Modeling Intra-Driver Behavioral Adaptation through Risk Sensitivity and Regime Transitions: Task Saturation Car-Following Model

Abstract

Over the last decade, there has been a growing trend toward integrating human factors (HF) into traffic flow models to better understand the complexities of human behavior and its effect on traffic dynamics. This study seeks to advance this trend by bridging the gap between traditional car-following models and the inherent variability in human driving behavior. By incorporating these elements, the models provide valuable insights into how driver behavior adaptation and risk-taking influence traffic flow dynamics. This study proposes a model, the Intelligent Driver Model with Task Saturation (IDMTS), that integrates behavioral adaptations and risk-taking strategies into the established Intelligent Driver Model, enriched by a cognitive layer based on Fuller’s Task Capability Interface model. This amalgamation enables interplay between driver behavior adaptation as the driving task saturates and drivers’ risk-taking strategy. First, steady state equilibria and fundamental diagrams of IDMTS are derived, and local and string stability are analyzed under rational driving constraints. Then, the IDMTS is calibrated and validated on three complementary data sets: controlled driving simulator experiments with normal and distracted driving, and naturalistic car-following trajectories from the Next Generation Simulation I-80 and Milan Trajectories data sets. The model is investigated for behavioral soundness and numerical soundness. The results demonstrate that the model effectively incorporates endogenous heterogeneity in driving behavior under varying task loads. The model generates two plausible HFs: (1) risk-taking; and (2) behavior adaptation. The findings of this study suggest a step toward a more realistic representation of driving behavior adaptation and risk-taking strategies.

Keywords

car-following human factors task capacity interface driver distraction behavioral adaptation

Introduction

Over the last decade, there has been a growing movement in the traffic flow theory community to integrate human factors (HF) into mathematical models of driving behavior ( 1 , 2 ). This reflects an increasing recognition that cognition, perception, and decision-making substantially influence a driver’s ability to maintain situation awareness and execute safe maneuvers. Incorporating such factors enables more realistic car-following models that respect the psychological mechanisms underlying longitudinal control ( 2 , 3 ). The practical stakes are considerable: car-following models underpin microsimulation platforms used for infrastructure design, safety analysis, and the evaluation of emerging technologies such as adaptive cruise control and connected automated vehicles. If these models rest on unrealistic assumptions about driver cognition, the downstream assessments they inform will likewise be unreliable ( 2 ).

Car-following theory has a rich history spanning several paradigms, stimulus-based, safe-headway, desired-measures, optimal velocity, and psycho-physical models, each distinguished by its assumptions about driver behavior ( 4 ). These formulations have proven valuable for traffic engineering; however, a fundamental limitation persists: they are premised on the strict assumption of unlimited cognitive processing capacity ( 1 ). This assumption is unrealistic because it fails to account for the inherent limitations of drivers’ task-processing capacity. In longitudinal driving, drivers continually adapt their behavior to compensate for increases in task demand. For instance, by increasing headway or reducing speed when the driving task becomes more difficult, such as when engaging in a secondary task. Such behavioral adaptation is not merely a reaction; it is a fundamental principle embodied in Fuller’s Task Capability Interface (TCI) model ( 5 ), which posits that drivers regulate control variables to maintain perceived task difficulty within tolerable bounds. This theoretical insight has motivated a growing body of work seeking to layer cognitive and perceptual mechanisms onto established car-following formulations.

Efforts to incorporate HFs into car-following modeling have broadly followed two methodological directions. The first develops behaviorally grounded car-following formulations from behavioral decision theories, such as prospect theory, where driver acceleration decisions are derived from risk perception and gain–loss evaluation ( 6 ). In these models, behavioral assumptions form the primary basis of the model structure rather than being embedded within an existing traffic flow formulation. A second line of research introduces human driver mechanisms within existing car-following models, most notably within the Intelligent Driver Model (IDM). These extensions incorporate features such as memory effects ( 7 ), variance-driven defensive behavior ( 8 ), reaction delays and multi-vehicle anticipation ( 9 ), psycho-physical action points ( 10 ), and time-varying desired time gaps ( 11 ). Although these models improve behavioral realism by representing specific perception and response processes, they typically address these mechanisms individually and do not explicitly represent driver task demand as a state variable governing behavioral adaptation. Therefore, existing models rarely provide a unified formulation in which evolving traffic conditions influence driver behavior through measurable task demand constructs. Moreover, empirical validation of behavioral mechanisms remains limited because relatively few models have been evaluated using controlled distraction experiments and naturalistic trajectory data.

To address these shortcomings, this study introduces the Intelligent Driver Model with Task Saturation (IDMTS), a behaviorally grounded car-following model that embeds HF mechanisms directly into the car-following dynamics. The model endogenizes behavioral adaptation using distinct driving regimes, free driving, car-following, and behavioral adaptation, derived from Fuller’s risk allostasis theory ( 12 ). The behavioral adaptation mechanism allows driver behavior to evolve dynamically in response to changing task demands. To achieve this, the well-established IDM is adapted ( 13 ), layering a cognitive dimension on top of the foundational physics of the original model. This additional layer is designed to ensure a fine balance between engineering and cognitive aspects of driving behavior. Together, these enhancements bridge the gap between classical car-following theory and cognitive behavioral modeling, yielding a mathematically tractable yet behaviorally realistic framework.

The specific contributions of this study are threefold. First, a task saturation measure expressed for observable traffic variables (speed, spacing, and desired time gap) is formulated that satisfies Fuller’s TCI theory, and embedded into the IDM to create endogenous regime transitions. Second, steady state equilibria and fundamental diagrams are developed for the IDMTS, analyzing both local and string stability under rational driving constraints, establishing the analytical tractability of the model. Third, the IDMTS is calibrated and validated on three data sets: controlled driving simulator experiments with normal and distracted driving, and naturalistic car-following trajectories from the Next Generation Simulation (NGSIM) I-80 and Milan Trajectories (MiTra) data sets and the model is evaluated for numerical and behavioral soundness at microscopic and macroscopic levels.

