Simulation-Oriented Analysis and Modeling of Distracted Driving

Abstract

Distracted driving significantly affects the efficiency and safety of traffic flow. Modeling distracted driving behavior in microscopic traffic flow simulation is essential for understanding its critical impacts on traffic flow. However, due to the influence of various external factors and the considerable uncertainties in behavior characteristics, modeling distracted driving behavior remains a challenge. This study proposed a model which incorporates distraction features into the microscopic traffic flow model to simulate distracted driving behavior. Specifically, the study first examines the characteristics of distracted driving, including the intervals and durations of distraction events, as well as the patterns and environments of distraction. It then introduces distraction parameters into the Intelligent Driver Model (IDM), including reaction time delays and perception deviations in both speed difference and following distance. These parameters are quantified by probabilistic distributions to reflect the uncertainty and individual differences in driving behavior. The model is calibrated and validated using 772 distracted following events from the Shanghai Naturalistic Driving Study (SH-NDS) data. Three patterns of distraction (excessive, moderate, mild) are distinguished and modeled separately. The results show that the model’s accuracy surpasses that of the IDM under various road types and traffic volumes, with an average improvement in model accuracy of about 11.30% on expressways with high traffic volume, 4.54% on expressways with low traffic volume, and 4.46% on surface roads. Meanwhile, the model can effectively simulate the variations in reaction times and perceptual deviations in both speed and following distance for different distraction modes at the individual level, maintaining consistency with reality. Finally, the study simulates distracted driving behavior under different road environments and traffic volumes to explore the impact of distracted driving on traffic flow. The simulation results indicate that an increase in the proportion of distraction reduces the efficiency and safety of traffic flow, which is consistent with real-world observations. Since the model considers human distraction factors, it can generate more dangerous driving scenarios in simulations, which holds significant importance for safety-related research. The findings from this study are expected to be helpful for understanding distracted driving behavior and mitigate its negative influence on the efficiency and safety of traffic flow.

1. Introduction

Efficiency and safety of traffic flow are two critical aspects and key challenges in traffic system management. In numerous engineering practices and scientific research activities, simulations based on microscopic traffic flow models are frequently employed to analyze traffic flow under various conditions. These models excel at simulating macroscopic characteristics of traffic flow, such as speed, density, and volume. However, traditional microscopic traffic flow models often fall short in their ability to simulate the microscopic driving behaviors that significantly impact traffic flow safety and efficiency. This limitation makes it challenging to use traditional microscopic traffic flow models to deeply analyze the mechanisms underlying traffic flow safety and efficiency. Consequently, there is a pressing need for models that can more accurately reflect the complexities of driver behaviors to improve the analysis and understanding of traffic flow dynamics.

Human factors are considered crucial in contributing to driving behavior, which affects traffic efficiency and safety. Studies have shown that drivers’ risk attitudes, driving preferences, and other personal traits play a central role in various traffic flow phenomena that significantly affect traffic efficiency [1,2], and human factors are the main causes of over 90% of traffic crashes [3,4,5]. These findings highlight the necessity of incorporating human factors into microscopic traffic flow models to more accurately simulate microscopic driving behavior, and thereby enhance the understanding of efficiency and safety of traffic flow [6,7,8,9,10,11,12,13,14,15,16].

Among human factors, distraction is a significant factor that poses a considerable threat to traffic flow. Distracted driving can cause driving performance degradation, traffic flow delays, oscillations, and other instability phenomena [17,18,19]. Distraction has been proven to affect both lateral and longitudinal driving performance. Several studies have investigated the impact of distraction on lateral performance, such as lane-keeping ability. Kantowitz et al. [20] demonstrated through simulations that the standard deviation of lane position increases when drivers are distracted. Lansdown et al. [21] found that distraction leads to an increase in lane exceedance (vehicles deviating from the lane) and lane deviation (tracking errors). Blanco et al. [22] also showed that high-information-density distraction tasks result in an increased number of lane deviations. Burns et al. [23] discovered that drivers performing distraction tasks (such as calibration and navigation tasks) exhibit a greater average deviation in lane-change paths compared to when they are not distracted. Additionally, numerous studies have demonstrated that distraction significantly impacts longitudinal car-following performance. Blanco et al. [22] found that during distraction tasks, the stability of drivers’ speed and following distance deteriorates. Lansdown et al. [24] showed that distraction results in significant compensatory speed reduction and adversely affects vehicle performance, such as significantly reduced headway and increased braking pressure. Ranney et al. [25] found that drivers increase following distance to compensate for the increased task demands associated with distraction. Distraction tasks also significantly degrade vehicle control, target detection, and car-following performance.

Approximately 14% of traffic crashes can be attributed to distraction [26]. In the United States, about 20,000 people died in distracted-driving crashes between 2014 and 2019 [27]. Despite this, research is still insufficient regarding microscopic traffic flow models under distracted-driving conditions. Van Lint and Calvert [28] proposed a multi-level microscopic traffic simulation framework in which distraction leads to erroneous driving behaviors. Saifuzzaman et al. [29] introduced a novel car-following modeling framework that incorporates driving task difficulty to model the impact of distraction on driver decision-making. Przybyla et al. [30] utilized the Dynamic Time Warping algorithm to identify characteristics of distracted driving behavior, and they expanded the Newell car-following model to include probabilities of driver distraction. Zatmeh-Kanj and Toledo [31] simulated distracted driving behavior to study its impact on car-following model parameters. Kim et al. [32] developed a driver-scanning model under a state of distraction and predicted the risk of crash by combining the driver-scanning model with the basic car-following model. These models typically use fixed parameters to simulate distracted driving, without considering the diversity and uncertainty of driver behavior. Moreover, they do not differentiate distracted driving behavior across different road environments and patterns. Finally, these models have not been applied to explore the impact of distraction on efficiency and safety of traffic flow.

