# Simulation-Oriented Analysis and Modeling of
## 4 Distracted Car-Following Model
### Distracted Car-Following Model  
For this study, we selected the Intelligent Driver Model (IDM) as the basic framework to simulate distracted car-following behavior. The IDM is a classical car-following model based on driving expectations. It has been confirmed by many other studies that the IDM is one of the best performance models in common use [24; 25; 38]. The expression of IDM is as follows:  
\[a_{n}(t)=a_{max}^{(n)}\left[1-\left(\frac{V_{n}(t)}{\widetilde{V}_{n}(t)} \right)^{\beta}-\left(\frac{\widetilde{S}_{n}(t)}{S_{n}(t)}\right)^{2}\right] \tag{4}\]  
\[\widetilde{S}_{n}(t)=S_{jam}^{(n)}+max\left(0,V_{n}(t)\widetilde{T}_{n}(t)- \frac{V_{n}(t)\Delta V_{n}(t)}{2\sqrt{a_{max}^{(n)}a_{conf}^{(n)}}}\right) \tag{5}\]  
where \(a_{max}^{(n)}\) is the maximum acceleration of the target vehicle, \(\widetilde{V}_{n}(t)\) is the desired speed, \(\beta\) is a constant representing the free acceleration exponent (usually \(\beta=4\). \(S_{n}(t)\) is the distance between two vehicles measured from the front bumper of the following vehicle to the rear bumper of the lead vehicle), \(\widetilde{S}_{n}(t)\) is the desired following distance, which depends on several factors: speed (\(V_{n}(t)\)), speed difference (\(\Delta V_{n}(t)\)), maximum acceleration (\(a_{max}^{(n)}\)), comfortable deceleration (\(a_{conf}^{(n)}\) ), distance at standstill (\(S_{jam}^{(n)}\)), and desired time headway (\(\widetilde{T}_{n}(t)\)).  
As previously discussed, when investigating car-following behavior under distraction, it is necessary to consider changes in driver attributes. Within the IDM, the stimulus perceived by the driver include the speed difference (\(\Delta V_{n}(t)\)) and the distance (\(S_{n}(t)\)) to the preceding vehicle, where these parameters represent the ideal perception capabilities. However, the perception deviation becomes more significant when the driver is distracted.  
Therefore, this paper proposes introducing two parameters (\(\lambda,\ \theta\)) to describe the perception deviation in distracted drivers. Moreover, the IDM does not account for the increase in the distracted driver's reaction time, which often leads to hazardous scenarios. Consequently, this paper integrates a reaction time parameter (\(\tau\)) into the IDM. The final car-following model that introduces distraction factors is called IDM-distraction, and its expression is as follows:  
\[a_{n}(t+\tau)=a_{max}^{(n)}\left[1-\left(\frac{V_{n}(t)}{\widetilde{V}_{n}(t)} \right)^{\beta}-\left(\frac{\widetilde{S}_{n}(t)}{S_{n}(t)+\lambda}\right)^{2}\right] \tag{6}\]  
\[\widetilde{S}_{n}(t)=S_{jam}^{(n)}+max\left(0,V_{n}(t)\widetilde{T}_{n}(t)- \frac{V_{n}(t)(\Delta V_{n}(t)+\theta)}{2\sqrt{a_{max}^{(n)}a_{conf}^{(n)}}}\right) \tag{7}\]  
where \(\tau\) represents the reaction time, \(\lambda\) is the perception deviation of the following distance, and \(\theta\) is the perception deviation of the speed difference.  
Despite our efforts to categorize distracted driving behavior by road environment and levels of distraction, there remains a considerable heterogeneity within distracted behaviors. This heterogeneity may be because the sources of distraction vary for the same driver, and different drivers react differently to the same sources of distraction. This variation suggests that a fixed parameter cannot sufficiently capture this heterogeneity. Therefore, the model considers representing the three newly introduced parameters with probability distributions, which can better reflect the actual distribution of behaviors than using a single statistical value.