The remainder of this paper is organized as follows. First, the relevant literature on car-following models is reviewed. Next, the TCI model is introduced and interpreted using key traffic flow variables, followed by the formulation of the proposed car-following model, with particular emphasis on drivers’ behavior adaptation and risk-taking strategies. The stability analysis of the model is then presented. Subsequently, the calibration and validation procedures are described. The results and discussion are then presented. Finally, the paper concludes with a summary of the main findings and directions for future research.

Literature Review

Since the seminal work of Pipes ( 14 ), five broad paradigms of car-following models have emerged ( 4 ): (1) stimulus-based models in which acceleration responds to position and speed variations of the leader ( 15 ); (2) safe-headway models, exemplified by Gipps ( 16 ), which ensure the follower can stop safely; (3) desired-measures models, such as the IDM ( 13 ), that minimize deviations from a preferred driving criterion; (4) optimal velocity models ( 17 ) in which a speed-dependent optimal velocity governs acceleration; and (5) psycho-physical models originating from Wiedemann ( 18 ) that partition the stimulus space into distinct perceptual regimes. Each paradigm captures important facets of longitudinal driving; all share a common limitation: the assumption of unlimited cognitive processing capacity ( 1 ). Drivers, in contrast, continuously regulate their behavior, increasing headway or reducing speed, to maintain the perceived task difficulty within tolerable bounds, a principle formalized in Fuller’s TCI model ( 5 ). This omission motivates the development of HF-informed extensions that explicitly acknowledge cognitive constraints.

Several studies introduced HFs into car-following formulations ( 3 ). Yang and Peng ( 19 ) proposed an error-able model that acknowledges perceptual inaccuracies whose magnitude grows with following distance. Hamdar et al. ( 6 ) incorporated risk-taking attitudes using prospect theory, linking subjective risk evaluation to acceleration decisions in congested settings. These approaches explicitly account for drivers’ risk perception and gain–loss evaluation; they primarily focus on behavioral responses to perceived collision risk. Kochi et al. ( 20 ) demonstrated that perceived task difficulty correlates with driving speed and that inexperienced drivers underestimate it. For model-level integration, Saifuzzaman et al. ( 21 ) introduced task difficulty as a scaling factor on desired spacing or acceleration within the IDM. This improves sensitivity to workload; however, it does not create an emergent behavioral response: there is no feedback loop from task difficulty to the driver’s evolving state, and the multiplicative scaling can generate unrealistic trajectories when task difficulty deviates substantially from unity. Van Lint and Calvert ( 22 ) proposed a more comprehensive multilevel framework based on TCI that endogenously incorporates heterogeneities in driver traits and cognitive capacity using fundamental diagrams of task demand. However, the framework’s input requirements, real time measures of task load, and driver awareness, remain abstract and difficult to observe, limiting its practical applicability to data-driven calibration.

A parallel line of research has focused on enriching the IDM with specific human driver mechanisms. Treiber and Helbing ( 7 ) introduced a memory effect by coupling an internal dynamical variable, the subjective level of service to the IDM’s time gap parameter, modeling a resignation effect in which drivers gradually become more tolerant of congested conditions. Treiber et al. ( 8 ) demonstrated that adapting time headways as a function of local velocity variance, a proxy for the driver’s cognitive load induced by strong inter-vehicle speed variations, reproduces the capacity drop and wide scattering in flow–density data, which can be interpreted as a direct implementation of the TCI and task difficulty homeostasis. In a complementary study, Treiber et al. ( 9 ) incorporated finite reaction times, estimation errors grounded in visual perception theory, and anticipation in time and over several vehicles into the IDM, showing that the destabilizing effects of delays can be compensated by experienced driver anticipation. Treiber and Kesting ( 10 ) introduced action points reflecting the psycho-physical observation that humans respond only when the stimulus has changed sufficiently, providing new insights into the relative importance of instabilities, noise, and perception thresholds in generating traffic oscillations. In a complementary empirical direction, Tian et al. ( 23 ) analyzed 25-car platoon experiments and proposed a mode-switching stochastic car-following model incorporating speed adaptation and spacing indifference, demonstrating that traffic instability arises from the competition between deterministic speed adaptation and cumulative stochastic perturbations rather than from classical deterministic instability alone. This work provides important evidence for mode-dependent driving behavior and regime-specific dynamics; the switching mechanism is governed by spacing thresholds defining an indifference region rather than by a cognitive construct linked to driver workload or task demand. Xiong et al. ( 11 ) proposed an alternative approach by allowing the desired time gap parameter to change over time in response to prevailing traffic conditions. More recently, Kashifi et al. ( 2 ) proposed a behavioral and numerical evaluation framework that benchmarks intra-driver heterogeneity car-following models across numerical soundness and behavioral soundness, demonstrating that improvements in numerical fit do not necessarily correspond to improvements in behavioral soundness.

The literature points to three unresolved issues. First, most classical car-following models do not explicitly represent cognitive limitations. Second, existing behavioral extensions tend to focus on perceived collision risk or to treat workload-related effects using parameter modification rather than by endogenous state-dependent adaptation. The conceptual distinction between risk perception under collision threat and workload regulation under task demand ( 5 , 12 ) has seldom been reflected in model design. Third, few models provide a parsimonious and calibratable formulation in which a dimensionless task saturation measure, constructed from observable traffic variables, functions as a state variable governing endogenous regime transitions between normal car-following and behavioral adaptation.

TCI Model and Car-Following Control Task

The TCI model explains driver decision-making by recognizing the inherent limitations of perception and vehicle control. It posits that drivers continuously regulate their behavior to ensure that task demands do not exceed their capabilities ( 5 ). The model is grounded in the principle of task difficulty homeostasis, where drivers adjust control variables such as speed and headway to maintain perceived difficulty within tolerable limits. When task demand increases, as in distraction, drivers may increase headway to reduce difficulty. In contrast, when the task feels too simple, such as driving on a straight road, they may increase speed to restore a desired level of challenge ( 21 ). Therefore, task saturation converges toward an acceptable range shaped by task capacity and motivational factors. The following variables are defined that are required for model development, following the definitions of Lint et al. ( 24 ), as given in Table 1.