It is challenging to simulate distracted driving behavior. Distracted driving is influenced by multiple factors, leading to significant diversities and uncertainties in its behavioral characteristics. First, the surrounding environment plays an important role in distraction. In these cases, drivers’ distraction performance may vary. For example, Horberry et al. [33] explored the impact of different traffic flow states on distraction. They observed degradation of driving performance due to in-vehicle distractions in both simple and complex highway environments. However, the average driving speed is lower in complex highway environments. Secondly, different patterns of distraction have different impacts on driving performance. Ersal et al. [34] analyzed the driver’s pedal positions under secondary tasks and found that distraction has three different levels of impact on drivers. These impacts are categorized as primary, moderate, and minor, based on which the distraction tasks are classified. Therefore, this paper attempts to consider the above influencing factors, including road type, traffic volume, and different patterns of distraction, when modeling distracted car-following behavior.

In addition, the impact of distraction on driving ability (i.e. perception ability and reaction time) involves significant uncertainties. Gao and Davis [35] found that drivers exhibit longer reaction times when distracted through analyzing naturalistic driving data. Choudhary and Velaga [36] also discovered that using a cell phone while driving leads to a decrease in situational awareness, thereby delaying drivers’ responses to events in the driving environment. Namian et al. [37] reported that distraction reduces the ability to perceive risks, leading to an underestimation of event risks. These characteristics all need to be considered in simulations to accurately describe distracted driving behavior.

This study focuses on a fundamental driving task: longitudinal car-following. This focus is due to the high proportion of longitudinal collisions reported in current accident statistics. For example, in Europe, rear-end collisions account for 16.75% of all accidents [38], and the injuries and fatalities from rear-end collisions exceed 13% of all casualties [39]. In the United States, rear-end collisions account for 29% of all accidents [40]. These data indicate that from a road safety perspective, longitudinal driving deserves more attention, and distraction is a major cause of longitudinal driving accidents. This motivates our investigation into distracted car-following behavior. Therefore, analyzing and modeling car-following behavior under distraction in this study is expected to provide new insights into the causes of accidents and potential countermeasures.

To address the research gap, this study aims to incorporate these distraction characteristics into microscopic traffic flow models and explore the impact of distraction on traffic flow through simulation. Specifically, given that rear-end collisions dominate in distraction-related traffic crashes, this research developed a car-following model under distraction. The contributions of this paper include:

(1) Distinguishing distractions in different road environments based on factors influencing distracted driving behavior, and further classifying different patterns of distraction.

(2) The distracted driving model considers the human factors of a distracted driver’s perception deviation regarding speed differences, perception deviation regarding following distance, and reaction time delay. By using the mathematical model to represent the distribution of distraction parameters, it reflects the heterogeneity of distracted driving behavior.

(3) Implementing the distracted driving model in microscopic traffic simulation, and exploring the impact of the proportion of distraction and distraction patterns on the efficiency and safety of traffic flow.

2. Data

2.1. Shanghai Naturalistic Driving Study (SH-NDS)

The SH-NDS includes GPS information, vehicle controller area network (CAN) data, longitudinal and lateral acceleration, surrounding vehicle speed, distance detected by radar, and video captured by four cameras. The videos include footage of the driver’s face, hand operations, the front view from the vehicle, and the rear view from the vehicle. Figure 1 illustrates the data information. The SH-NDS includes 50 drivers; the distributions of gender, age, and driving experience are consistent with those of the general driver population. A total of 7400 hours of driving data was collected, covering a distance of 200,000 kilometers. All data collected from these drivers were pooled together for the analysis presented in this study.

2.2. Participants

In the SH-NDS, a total of 60 drivers initially participated in the study. However, due to incomplete data from 10 drivers, their data was excluded from subsequent analysis. In total, 7400 hours of driving data were collected, covering a distance of 200,000 kilometers. Among the participating drivers, 70% were male, and 30% were female. Drivers under the age of 25 accounted for 10%, those aged 25–50 made up 74%, and those over 50 comprised 16%. This distribution aligns with the gender and age proportions of registered drivers in China. Additionally, the occupational distribution of the drivers was balanced, including fields such as education, engineering, finance, and media, as well as government employees and homemakers. Therefore, the data are both representative and generalizable. All data collected from these drivers were pooled together for analysis in this study.

2.3. Distracted Car-Following Event

The Distracted Car-following Event data were extracted from the SH-NDS dataset. Specifically, the SH-NDS dataset includes various driving scenarios, from which we identified and isolated events wherein drivers were following another vehicle and were distracted. Some studies have proposed methods to extract car-following segments [41,42,43]. Firstly, car-following segments were extracted from the SH-NDS dataset based on existing standards. The conditions for extracting car-following segments in this study are defined as follows:

(1) The radar target identification number remains unchanged, ensuring that the vehicle continuously follows the same lead vehicle.

(2) There is no lateral displacement greater than 1.5 meters between adjacent timestamps, ensuring that the vehicle does not change lanes or overtake.

(3) The longitudinal distance to the lead vehicle is less than 100 meters, eliminating completely unconstrained car-following states.

(4) The test vehicle speed is greater than 20 km/h, excluding low-speed waiting events.

(5) The car-following duration is equal to 10 seconds, ensuring that each car-following segment contains enough sequences for subsequent algorithm analysis.

On this basis, distracted car-following events were extracted according to the following criteria; when a car-following segment meets any of these criteria, it can be considered as a candidate for distracted car-following events [44]:

(1) The distance between the leading and following vehicles is getting larger and larger.

(2) The distance between the leading and following vehicles is unexpectedly small.

(3) The vehicle’s lateral swing becomes unstable.

The distracted car-following events filtered out by the criteria may be misclassified due to data anomalies or individual differences between drivers. Therefore, each car-following event was further manually reviewed via video to ensure that the driver was distracted. During the manual review of distracted car-following events, specific criteria were used to confirm distraction incidents. The review criteria included instances where the driver’s gaze was away from the road ahead (e.g., using a mobile phone) or where the driver’s gaze remained on the road, but they were engaged in non-driving tasks (e.g., talking on the phone, conversing with passengers). If the manual check did not identify any abnormal behavior, the event was considered normal driving and subsequently removed from the distraction database. Ultimately, 772 distracted car-following events were extracted.

