}{\lambda l_{i}}\right]\approx\frac{Q_{i}}{U_{i}}=K_{i}\text{, where }\lim_{U_{i}\rightarrow\infty}1-e^{-}\frac{1}{U_{i}}=\frac{1}{U_{i}} \tag{6}\]  
Traffic Density Estimation by Using PLU Events  
For mobility management, the MS periodically performs the PLU event for the core network in cellular networks. Some MSs may perform the PLU event across the cell in Figure 2. Therefore, the probability models shown in Equations (7) and (8) are proposed to provide estimated vehicle density \(k_{i,p}\) (_i.e._, the number of MS) on the road segment covered by Cell\({}_{i}\) according to the number of PLU events. This model had been proven in [24; 38], and this study summarizes the assumptions in this model as follows. * The actual vehicle density \(K_{i}\) and traffic speed \(U_{i}\) can be obtained from VD on the road. Furthermore, the length of a road segment covered by the cell is \(l\). * The call arrival rate to a cell is \(\lambda\), and the call arrival process is assumed to be a Poisson process. * The cycle time of PLU is \(b\), and the number of PLU events is \(p_{i}\).