Table 1.

Human Factor Constructs used by IDMTS

Human factor	Definition
Nominal task capacity, TC	Standard capacity of a typical driver to process information for safe and effective task performance. This capacity is represented as TC = 1 (or 100%) ( 24 )
Driver task capacity	Processing capacity of a specific driver, measured for the standard task capacity
Driver task demand	Amount of information processing required by a specific driver to safely and efficiently perform a certain task (i.e., a), denoted in units of TC as ${TD}_{i}^{a} (t)$
Total task demand	The cumulative task demand from all tasks for a specific driver, as given in Equation 1 ${TD}_{i} (t) = \sum_{a} {TD}_{i}^{a} (t)$ (1)
Driver task saturation $T S_{i} (t)$	Ratio of the total task demand ${TD}_{i} (t)$ for a specific driver to their individual task capacity ${TC}_{i} (t)$ . It indicates how saturated or occupied a driver is in relation to their capacity, as in Equation 2 ${TS}_{i} (t) = \frac{{TD}_{i} (t)}{{TC}_{i} (t)}$ (2)

Note: IDMTS = Intelligent Driver Model with Task Saturation; a = maximum acceleration; i = specific driver; t = time.

According to Fuller ( 25 ), drivers continuously regulate their behavior to maintain an acceptable balance between task demand and capacity. Task demand at any moment depends primarily on the vehicle’s speed and the spacing to the leader. Therefore, task saturation can be expressed for traffic variables under three behavioral assumptions.

Task demand increases with speed. At higher speeds, the time available to perceive hazards, process information, and execute control actions diminishes, increasing cognitive and physical workload.

Task demand decreases with spacing. Smaller headways reduce the reaction time available in the event of a sudden maneuver by the leader, increasing perceived difficulty ( 21 ). In general, higher speeds or shorter spacings (or both) narrow the safety margin and elevate the instantaneous task demand.

Drivers with greater task capacity select shorter desired time gaps. The desired time headway reflects the temporal buffer a driver maintains to respond to unexpected events such as abrupt braking. Empirical evidence shows that drivers with higher cognitive capability or confidence often accept shorter headways, perceiving them as manageable (26 –28). Nevertheless, this tendency is moderated by individual risk attitude, traffic context, and distraction level, implying that the relationship between task capacity and preferred headway represents a behavioral tendency rather than a strict rule.

Let $v_{i} (t)$ represent the follower’s speed, $s_{i} (t)$ represents the space headway to the leader, and $T_{i}$ represent the driver’s desired time headway. To employ the previous assumptions (1–3), a scalar task saturation measure ${TS}_{i} (t)$ is required, which increases with speed, decreases with spacing, and because higher capacity tends to choose smaller desired headways, increases with $T_{i}$ so that selecting shorter $T_{i}$ corresponds to lower perceived difficulty for that driver.

\frac{\partial {TS}_{i}}{\partial v_{i}} > 0, \frac{\partial {TS}_{i}}{\partial s_{i}} < 0, \frac{\partial {TS}_{i}}{\partial T_{i}} > 0

(3)

These signs operationalize Fuller’s idea that higher speed compresses perception–decision–action time, smaller spacing reduces safety margin, and a driver’s chosen (larger) desired headway reflects a tendency to keep the task easier; therefore, increasing $T_{i}$ should raise the perceived task saturation for that same driver, while choosing a shorter $T_{i}$ (typical of higher capacity) lowers it. A dimensionless and scale-free measure satisfying this is,

{TS}_{i} (t) = \frac{v_{i} (t) T_{i}}{s_{i} (t) + ε}

(4)

where $ε > 0$ prevents singularities as $s_{i} \to 0$ (in simulation $ε \approx 0.1 m$ ). For $v_{i}, s_{i}, and T_{i} > 0$ ,

\begin{matrix} \frac{\partial {TS}_{i}}{\partial v_{i}} = \frac{T_{i}}{s_{i} + ε} > 0, \frac{\partial {TS}_{i}}{\partial s_{i}} = - \frac{v_{i} T_{i}}{(s_{i} + ε)^{2}} < 0 and \\ \frac{\partial {TS}_{i}}{\partial T_{i}} = \frac{v_{i}}{s_{i} + ε} > 0 \end{matrix}

(5)

Therefore, Equation 4 satisfies the design rules by construction (Equation 3) while remaining interpretable.

Model Formulation and Behavioral Extensions

To incorporate HFs into a mechanistic framework, the Intelligent Driver Model Plus (IDM+) ( 29 ) is adopted as the foundation for the proposed model. IDM+ extends the original IDM ( 13 ) by separating free driving and car-following components and selecting the minimum of their respective accelerations, rather than superimposing them. This modification yields more realistic capacity estimates and a smoother transition between traffic regimes.

The instantaneous acceleration of vehicle $i$ in IDM+ is given by,

\begin{matrix} \frac{dv}{dt} = a_{i} \cdot \min [1 - {(\frac{v_{i} (t)}{v_{0 i}})}^{4}, 1 - {(\frac{s_{i}^{*} (v, Δ v)}{s_{i} (t)})}^{2}] \end{matrix}

(6)

where the desired dynamic spacing is,

s_{i}^{*} (v, Δ v) = s_{i} (t) + v_{i} (t) T_{i} + \frac{v_{i} (t) Δ v_{i} (t)}{2 \sqrt{a_{i} b_{i}}}

(7)

where $a_{i}$ represents the maximum acceleration, $b_{i}$ represents the maximum comfortable deceleration, $v_{0 i}$ represents the desired speed, $T_{i}$ represents the desired headway, $s_{0}$ represents the stopping distance, and $i$ represents a specific driver. In all equations, the symbol “•” denotes standard scalar multiplication. The model is formulated on certain assumptions, including maintaining a safe distance from the vehicle ahead, aiming to drive at a desired speed, and favoring acceleration/deceleration within a comfort zone. In addition, it considers kinematic factors such as the squared relationship between braking distance and speed. The model stimuli are the speed of the ego vehicle, the speed difference between the ego vehicle and the leading vehicle, and the spacing between the ego vehicle and the leading vehicle.