3. Characteristics of Distracted Car-Following Events

The interval between two consecutive events and the duration of a distracted car-following event need to be determined for a microscopic-traffic-flow-level simulation. By statistically modeling the distributions of duration and interval, the simulation can more accurately reflect the actual distracted driving patterns on the road. In this paper, the Kolmogorov–Smirnov (K–S) test and the chi-squared test were utilized to select the best-fit distributions of the interval and the duration of the distracted driving, with the p-values from these tests used to determine if there is a significant difference between the hypothesized distribution and the observed data.

3.1. Interval between Events

The interval between two consecutive distracted car-following events indicates the event frequency for one particular driver. Distracted driving is common, such that a driver may experience it several times during a journey. The goodness-of-fit of interval distribution is shown in

Table 1. Distribution of the interval between two distracted car-following events.

Possible Distributions	Distraction Event Interval		Distribution Diagram
	K–S Test Statistic	Chi-Squared Test Statistic
Normal distribution	<0.001	<0.001
Exponential distribution	0.003	0.002
Extreme value distribution	<0.001	<0.001
Burr distribution	<0.001	0.015
α-Stable distribution	0.005	0.017
Gamma distribution	0.542	0.220

. Results indicate that the p-values for most distributions are less than 0.05, leading to the rejection of the null hypothesis that the distribution fits the data. However, the gamma distribution has a p-value of 0.542 in the K–S test and 0.220 in the chi-squared test, both of which are significantly higher than 0.05. Consequently, it is concluded that the interval between events follows a gamma distribution, which has the form of Equation (1):

$$f\left(x,\beta ,\alpha \right)=\frac{{\beta }^{\alpha }}{\mathsf{\Gamma }\left(\alpha \right)}{x}^{\alpha -1}{e}^{-\beta x},x>0$$

(1)

where $\alpha$ = 0.7176 and $\beta$ = 10,777.13.

3.2. Duration of the Events

The duration of the distracted car-following events is mostly between 10 s and 30 s. The goodness-of-fit of duration distribution is shown in

Table 2. Distribution of distraction event duration.

Possible Distributions	Distraction Event Duration		Distribution Diagram
	K–S Test Statistic	Chi-Squared Test Statistic
Normal distribution	<0.001	<0.001
Exponential distribution	0.012	0.045
Extreme Value distribution	<0.001	<0.001
Burr distribution	0.443	0.086
a-Stable distribution	0.005	0.017
Gamma distribution	0.032	0.057

. The results indicate that the p-values for the normal distribution, exponential distribution, and extreme value distribution are below 0.05 in both tests, suggesting that they do not fit the data well. Although the p-values for the α-stable distribution and gamma distribution are higher, one or both tests still have p-values below 0.05, suggesting they may not be the best choices. The Burr distribution, however, has a p-value of 0.443 in the K–S test and 0.086 in the chi-squared test, which are greater than 0.05. Therefore, the duration of the event is best described by the Burr distribution, which has the form of Equation (2):

$$f\left(x\right)=\frac{\alpha \gamma (x/\theta {)}^{\gamma }}{x{\left[1+(x/\theta {)}^{\gamma }\right]}^{\alpha +1}}$$

(2)

where $\alpha$ = 95.94, $\theta$ = 62.85, and $\gamma$ = 0.03.

3.3. Distracted Car-Following Pattern

Distracted car-following events have been classified into different patterns to allow for more accurate simulations for each pattern. Using the GPS information in the dataset, 593 distracted car-following events on expressways and 179 events on surface roads were distinguished. A scenario is considered to have a high traffic volume when the number of vehicles in the same direction on the lane exceeds 6 within a 5-second observation period [45]; otherwise, it is considered to have a low traffic volume. It should be noted that by “scenario”, this study refers to the observation period under actual traffic conditions, rather than a part of a simulation. By analyzing the traffic flow on different types of roads, it can be observed that the traffic volume on surface roads consistently remains at a high level, whereas the traffic flow on expressways varies more significantly. Therefore, distracted car-following events on expressways are further divided into two groups based on traffic volume. Finally, there are 67 distracted car-following events with a high traffic volume and 526 with a low traffic volume.

The Fuzzy C-Means clustering method was utilized to categorize distracted following behavior. This method was chosen for its flexible clustering outcomes and robustness against noise and outliers [46]. Initially, Spearman correlation analysis was employed to select critical clustering parameters from 20 common parameters of car-following behavior, including statistical indicators (mean, standard deviation, maximum, and minimum values) for distance, speed, time headway, acceleration, and deceleration. The results confirmed that the standard deviation of speed, maximum acceleration, and maximum deceleration exhibit the lowest correlation among these variables. Thus, these three parameters were selected for clustering in the Fuzzy C-Means method.

The clustering results revealed three patterns of distracted driving behavior, including excessive distraction driving, moderate distraction driving, and mild distraction driving. Specifically, excessive distraction driving refers to distractions that significantly impact driving safety. As seen in Figure 2, drivers in this category show considerable speed fluctuations and exhibit higher accelerations and decelerations. Moderate distraction driving refers to distractions that moderately affect driving safety, with less speed fluctuation than excessive distraction driving. Mild distraction driving refers to distractions that minimally impact driving safety, in which drivers experience minor speed fluctuations and lower accelerations/decelerations. Additionally, the clustering results under different road environments were visualized through the t-distribution Stochastic Neighbor Embedding (t-SNE) dimensionality reduction method. Figure 2 shows that the clustering results identify three distinct groups, each with its own cohesive characteristics and almost no overlap between groups. This indicates that different patterns of distraction can be effectively distinguished.