Behavioral Adaptation: Task Difficulty Homeostasis

The fundamental idea underpinning behavior adaptation is the task difficulty homeostasis theory. This theory posits that drivers consistently make instantaneous decisions to ensure that the perceived level of task saturation of the driving task remains within acceptable limits. They achieve this by modulating control elements such as speed and headway ( 5 ), which is called driver behavior adaptation.

The behavior adaptation effects emerge because of the dynamic interplay between the total task demands and the capacity of the driver. Therefore, when the driver perceives that the task saturation exceeds their tolerable limit, such as driving while being distracted by a cell phone call, they tend to reduce their speed to bring the task saturation back within their acceptable range. In contrast, when the driving task appears overly uncomplicated, as when driving along a straight highway in free-flow conditions, the driver may accelerate to increase the challenge. The scope of task difficulty the driver aims for is determined by their perception of their own abilities and their desire to engage at a specific level of task saturation.

The driving parameters influencing the task demand are the driving speed and spacing to the leading vehicle ( 25 , 30 ). These two parameters influence the available time for decision-making and response to stimuli that, ultimately, influence the task demand. The higher speed and short spacing mean a short time available for decision-making and response, which leads to a higher task demand ( 21 ). If the driver’s task capacity exceeds the task demand, the driving task is straightforward. However, if task demand exceeds the driver’s task capacity, loss of control may occur ( 5 ).

The original IDM might not offer a comprehensive understanding of behavior adaptation effects in longitudinal driving behavior. This limitation stems from the IDM’s lack of incorporation of crucial HFs ( 26 ). This suggests that for a more accurate representation of driving behavior adaptations, especially in scenarios of driver distraction, models should integrate and reflect human behaviors and characteristics more explicitly.

Risk Sensitivity and Behavioral Heterogeneity

Drivers’ behavioral adaptation occurs when they sense a need to adapt their behavior, especially when faced with an oversaturated driving task. Of interest, not all drivers respond in the same manner to these heightened driving tasks. Some proactively adapt their behavior; others remain unresponsive to the escalating risks, continuing their journey without any behavioral adaptations. This variation underscores the importance of acknowledging the diverse range of responses and factoring in the heterogeneity of driving behaviors.

This analysis suggests two predominant car-following strategies. The first group, risk-taking drivers, tend not to adapt their behavior even when task demands exceed their capacity. Their safety margins are narrow, often operating closer to the boundaries of risk. In contrast, risk-averse drivers adapt their behavior when task demands exceed their capacity. They prioritize safety by creating larger buffers or headways, ensuring sufficient reaction time even under challenging conditions. The behavior adaptation regime (BAR) in this model is designed to capture these divergent strategies, providing a more nuanced representation of driver heterogeneity in risk-taking behavior.

Mathematical Formulation

Building on the behavioral concepts introduced earlier, the proposed IDMTS is formalized by specifying the acceleration function as a regime-dependent control law. In this model, behavioral adaptation and risk-taking are endogenized. This allows behavior to evolve dynamically rather than being imposed externally. Task saturation triggers a BAR when the driving task becomes challenging. The central idea is that drivers do not respond uniformly across all contexts but operate within three qualitatively distinct regimes. Therefore, acceleration is expressed as a function of spacing $s$ , speed $v$ , and relative speed $Δ v$ as $a (t) = f (s, v, Δ v)$ , where the functional form of $f (\cdot)$ depends on the prevailing regime. The mathematical formulations for each regime are detailed as follows.

Free Driving Regime

When the leader is distant and its influence is negligible, the driver accelerates toward the desired speed $v_{0 i}$ .

{(\frac{dv}{dt})}_{FDR} = a_{i} [1 - {(\frac{v_{i} (t)}{v_{0 i}})}^{4}]

(8)

This term reflects the natural tapering of acceleration as the desired speed is approached.

Car-Following Regime

When interaction with the leader is dominant, acceleration depends on the ratio between desired and actual spacing.

{(\frac{dv}{dt})}_{CFR} = a_{i} [1 - {(\frac{s_{i}^{*} (v, Δ v)}{s_{i} (t)})}^{2}]

(9)

with $s_{i}^{*}$ given in Equation 6.

Behavior Adaptation Regime

This regime encapsulates two interrelated mechanisms: (1) driver adaptation in response to variations in task saturation; and (2) risk sensitivity that governs how strongly this adaptation is expressed. Following the modeling principles of Treiber et al. ( 31 ), introducing minimal new parameters, preserving the interpretation of existing ones, and maintaining parsimony, the BAR extends the IDM + model with two additional behavioral parameters: (1) the adaptation smoothness exponent $γ_{i}$ ; and (2) risk sensitivity $δ_{i}$ .

Formulating Driver Behavior Adaptation in Response to Task Saturation

Drivers consistently make instantaneous decisions to ensure that the perceived level of driving task saturation remains within their acceptable limit. Drivers maintain their desired level of task saturation based on “task difficulty homeostasis” of Fuller ( 5 ). In essence, drivers are inclined to decelerate (reduce speed or increase the gap between vehicles) when they feel overwhelmed by task saturation. In contrast, they accelerate when they sense that task saturation is below their capability. Therefore, the rate of acceleration in the BAR is inversely proportional to the task saturation. The task saturation, articulated for speed, spacing, and desired time gap, referenced in Equation 4, is integral to the BAR in this model. A noteworthy aspect is the drivers’ inclination toward a certain deceleration curve when they perceive high task saturation. The smoothness factor $γ_{i}$ modulates this curve (see Equation 10). A higher value of $γ_{i}$ implies a more abrupt deceleration (or acceleration) in response to high (or low) task saturation. In contrast, lower $γ_{i}$ signifies a more gradual deceleration (or acceleration) pattern. In simpler terms, the larger $γ_{i}$ is the quicker drivers adapt their acceleration as they approach their preferred task saturation level, and vice versa.

Formulating Risk-Taking Strategy and Behavior Adaptation

The degree to which drivers adapt their behavior depends on their individual risk-taking strategies. Therefore, could every driver possess a unique risk-taking strategy when exposed to higher task saturation, symbolized by $δ_{i}$ (see Equation 10), a parameter that captures risk sensitivity. It is a personal metric reflecting a driver’s assessment of driving-associated risks. A higher $δ_{i}$ suggests that drivers are more risk-averse and more inclined to adapt their behavior when a driving task becomes difficult. In contrast, with lower $δ_{i}$ , drivers are less sensitive to risks and less likely to adapt their response. The parameter spans $0 \leq δ_{i} < 1$ . Of note, the endogenous behavioral adaptation in the IDMTS does not arise from temporal variation in the risk sensitivity parameter $δ_{i}$ itself. Instead, endogeneity emerges from the interaction between the driver-specific parameter $δ_{i}$ and the dynamically evolving task saturation state. Because task saturation is a nonlinear function of instantaneous speed, spacing, and desired time gap, it varies continuously as traffic conditions evolve. When the task saturation level exceeds the driver’s tolerance threshold, the control law activates the BAR, leading to compensatory deceleration. Therefore, the same driver (with fixed $δ_{i}$ ) may operate in different regimes at different moments. A comprehensive discussion of the nuances of $δ_{i}$ and the underlying risk-taking strategies of drivers are discussed in subsequent sections. Building on this discourse, the BAR of this model is mathematically derived in Equation 10.

{(\frac{dv}{dt})}_{BAR} = a_{i} [1 - \frac{1}{(1 - δ_{i})} {(\frac{v_{i} (t) T_{i}}{s_{i} (t) + ε})}^{γ_{i}}]

(10)

where $δ_{i}$ represents an individual driver’s unique risk sensitivity parameter and $γ_{i}$ represents a smoothing factor that shows the drivers how smoothly they decelerate to achieve their desired task saturation. In Equation 10, vehicles decelerate proportionally to the gap between the current TS and a driver-specific tolerable level, with $γ_{i}$ shaping the smoothness of approach and $δ_{i}$ governing how early adaptation starts. This maintains the IDMTS responsive yet numerically well-behaved under bounded $a, b$ . Assembling the three regimes gives the IDMTS in Equation 11.

\begin{matrix} \frac{dv}{dt} = a_{i} \cdot \min \\ [\underset{\begin{matrix} Free Driving \end{matrix}}{\underset{︸}{1 - {(\frac{v_{i} (t)}{v_{0 i}})}^{4}}}, \underset{\begin{matrix} Car Following \end{matrix}}{\underset{︸}{1 - {(\frac{s_{i}^{*} (v, Δ v)}{s_{i} (t)})}^{2}}}, \underset{\begin{matrix} Behavior Adaptation \end{matrix}}{\underset{︸}{1 - \frac{1}{(1 - δ_{i})} {(\frac{v_{i} (t) T_{i}}{s_{i} (t) + ε})}^{γ_{i}}}}] \end{matrix}

(11)

To ensure that acceleration remains within physical and behavioral limits $[- b_{i}, a_{i}]$ , the BAR term must satisfy,

- b_{i} < a_{i} [1 - \frac{1}{(1 - δ_{i})} {(\frac{v_{i} (t) T_{i}}{s_{i} (t) + ε})}^{γ_{i}}] < a_{i}

(12)

Solving for the admissible range of $δ_{i}$ yields,

δ_{i} < 1, δ_{i} < \frac{1}{1 + \frac{b_{i}}{a_{i}}} [1 - {(\frac{v_{i} (t) T_{i}}{s_{i} (t) + ε})}^{γ_{i}}]

(13)

Values of $δ_{i}$ approaching unity results in excessive braking and potential instability; therefore, $0 \leq δ_{i} < 1$ is enforced in all simulations.

Illustration of Behavioral Adaptation and Risk-Taking Strategy With an Example of Distracted Driver

Figure 1 shows the driver behavior adaptation, emphasizing the difference between risk aversion and risk-taking strategies. In the first scenario presented in Figure 1a, the driver is risk-averse and characterized by high risk sensitivity (i.e., high risk sensitivity parameters δ). The driver’s journey can be broken down into specific phases.

Figure 1.

Driver behavioral adaptation and risk-taking strategy in the IDMTS model: (a) a risk-averse driver (high risk sensitivity $δ$ ) who adapts headway and speed when task demand exceeds capacity; and (b) a risk-taking driver (low $δ$ ) who sustains higher speed and shorter spacing under distraction.

This figure links model control terms, task demand profiles, and traffic situations to highlight the integration of behavioral regimes (free driving regime [FDR], car-following regime [CFR], and BAR) within the proposed framework.

Moving without the interaction phase (time t₀). The driver moves effortlessly at their desired speed, free from interaction with the leading vehicle, resulting in an uncomplicated drive that is well within their control. In this instant, the driving behavior is modeled by the FDR of IDMTS (i.e., the first term of the model).

Merging vehicle in front phase (time t₁). A new element is introduced as another vehicle merges directly ahead of the ego driver, inducing a change in dynamics. The short headway with this new vehicle increases task demand on the ego vehicle, though the driving task is still sufficiently within their skill set to manage, as the driving task is less than the driver’s capacity. In this phase, the interaction with the leading vehicle is controlled by the CFR of this model (i.e., the second term of the model).

Distraction phase (time t₂). The driver receives a phone call. This external distraction acts as an additional layer of task demand to the already evolving driving scenario. The total task demand outweighs the driver’s capacity, intensifying the challenge of the driving task. In this phase, the transition from CFR to BAR occurs, and the third term of the IDMTS model controls the driving behavior.

Behavior adaptation phase (time t₂–t₄). Adopting a risk-averse strategy, the driver adapts their behavior in this situation by reducing speed, naturally increasing the headway, and decreasing total task demand. This risk aversion strategy is maintained consistently until the phone conversation ends (t₄). Therefore, the BAR of this model controls the driving behavior in this period until the driver begins the recovery to normal driving behavior.