4. Distracted Car-Following Model

4.1. Driver Attributes Affected by Distraction

The process of driving behavior can be regarded as a stimulus–response control system [37,47]. Traffic information either on the road or in the vehicle is a stimulus to the system. After the driver perceives the stimulus, the driver generates the observable driving behavior. Therefore, this paper introduces the driver stimulus-response framework to model abnormal behavior under distraction. This framework can be expressed as:

$$a_i\left(t+{\tau }_i\left(t\right)\right)=f\left(S_i\left(t\right),{\theta }_i\left(t\right),{\omega }_i\left(t\right)\right)$$

(3)

where $a_i\left(t\right)$ represents acceleration, ${\tau }_i\left(t\right)$ is reaction time, $S_i\left(t\right)$ is the stimulus, ${\theta }_i\left(t\right)$ is the driver’s sensitivity to stimulus, and ${\omega }_i\left(t\right)$ represents the driving environment characteristics. The spacing and speed difference observed by the driver between the leading vehicle and the following vehicle are the perceived stimulus, while the driver’s subsequent acceleration or deceleration are the driver’s response, and the response time is the time required for the driver to perceive the stimulus and make corresponding driving operations.

Within the stimulus–response framework, the attributes of drivers undergo significant changes under the distraction. For example, Amini et al. [48] found that drivers’ vision worsens when they are distracted, leading to a decreased perception of stimuli [48]. Similarly, Cvahte Ojsteršek and Topolšek [49] discovered that the perceptual abilities of a driver change under a state of distraction, which is reflected in their sensitivity to stimuli within the stimulus–response framework. Furthermore, Kim and others [50] showed that reaction times for drivers increase when they are distracted, manifesting as longer reaction times within this framework. Therefore, when modeling distracted driving behaviors, it is important to include the increased reaction time and changes in drivers’ sensitivity to stimuli within the framework.

4.2. 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:

$$a_n\left(t\right)=a_{max}^{\left(n\right)}\left[1-{\left(\frac{V_n\left(t\right)}{{\tilde{V}}_n\left(t\right)}\right)}^{\beta }-{\left(\frac{{\tilde{S}}_n\left(t\right)}{S_n\left(t\right)}\right)}^2\right]$$

(4)

$${\tilde{S}}_n\left(t\right)=S_{jam}^{\left(n\right)}+max\left(0,V_n\left(t\right){\tilde{T}}_n\left(t\right)-\frac{V_n\left(t\right)\mathsf{\Delta }{V}_n\left(t\right)}{2\sqrt{a_{max}^{\left(n\right)}{a}_{comf}^{\left(n\right)}}}\right)$$

(5)

where $a_{max}^{\left(n\right)}$ is the maximum acceleration of the target vehicle, ${\tilde{V}}_n\left(t\right)$ is the desired speed, $\beta$ is a constant representing the free acceleration exponent (usually $\beta$ = 4. $S_n\left(t\right)$ is the distance between two vehicles measured from the front bumper of the following vehicle to the rear bumper of the lead vehicle), ${\tilde{S}}_n\left(t\right)$ is the desired following distance, which depends on several factors: speed ($V_n\left(t\right)$), speed difference ($\mathsf{\Delta }{V}_n\left(t\right)$), maximum acceleration ($a_{max}^{\left(n\right)}$), comfortable deceleration ($a_{comf}^{\left(n\right)}$), distance at standstill ($S_{jam}^{\left(n\right)}$), and desired time headway (${\tilde{T}}_n(t$)).

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 ($\mathsf{\Delta }{V}_n\left(t\right)$) and the distance ($S_n\left(t\right)$) 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:

$$a_n\left(t+\tau \right)=a_{max}^{\left(n\right)}\left[1-{\left(\frac{V_n\left(t\right)}{{\tilde{V}}_n\left(t\right)}\right)}^{\beta }-{\left(\frac{{\tilde{S}}_n\left(t\right)}{S_n\left(t\right)+\lambda }\right)}^2\right]$$

(6)

$${\tilde{S}}_n\left(t\right)=S_{jam}^{\left(n\right)}+max\left(0,V_n\left(t\right){\tilde{T}}_n\left(t\right)-\frac{V_n\left(t\right)\left(\mathsf{\Delta }{V}_n\left(t\right)+\theta \right)}{2\sqrt{a_{max}^{\left(n\right)}{a}_{comf}^{\left(n\right)}}}\right)$$

(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.

5. Model Calibration

5.1. Model Calibration Results

The genetic algorithm (GA) was used to calibrate a set of parameters for each type of distracted car-following model. GA is a heuristic optimization algorithm, which is often used in the calibration of parameters of car-following models [39]. According to previous studies [24,38], minimizing the calibration error regarding the car-following distance can result in a better model, so the objective function of GA in the calibration process is set as the Root Mean Square Normalized Error (RMSNE) of the car-following distance (i.e., spacing):

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{N}\mathrm{E} \mathrm{o}\mathrm{f} \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{n}\mathrm{g}=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(\frac{s_i^{sim}-s_i^{obs}}{s_i^{obs}}\right)}^2}$$

(8)

The calibration was carried out in MATLAB ^® v. R2022a (Natick, MA, USA). The parameters of GA are shown in

Table 3. Genetic algorithm parameter setting.

Parameter	Value
Population size	300
Number of generations	1000
Number of stall generations	100

. In addition, since GA is a random search algorithm, the output of each run may be different; therefore, the algorithm was repeated five times, and the one with the smallest error was taken as the calibration result. The parameters and their specific meanings, boundary constraints, and initial searching values are shown in

Table 4. Bounds and initial searching values of Intelligent Driver Model (IDM) parameters.

Parameter	Description of Parameter	Boundary	Initial Searching Value
$a_{max}^{\left(n\right)}$	Maximum acceleration	[0.1, 5]	2
${\tilde{V}}_n\left(t\right)$	Desired speed	[1, 50]	30
$a_{comf}^{\left(n\right)}$	Comfortable deceleration	[0.1, 5]	2
$S_{jam}^{\left(n\right)}$	Distance at standstill	[0.5, 10]	5
${\tilde{T}}_n\left(t\right)$	Desired time headway	[0.1, 5]	1.5

. The calibration results are shown in

Table 5. Calibration results of IDM-Distraction.

Scenarios	Expressway with High Traffic Volume			Expressway with Low Traffic Volume			SURFACE ROAD
Pattern	Excessive	Moderate	Mild	Excessive	Moderate	Mild	Excessive	Moderate	Mild
$S_{jam}^{\left(n\right)}$	3.614	3.745	2.663	2.974	0.911	3.854	2.103	6.074	6.840
${\tilde{T}}_n\left(t\right)$	0.671	0.437	1.223	0.814	0.875	0.621	0.886	0.140	1.032
$a_{max}^{\left(n\right)}$	0.917	0.585	0.275	1.080	0.494	0.144	0.822	0.237	0.658
$a_{comf}^{\left(n\right)}$	0.906	1.828	0.716	5.000	0.840	2.966	1.311	2.000	0.973
${\tilde{V}}_n\left(t\right)$	30.638	25.467	23.795	32.016	29.605	27.007	25.292	23.115	18.793
$\tau$	GMM μ = [4.09, 1.58], σ = [0.18, 1.13], α = [0.50, 0.50]	Normal μ = 2.77, σ = 1.53	GMM μ = [4.62, 0.94], σ = [0.04, 0.48], α = [0.17, 0.83]	Normal μ = 2.46, σ = 1.53	GMM μ = [1.68, 4.11], σ = [0.73, 0.33], α = [0.65, 0.35]	GMM μ = [3.93, 1.42], σ = [0.39, 0.71], α = [0.56, 0.44]	Normal μ = 3.28, σ = 1.66	α_stable α = 0.40, β = –0.99, γ = 0.76, δ = 4.49	Normal μ = 3.28, σ = 1.66
$\theta$	Normal μ = 0.11, σ = 13.53	GMM μ = [8.33, −14.39], σ = [24.98, 22.18], α = [0.64, 0.36]	Normal μ = −1.43, σ = 10.57	GMM μ = [15.32, −6.51], σ = [49.69, 22.57], α = [59.62, 12.11]	GMM μ = [8.10, −11.03], σ = [22.57, 49.69], α = [0.43, 0.57]	GMM μ = [12.72, −5.71], σ = [62.08, 18.17], α = [0.32, 0.68]	α_stable α = 1.41, β = −0.74, γ = 3.80, δ = 0.42	α_stable α = 1.58, β = −0.67, γ = 4.21, δ = –2.09	α_stable α = 2, β = 0.94, γ = 8.01, δ = −2.00
$\lambda$	α_stable α = 0.40, β = 0.47, γ = 0.04, δ = −0.09	GMM μ = [3.75, −3.68], σ = [0.08, 0.04], α = [0.64, 0.36]	α_stable α = 0.4, β = 0.96, γ = 1.01, δ = −3.14	GMM μ = [3.17, −3.15], σ = [0.14, 0.20], α = [0.45, 0.55]	GMM μ = [−0.86, 0.32], σ = [0.01, 3.34], α = [0.63, 0.37]	GMM μ = [0.49, −5], σ = [2.27, 0.01], α = [0.95, 0.05]	GMM μ = [5, −2.16, 3.67], σ = [0, 4.67, 1.17], α = [0.26, 0.42, 0.32]	α_stable α = 2, β = −0.97, γ = 2.59, δ = 0.75	α_stable α = 0.4, β = −0.38, γ = 0.46, δ = 2.22

.

The results show that drivers who are excessively distracted exhibit the highest desired speeds and greater accelerations/decelerations; this indicates that these drivers might fail to reduce their speeds in time, and they have many intensive changes in their speeds. For moderately and mildly distracted driving patterns, more variations can be observed in different road environments, which demonstrates the complexity of distracted driving behavior.

5.2. Model Performance at the Population Level

Introducing distraction parameters has improved the calibration accuracy of IDM-distraction by an average of 7.14% compared with IDM, as demonstrated in Figure 3. Specifically, on expressways with high traffic volume, the model accuracy for mild distraction in IDM-distraction showed the most significant improvement, with an increase of 18.8% compared to IDM. The accuracy for excessive and moderate distractions also saw improvements. On surface roads, IDM-distraction achieved the highest model accuracy for moderate distraction, with a 9.7% increase compared to IDM. Although the performance of the two models under excessive and mild distraction patterns is similar, IDM-distraction slightly outperforms IDM. On expressways with low traffic volume, IDM-distraction exhibited lower RMSNE values for both excessive and mild distraction patterns, indicating better accuracy in these environments. However, for moderate distraction, the accuracy of IDM-distraction was slightly lower than that of IDM. This may be due to the moderate distraction pattern representing a “gray area” where drivers’ behaviors become more unpredictable. As illustrated in Figure 2c, the standard deviation of speed for drivers under moderate distraction is significantly larger on expressways with low traffic volume, and the ranges for maximum acceleration and deceleration are also significantly broader compared to other traffic conditions and distraction patterns.

Based on Figure 4, we can analyze the distribution of the three distraction parameters.

(1) Reaction times

Firstly, on expressways with high traffic volume, drivers experiencing excessive distraction have significantly prolonged reaction times compared to the other two distraction patterns. In contrast, on expressways with low traffic volume, the differences in reaction times among the three distraction patterns are minor, suggesting that low traffic volume has a smaller impact on distracted driving behaviors. On surface roads, reaction times are longer across all three distraction patterns, likely because the complex environment of surface roads increases driver workload, requiring more time to respond to emergencies when distracted.

(2) Differences in speed perception

Regarding the differences in speed perception, whether on expressways with high or low traffic volumes, the distribution under different distraction patterns is similar, indicating that traffic volume has a lesser impact on drivers’ speed perception on expressways. However, on surface roads, the distribution of speed perception deviation is broader among excessively distracted drivers, suggesting that these drivers are more prone to extreme perception deviations.