Back to normal phase (time t₄). When the conversation ends, the driver wants to return to a normal situation. In this phase, the transition from BAR to CFR occurs. Therefore, the model shifts from BAR to CFR and the second term of the IDMTS controls the driving behavior.

Recovery phase (time t₅). The driver wants to recover the lost time by increasing speed. The second term of this model controls the driving behavior.

In contrast, Figure 1b shows a risk-taking driver scenario. While the initial conditions mirror the previous scenario, a notable divergence occurs when the driver receives a call at time t₂. Unlike the risk-averse driver, this driver does not adapt their behavior because of increased task demand, maintaining their speed and distance from the vehicle ahead. This lack of behavior adaptation makes driving tasks risky during the phone call conversation (from t₂ to t₄). This choice, coupled with the added distraction, elevates the difficulty and risk during the conversation, signifying a driving strategy marked by lower risk sensitivity (i.e., low risk sensitivity δ).

Steady State Equilibrium and the Fundamental Diagram of IDMTS

In microscopic models, it is essential to have uniform driver–vehicle combinations operating on a consistent road environment to describe a stable equilibrium state. This essentially means that all drivers and vehicles are characterized by the same model parameters. The acceleration or speed metrics relevant to the model are not distinguished by the type or identity of the vehicle. From a modeling perspective, this steady equilibrium can be defined by two primary conditions that are homogeneous traffic and no accelerations ( 31 ). When considering that $\frac{dv}{dt} = 0$ and $Δ v = 0$ in Equation 10 ( $ε$ is neglected to simplify notation.), it leads to a relationship between speed and spacing as in Equation 14.

V^{e} (s) = {\begin{matrix} \min [v_{0}, \frac{s - s_{0}}{T}, (1 - δ) \frac{s}{T}] s > s_{0} and δ < 1 \\ 0 otherwise \end{matrix}

(14)

where $V^{e}$ represents equilibrium speed, and the other variables and parameters have already been described.

From Equation 14, in the FDR, a driver’s steady state equilibrium speed aligns with their desired speed $v_{0}$ . In contrast, under the CFR, the equilibrium speed is determined by the driver’s spacing and their desired headway between vehicles. Delving further into the BAR scenario, the speed becomes a function of the spacing and desired headway and is influenced by the risk sensitivity parameter $δ$ . Of interest, the risk sensitivity parameter affects the equilibrium speed when the risk sensitivity is high (i.e., risk-averse drivers) and it will not affect when the risk sensitivity is low (i.e., risk-taking drivers). Therefore, the degree to which a driver adapts their speed (and headway, or both) depends on the magnitude of their risk sensitivity parameter. This highlights the intricate interplay between individual driver characteristics and broader traffic dynamics. To establish a micro–macro connection between the micro-level spacing and the macro-level density, $ρ = 1 / s$ , $ρ_{\max} = 1 / s_{0}$ , and $q_{\max} = 1 / T$ are replaced in Equation 14.

\begin{matrix} V^{e} (ρ) = \\ {\begin{matrix} \min (v_{0}, q_{\max} (\frac{1}{ρ} - \frac{1}{ρ_{\max}}), (1 - δ) \frac{q_{\max}}{ρ}) ρ < ρ_{\max} and δ < 1 \\ 0 otherwise \end{matrix} \end{matrix}

(15)

where ρ and $ρ_{\max}$ represent the density and maximum density, respectively, and $q_{\max}$ represents the maximum flow. The equilibrium flow can be obtained as $Q^{e} (ρ) = ρ V^{e} (ρ)$ . Replacing $V^{e} (ρ)$ from Equation 15, this yields Equation 16, as follows:

\begin{matrix} Q^{e} (ρ) = \\ {\begin{matrix} \min (v_{0} ρ, q_{\max} (1 - \frac{ρ}{ρ_{\max}}), (1 - δ) q_{\max}) ρ < ρ_{\max} and δ < 1 \\ 0 otherwise \end{matrix} \end{matrix}

(16)

Figure 2 shows the equilibrium surfaces of the IDMTS model as a function of density and risk sensitivity. Figure 2a shows the flow–density–risk sensitivity surface, which retains the expected unimodal fundamental diagram shape. The influence of the risk sensitivity parameter $δ$ on macroscopic traffic behavior is exhibited primarily through equilibrium modification. Higher values of $δ$ shift the system toward lower equilibrium speeds and reduced maximum flow, effectively advancing the onset of capacity drop. Figure 2b shows the corresponding speed–density–risk sensitivity surface. Speed decreases monotonically with density, with sharper reductions for higher values of $δ$ . These results highlight that greater risk sensitivity (i.e., risk-averse drivers) is associated with lower equilibrium speeds and flows, and lower $δ$ values (i.e., risk-taking drivers) sustain higher flow and speed by tolerating elevated task demands.

Figure 2.

Equilibrium surfaces of the IDMTS model under varying densities and risk sensitivities and IDM +( $δ$ = 0): (a) flow–density–risk sensitivity surface; and (b) speed–density–risk sensitivity surface.

From a macroscopic behavioral perspective, the IDMTS demonstrates behavioral coherence via equilibrium restructuring. Higher risk sensitivity $δ$ leads to earlier activation of behavioral adaptation, resulting in lower equilibrium speeds and reduced maximum flow. This demonstrates an earlier capacity drop for risk-averse driver populations. These macroscopic outcomes are consistent with behavioral expectations: conservative drivers maintain larger safety margins, reducing flow and density under elevated task demand. Therefore, the IDMTS is behaviorally sound from a macroscopic perspective.