(3) Deviation in the perception of following distance

Excessively distracted drivers exhibit noticeable perception biases regarding following distance both on expressways with low traffic volume and on surface roads. This indicates that excessive distraction leads to greater perceptual deviation in following distance, affecting vehicle control. The distribution of perception deviation under moderate distraction is wider than under mild distraction, indicating more extreme values and thus more unstable perceptual abilities. On expressways with low traffic volume, perception deviation of following distance for excessive distraction is larger, while it is smaller for mild distraction. This shows that the more severe the distraction pattern in low traffic conditions is, the greater is its impact on distance perception deviation. On surface roads, the perception deviation of following distance for excessive distraction is unstable, reflecting greater uncertainty for excessively distracted drivers in estimating following distance. Compared to expressways, perception deviations, even under moderate and mild distraction, are larger on surface roads, likely due to the complex surface road environment increasing drivers’ workload and resulting in larger perception errors.

This analysis highlights the varying impact of different road environments on distracted driving behaviors and underscores the importance of considering these factors in traffic management and safety interventions.

5.3. Model Performance at the Individual Level

(1) On expressways with high traffic volume

As shown in Figure 5a, on expressways with high traffic volume, the speed of the following vehicle gradually increases as the level of distraction changes from mild to excessive. For those experiencing excessive distraction, the following speed remains higher than that of the leading vehicle, without timely adjustments. This can be attributed to increased reaction times, preventing the quick speed adjustments necessary to maintain a safe following distance. In this environment, the following behavior during mild distraction aligns most closely with the leading vehicle, likely because high traffic volume requires drivers to stay more vigilant to respond to frequently changing traffic conditions.

(2) On expressways with low traffic volume

As depicted in Figure 5b, on expressways with low traffic volume, the following speeds in moderate and mild distraction patterns are consistently slightly higher than that of the leading vehicle. However, under excessive distraction, the speed increases more rapidly over the duration of the distracted epochs. This indicates that drivers under excessive distraction have poorer speed perception, which affects their ability to match the speed of the leading vehicle accurately.

(3) On surface roads

As illustrated in Figure 5c, on surface roads, when the leading vehicle’s speed increases smoothly, following vehicles under excessive distraction accelerate noticeably, whereas those under moderate and mild distraction opt for slower acceleration to maintain the following distance. This suggests that on surface roads, excessively distracted drivers perceive following distance quite differently compared to those who are moderately or mildly distracted.

6. Micro-Simulation Implementation

6.1. Distracted Driving Simulation in Traffic Flow

Due to the complexity of traffic systems, studying the impact of driving distractions on traffic flow becomes a challenging task. Traffic crashes caused by distractions are difficult to replicate through naturalistic driving experiments. Therefore, this paper proposes a framework to simulate distracted driving behavior and explore its impact on traffic flow. The simulation framework is shown in Figure 6. First, probability distributions of distraction patterns in different road environments are derived from the naturalistic driving dataset. Then, these probability distributions are used to set the simulation parameters, including the interval and duration of distraction events. When a vehicle is set to be distracted, it will choose the corresponding IDM-distraction model according to the type of distraction; otherwise, IDM will be executed for the vehicle.

The above simulation framework was applied to the Shenjiang Road network in Shanghai, which is a segment including expressway and surface road with 5.4 km length. The simulation was conducted for peak and off-peak periods. For the expressway, the traffic volume was set as 1800 pcu/h for high traffic volume and 1000 pcu/h for low traffic volume. Since the dataset in this study did not differentiate the traffic flow on surface roads, the traffic volume on surface roads was set to 1400 pcu/h in the simulation. According to Sullman [51], about 15% to 25% of drivers on the roads are distracted; therefore, in the simulation, scenarios with different fractions of distracted drivers (15%, 20%, 25%) were run to explore the impact of various fractions of distraction.

To compare the traffic flow characteristics under different conditions, this paper selected the average speed and the coefficient of variation (COV) of speed as measurement indicators. The average speed can measure the operational efficiency of the road segment, while the COV can reflect changes in speed, thereby indicating road safety.

6.2. Impact of Distraction on Traffic Flow

6.2.1. Impact of Distraction Fractions on Traffic Flow

Figure 7 compares the differences in the confidence intervals of drivers’ speeds under different distraction fractions. When the confidence intervals between two groups of data do not overlap, it indicates that there is a significant difference between these two groups of data. It is observed that on both expressways and surface roads, the average speed significantly decreases when the fraction of distraction increases from 15% to 25%. This decrease can be attributed to the disruption of traffic flow stability as the distraction fraction increases, compelling drivers to adopt slower speeds in response to hazardous scenarios induced by distractions (there are other factors that may also contribute, but it is the effect of distractions highlighted here). Furthermore, it is also evident that the speeds on surface roads are significantly lower than those on expressways, which aligns with real-world driving scenarios.

As shown in Figure 8a, on expressways with high traffic volume, the average speed decreases and the COV of speed increases as the fraction of distracted driving rises. This indicates that an increase in the fraction of distracted drivers significantly reduces overall traffic flow efficiency and safety. Similarly, Figure 8b,c demonstrate that distraction significantly lowers traffic flow efficiency and safety. Moreover, by comparing Figure 8a,b, it can be observed that there is a broader distribution of drivers’ average speeds on expressways with lower traffic volume. This is because drivers have more freedom in a less congested environment, leading to a wider range of speed choices. Figure 8c reveals that on surface roads, the complexity of the road environment prompts drivers to adopt a more conservative driving style. Furthermore, because surface roads often have sections where vehicles and non-motorized traffic mix together, drivers’ speeds are significantly lower than those on expressways.

6.2.2. Impact of Distraction Fractions on Intersections and Segments on Surface Road

Based on Section 6.2.1, it is known that distraction leads to poorer efficiency and safety on surface roads, which may be due to the complex road conditions on these roads. Therefore, this section distinguishes between segments and intersections of surface roads to investigate the impacts of distraction on traffic flow. As shown in

Table 6. Significance analysis of the impact of distraction on different road sections.

Distraction Fractions	p-Value
15%	<0.001
20%	<0.001
25%	<0.001

, when the distraction fraction increases from 15% to 25%, the impact of distraction on the speed of road segments and intersections is significantly different.

For road segments and intersections, since the expected speeds differ, comparing their speeds may not be very meaningful. Therefore, this section considers the ratio of actual speed to expected speed as the efficiency indicator, while the safety indicator continues to be represented by the COV of speed. From Figure 9a, it can be observed that the efficiency of both intersections and road sections decreases with an increase in the distraction fraction. Additionally, Figure 9a also shows that at the same level of distraction, the throughput efficiency of road sections is significantly higher than that of intersections, and the distribution error of throughput efficiency at intersections is significantly greater than that of road sections. This is due to the fact that signal timing at intersections further affects throughput efficiency. Figure 9b demonstrates that the COV of speed at both intersections and road sections increases with an increase in the distraction fraction, implying that an increase in the distraction fraction reduces safety at both intersections and road sections. Moreover, the figure also reveals that the COV of speed at intersections is significantly higher than that on road sections, indicating that safety at intersections is significantly lower than on road sections. This is because vehicle interaction behaviors at intersections are more complex, thereby increasing the risk of accidents at intersections.

6.3. Pre-Crash Scenario Simulation

For many classical car-following models like IDM, generating actual accidents or dangerous scenarios is challenging due to the model’s inherent structural limitations. Given the non-zero minimum spacing at standstill in the IDM requires the driver’s desired following distance to be greater than zero, the IDM inherently cannot simulate collision scenarios. This paper incorporates distractions into the IDM to introduce more dangerous reaction times and perception deviation, thereby increasing interference in traffic flow. This approach enhances the simulation’s ability to produce more dangerous scenarios. This section selects Time to Collision (TTC) to demonstrate the changes in traffic safety caused by distraction, thereby exploring the simulation capability of IDM-distraction for pre-crash scenarios [52]. TTC is defined as the time constant at which a follower would collide with his leader given the speeds of both vehicles stay constant, and is a longitudinal indicator often used to measure the danger of a scenario.

$${\mathrm{TTC}}_n\left(t\right)=\frac{S_n\left(t\right)}{\mathsf{\Delta }{V}_n\left(t\right)},\mathsf{\Delta }{V}_n\left(t\right)>0$$

(9)

Table 7. Distribution of dangerous scenario levels in simulations. TTC: Time to Collision.

Distraction Fraction	Dangerous (TTC < 1 s)	Serious Conflict (1 s ≤ TTC< 1.5 s)	Mild Conflict (1 s ≤ TTC < 2 s)	Safe (TTC ≥ 2 s)
15%	4.52%	3.75%	4.59%	87.14%
20%	7.24%	3.56%	5.25%	83.95%
25%	8.60%	4.42%	5.07%	81.91%

indicates that when the fraction of distracted drivers is 25%, the fraction of dangerous scenarios (TTC < 1 s) is the highest, at 8.60%. Then, as the distraction fraction decreases, the fraction of dangerous scenarios also declines. For serious conflicts (1 s ≤ TTC < 1.5 s), the scenario fraction is still the largest when the distraction fraction is 25%. The fraction of safe scenarios (TTC ≥ 2 s) decreases as the distraction fraction increases.

For comparative analysis, we selected 360 car-following pairs from a naturalistic driving dataset and calculated the distribution of their TTC. Statistical analysis shows that the fraction of dangerous scenarios in naturalistic driving dataset is 8.11%, the fraction of serious conflicts is 0.78%, the fraction of mild conflicts is 1.10%, and the fraction of safe scenarios is 90.01%. Comparison reveals that the fraction of dangerous scenarios generated in simulation experiments when the distraction fraction is 15% is closest to real scenarios. However, the fractions of serious and mild conflicts in simulation experiments are both higher than in real scenarios, and the fraction of safe scenarios is lower than in real scenarios.

This may be because the driver model used in the simulation does not simulate the evasive actions taken by drivers in dangerous situations (nor the contributions of any active safety systems with which vehicles may be equipped), leading to an overestimation of the fraction of conflict scenarios. Moreover, distracted behavior in the real world may be more random and irregular; however, the simulation of types of distraction in simulation experiments is limited.

7. Discussion and Conclusions

This study proposes a microscopic traffic flow model (i.e. IDM-distraction) that considers distraction characteristics to simulate distracted driving behavior. Two types of distraction parameters are considered in the model, namely reaction time delay and perception deviation in speed difference and following distance. Compared to existing deterministic models, IDM-distraction uses probability distributions to represent these parameters, capturing the inherent uncertainty and individual differences in distracted driving behavior.

By distinguishing road environments and categorizing distracted driving behaviors, behavioral characteristics under different road environments and distraction patterns were analyzed in detail. For example, the probabilistic distribution of distraction parameters shows that the impact on reaction time is the greatest under distracted conditions on expressways with high traffic volume. This may be because, in such environments, drivers need to constantly react to rapid changes around them, such as sudden lane changes or emergency braking by other vehicles. Distractions can divert the driver’s attention from the road, significantly increasing reaction time in high-speed and complex traffic environments, as drivers need more time to re-focus on the driving task.