To illustrate the transition between regimes and understand the task saturation that triggers regime transition, the IDMTS model was simulated for variations in speed and spacing. Figure 3 shows the phase diagram for a risk-averse driver ( $δ$ = 0.8), a moderate driver ( $δ$ = 0.5), and a risk-taking driver ( $δ$ = 0.2). The simulation used parameter values $a$ = 1.0 m/ $s^{2}$ , $b$ = 1.5 m/ $s^{2}$ , $s_{0} =$ 2.0 m, $v_{0}$ = 120 km/h, and $T$ = 1.2 s, with 2,500 combinations of speed (1–25 m/s) and spacing (10–100). Each dot in Figure 3 indicates a simulation experiment. Of note, drivers are considered to have identical driving skills (e.g., the same task capacity) aside from risk sensitivity. For the risk-averse driver (Figure 3a), the BAR (blue) dominates a larger region of the speed–spacing space, particularly at moderate-to-high speeds and shorter spacings. This indicates that risk-averse drivers adapt more readily under elevated task demand, reducing speed and increasing spacing to maintain acceptable task saturation. Therefore, the FDR (red) is confined to lower speeds and larger spacings, and the CFR (green) contracts compared with the risk-taking case. Task saturation is generally lower and more evenly distributed, reflecting conservative behavior and stronger compensatory adaptations.

Figure 3.

Regime phase diagrams of the IDMTS model under varying risk sensitivities. As δ increases, BAR dominates a larger $(v, s)$ region showing behavior adaptation.

In contrast, the risk-taking driver (Figure 3c) shows BAR in a limited portion of the space. Instead, the CFR dominates at moderate spacings, and the FDR extends over a wider range of speeds and headways. This behavior highlights that risk-taking drivers sustain higher speeds and shorter spacings under elevated task demands without significant adaptation, resulting in consistently higher task saturation. The moderate driver is somehow between the risk-taking and risk-averse drivers (Figure 3b). The comparison underscores that higher risk sensitivity induces earlier behavioral adaptation and greater safety margins, and lower risk sensitivity sustains aggressive maneuvers with elevated task saturation. These differences align with Fuller’s TCI framework: risk-averse drivers adjust control variables (speed and headway) to reduce overload ( 25 ), and risk-taking drivers tolerate or even ignore overload, accepting narrower safety margins.

Stability Analysis and Rational Driving Constraints

Since the early development of car-following models, researchers have examined their stability by analyzing how small perturbations evolve over time (local stability) and across a platoon (string stability). This analysis primarily focuses on the progression of a minor disturbance from a leading vehicle temporally and spatially. Two types of stability are investigated: (1) local linear stability; and (2) string linear stability. Local stability concerns the behavior of a vehicle platoon over time when subjected to a small disturbance. A system is considered locally stable if a perturbation does not grow over time, allowing the system to return to equilibrium. String linear stability deals with the spatial stability of a vehicle platoon under a small disturbance originating from the lead vehicle. A system is deemed string stable if the disturbance decreases progressively across the vehicles; otherwise, it is string unstable ( 32 ). In this study, these two types of stability are discussed and the rational driving constraints that a car-following model should possess.

Local Stability and Rational Driving Constraints

Local instability is a particularly severe form of instability, and it is fundamental to design car-following models to avoid it. To analyze the local stability of our proposed car-following model, we consider the steady state equilibrium flow. The car-following model is structured as ${\overset{\cdot}{v}}_{i} (t) = f {(s_{i}, v_{i}, Δ v_{i})}_{t}$ and this model is formulated as $f (s_{e}, v_{e}, 0)_{t} = 0$ in the steady state equilibrium condition, where $s_{e}$ and $v_{e}$ are the equilibrium spacing and speed, respectively. Then, the equilibrium state is perturbed as follows:

y_{i} (t) = s_{i} (t) + s_{e}

(17)

u_{i} (t) = v_{i} (t) + v_{e}

(18)

where $s_{i} (t)$ and $v_{i} (t)$ represent small spacing and speed perturbation. The system is stable if the disturbances approach zero over time. For the local stability analysis with small perturbations, a multivariate first-order Taylor expansion is performed on the right-hand side around the equilibrium point, which results in the linearization of the system (further details can be found in the literature [ 33 ]). Therefore, the stability criterion, the local stability criterion of a car-following without time delay, can be expressed.

f_{v} - f_{Δ v} < 0

(19)

where $f_{v}$ and $f_{Δ v}$ represent the Taylor expansion coefficients of the acceleration function at the steady state, for different variables after first-order linearization. To describe how the acceleration changes with respect to small perturbations in spacing and speed around the equilibrium point, these coefficients are expressed as follows:

f_{s} = \frac{\partial f}{\partial s} |_{e}, f_{v} = \frac{\partial f}{\partial v} |_{e}, f_{Δ v} = \frac{\partial f}{\partial Δ v} |_{e}

(20)

In addition to the local stability requirement for a newly developed car-following model, the rational constraints of car-following models are fundamental to ensuring that the model’s dynamics are rational. Therefore, Wilson et al. ( 34 ) investigated the rational driving constraints of car-following models and concluded that the partial derivatives of sensible car-following models should satisfy the following conditions:

f_{s} > 0, f_{Δ v} > 0 and f_{v} < 0

(21)

Therefore, in two scenarios with all other factors identical, greater spacing should lead to greater acceleration (or less braking). A higher (more positive) relative velocity should result in increased acceleration (less braking) as the leading vehicle is moving away. In addition, as a vehicle’s own velocity increases, the tendency to accelerate should decrease (or the tendency to brake should increase) ( 34 ). Theoretically, Equation 21 should be modified to allow $f_{s} = 0 and f_{v} = 0$ at larger spacings, where the driving behavior is largely independent of the leading vehicle. Therefore, the model should behave as follows under conditions where there is no interaction with leading vehicles:

f_{s} = 0 and f_{v} = 0

(22)

For reasonable car-following models that satisfy these rational driving constraints, which are shown in Equations 21 and 22, the local stability of such car-following models is automatically guaranteed ( 34 ). Therefore, the rational driving constraints expressed in Equations 21 and 22 are investigated to satisfy the local stability and rational constraints at the same time. The partial derivatives for each regime of the car-following model are taken separately. For FDR, Equation 21 leads to the Taylor expansion coefficients given in Equation 23.