In contrast, on expressways with low traffic volume, there might be fewer interactions between vehicles, and the driving task may not require as much continuous attention as in high-traffic environments. In these settings, distraction mainly affects drivers’ perception of speed differences and following distances. This might be because at lower traffic volumes, drivers may more easily fall into their own rhythm, with perceived speed and following distance more reliant on their subjective assessment rather than frequently making decisions based on external stimuli. Distraction could distort this assessment, leading drivers to misjudge their relative speed and distance to the vehicle ahead, which could result in inappropriate driving responses in emergencies. On surface roads, due to the more complex environment, which includes more frequent intersections, pedestrian crossings, stopping and starting vehicles, as well as more road signs and traffic lights, drivers are required to allocate more attention to process diverse information and react accordingly. When drivers are distracted, the time needed to deal with these complex situations increases, thus lengthening the reaction time. Furthermore, as the traffic conditions on surface roads are more variable, drivers’ perceptual biases exhibit a wider range of inconsistencies, manifested by a greater number of extreme values in the perception of speed differences and following distance.

By applying the model in the simulation environment, effects of distraction on traffic flow were analyzed. The efficiency of traffic flow largely depends on drivers’ reaction times and their ability to perceive speeds and following distances. Distraction typically increases reaction times, leading to delayed responses to changing traffic conditions. Additionally, perceptual deviation regarding speed and following distances may lead to less than ideal spacing between vehicles, creating a domino effect that disrupts traffic and reduces overall traffic efficiency. The results of the simulation also indicate that as the fraction of distraction increases, the average speed of traffic flow decreases, suggesting a decline in traffic efficiency as a result. Furthermore, the increased variability in driver behavior due to distraction can lead to uneven traffic flow, thereby lowering efficiency. Similarly, as the distraction fraction increases, the coefficient of variation (COV) of speed gradually increases, indicating an increase in speed variability, which in turn affects traffic flow efficiency. As simulation results show, an increase in the distraction fraction leads to a rise in the proportion of dangerous scenarios. More distraction contributes to greater variability in traffic flow speeds on both expressways and surface roads, thus diminishing road safety.

To further elaborate, the IDM-distraction model’s unique approach to capturing the stochastic nature of distracted driving behaviors through probability distributions is a significant advancement. Traditional models often fail to accommodate the variability in human behavior, which is particularly pronounced in distracted driving scenarios. By incorporating variability in reaction times and perception deviations, the IDM-distraction provides a more realistic representation of driver behavior under distraction, which is crucial for developing effective traffic management strategies and safety interventions. The detailed analysis of different road environments and distraction patterns highlights the context-dependent nature of distracted driving. For instance, the increased reaction time in high-traffic environments on expressways underscores the need for tailored interventions that address the specific challenges posed by these settings. Similarly, understanding how low-traffic environments affect speed perception and following distances can inform the design of driver assistance technologies that mitigate these specific risks. Moreover, the findings that distraction leads to increased speed variability and decreased traffic flow efficiency have practical implications for traffic management. Traffic authorities can use these insights to develop policies that minimize distractions, such as stricter regulations on mobile phone use while driving or the implementation of technologies that alert drivers when their attention wanes. The link between increased distraction and a higher proportion of dangerous scenarios also supports the development of advanced safety systems that can detect and respond to distracted driving behaviors in real-time.

However, the current driver model used in our simulations does not fully capture the range of evasive actions drivers might take in dangerous situations. This limitation can lead to an overestimation of the fraction of conflict scenarios. Moreover, real-world distracted behavior is inherently more random and irregular compared to the controlled conditions in our experiments. To address these issues and enhance the realism of our simulations, future simulations should incorporate models that can simulate a wider range of evasive actions, including quick steering adjustments, braking responses, and lane changes under high-risk conditions. Incorporating dynamic interaction models that account for interactions between multiple vehicles can further improve the simulation’s ability to replicate real-world traffic scenarios. These models can simulate how distracted behavior affects not just the distracted driver, but also the surrounding traffic environment. Additionally, including mechanisms for behavioral adaptation in the simulation can enhance its realism. Drivers often adapt their behavior in response to repeated exposure to distractions or after near-miss incidents. This involves understanding the cognitive processes of drivers. By analyzing and modeling the cognitive aspects of distracted drivers, this study can gain a deeper understanding of their behavioral adaptation mechanisms, thereby providing a more comprehensive insight into distracted driving.

In cognitive psychology, people’s attentional resources are limited, and when drivers are distracted, their cognitive resources are divided among multiple tasks. This means that the cognitive resources available for performing the main driving tasks are reduced, thereby affecting the quality of execution of these tasks [53]. Therefore, understanding the cognitive mechanisms behind distracted driving and exploring drivers’ cognitive decision-making processes while distracted is crucial for reducing the dangers of distraction. With the advancement of brain cognitive science, the cognitive theory of driving behavior is becoming increasingly rich. Cognitive models allow for a deeper exploration and quantification of how cognitive phenomena like cognitive load influence distracted driving behaviors. This allows for a more microscopic simulation of the distraction process, thereby improving the accuracy of modeling of distracted driving behaviors. Future research might explore more complex distraction scenarios, including the impact of different distraction sources on distracted driving behavior and the cognitive mechanisms during the distraction process. These efforts are expected to contribute to understanding and reducing the risks associated with distracted driving, ultimately improving the efficiency and safety of the traffic flow.

In summary, this study proposes the IDM-distraction, which incorporates human factors into microscopic traffic flow models to more accurately simulate distracted driving behavior. By introducing distraction characteristic parameters (reaction time, perception bias of speed difference and following distance), the model better captures driving behavior in distracted states, thereby improving the accuracy of traffic flow simulation. In addition, by incorporating the probability distribution of distracting event characteristics (duration, interval) into the simulation, the model can adapt to distracted driving behavior in different road environments and capture the variability and individual differences of distracted driving. This provides a more realistic and detailed analysis of traffic flow safety and efficiency. The simulation results show that distraction will cause the traffic flow speed to decrease and the COV of the speed to increase, indicating that distraction leads to a decrease in traffic safety and efficiency. Furthermore, the findings highlight practical implications for traffic management. Future research can further enhance these models by integrating evasive maneuvers, vehicle interactions, and behavior adaptation mechanisms based on cognitive theory. This will enhance the realism of the models and ultimately improve the understanding and management of traffic systems, leading to better road safety and efficiency.