{(f_{s})}_{FDR} = 0, {(f_{v})}_{FDR} = - \frac{4 {av}_{e}^{3}}{v_{0}^{4}}, {(f_{Δ v})}_{FDR} = 0

(23)

Similarly, for CFR, the Taylor expansion coefficients are as follows:

{(f_{s})}_{CFR} = \frac{2 a}{s_{e}} {(\frac{s_{0} + {Tv}_{e}}{s_{e}})}^{2}

(24)

{(f_{v})}_{CFR} = - a [\frac{2 T (s_{0} + {Tv}_{e})}{s_{e}^{2}}]

(25)

{(f_{Δ v})}_{CFR} = \sqrt{\frac{a}{b}} \frac{v_{e}}{s_{e}} \frac{s_{0} + {Tv}_{e}}{s_{e}}

(26)

Finally, the Taylor expansion coefficients for the BAR regime are represented by Equations 27–29.

{(f_{s})}_{BAR} = \frac{a γ v_{e} T^{γ}}{(1 - δ)} s_{e}^{- γ - 1}

(27)

{(f_{v})}_{BAR} = - \frac{a γ}{(1 - δ)} (\frac{v_{e} T}{s_{e}})^{γ - 1} \frac{T}{s_{e}}

(28)

{(f_{Δ v})}_{BAR} = 0

(29)

The local stability of each regime will be evaluated. In the FDR, there is no interaction with leading vehicles. Therefore, the vehicle relative driving concerning spacing and relative speed must be zero according to Equation 22. Comparing the stability condition expressed in Equation 30, the stability condition is expressed as follows:

{(f_{s})}_{FDR} = 0, {(f_{v})}_{CFR} = - \frac{4 {av}_{e}^{3}}{v_{0}^{4}} < 0, {(f_{Δ v})}_{FDR} = 0

(30)

Equation 30 meets the stability condition in Equation 22. Therefore, the stability and rational constraints criteria are met for the FDR. The stability criteria for CFR can be met by the following:

{(f_{s})}_{CFR} = \frac{2 a}{s_{e}} {(\frac{s_{0} + {Tv}_{e}}{s_{e}})}^{2} > 0

(31)

{(f_{v})}_{CFR} = - a [\frac{2 T (s_{0} + {Tv}_{e})}{s_{e}^{2}}] < 0

(32)

{(f_{Δ v})}_{CFR} = \sqrt{\frac{a}{b}} \frac{v_{e}}{s_{e}} \frac{s_{0} + {Tv}_{e}}{s_{e}} > 0

(33)

Because all parameters and variables are positive inequalities, verifying that the previous inequalities are true for all values of parameters and variables is easy. Therefore, the CFR meets the local stability and rational driving constraints accordingly. Finally, the BAR local stability and rational driving constraints are expressed as follows:

{(f_{s})}_{BAR} = \frac{a γ v_{e} T^{γ}}{(1 - δ)} s_{e}^{- γ - 1} > 0

(34)

{(f_{v})}_{BAR} = - \frac{a γ}{(1 - δ)} (\frac{v_{e} T}{s_{e}})^{γ - 1} \frac{T}{s_{e}} < 0

(35)

{(f_{Δ v})}_{BAR} = 0

(36)

If $δ < 1$ , all inequalities are true for the positive values of parameters and variables. The proposed car-following model meets the local stability criteria and the rationale driving constraints.

String Stability

Stability analysis is a fundamental requirement for evaluating the plausibility of car-following models because it ensures that small perturbations in speed or spacing do not escalate into unrealistic traffic oscillations. The general stability of car-following models (with no time delay) is indicated in the following equation:

\frac{1}{2} - \frac{f_{Δ v}}{f_{v}} - \frac{f_{s}}{f_{v}^{2}} > 0

(37)

Following the Sun et al. (2018) setup, the string stability of each regime in this model was evaluated. The length of the vehicle was $l$ = 5 m, minimum gap $s_{0}$ = 2 m, and the desired speed of $v_{0}$ = 50 and 120 km/h. To observe the string stable and unstable regions, the desired time headway $T$ (0.1–4.0 s), and maximum acceleration $a$ (0.1–4.0 m/s²) were varied for CFR. In total, 2,500 combinations of parameters were considered to observe and visualize stability regions. The smoothing factor $γ$ (0.1–4.0) and risk sensitivity $δ$ (0.0–0.9) were varied to observe the effect of these parameters on the stability of the BAR.

The string stability condition of FDR, following Equation 37, leads to $\frac{1}{2} > 0$ . The stability criterion is satisfied for all parameters, implying that the system is stable under the given conditions. Therefore, the stability criterion for the given values is always met, indicating that this regime is string stable. This is plausible because, in this regime, the vehicle does not interact with other vehicles, and string stability is not an issue. The CFR following Equation 37 gives the stability of this regime as Equation 38.

1 + \frac{v_{e}}{T} \sqrt{\frac{1}{ab}} - \frac{s_{e}}{{aT}^{2}} > 0

(38)

To assess the effect of each parameter, the stability regions for the CFR for various values of the desired headway and maximum acceleration were plotted (Figure 4). Figure 4 shows that increasing either the maximum acceleration or the desired time headway improves the string stability. Further, higher speed tends to stabilize the system, provided the other parameters remain constant.

Figure 4.

String stability region of the IDMTS model in the car-following regime (CFR) for two desired speeds: (a) 50 km/h; and (b) 120 km/h.

The solid red line separates stable from unstable regions. Each point represents a parameter combination that yields string instability in simulation, while the empty region corresponds to string-stable parameter sets.

Simplifying the string stability condition for the BAR leads to $a γ v_{e}^{2 γ - 3} T^{γ} > 2 (1 - δ)$ . To assess the influence of each parameter on the string stability of this regime, the stability condition for various risk sensitivity $δ$ and the smoothing factors $γ$ are plotted because these two factors play a more significant role in the stability of this regime (Figure 5). The string stability analysis reveals that the smoothing factor $γ$ exerts a stronger direct influence on perturbation damping than the risk sensitivity parameter $δ$ . This result warrants clarification on the distinct functional roles of these parameters. The parameter $γ$ primarily governs the dynamic smoothness of behavioral adaptation. Through its direct appearance in the acceleration derivatives, it shapes how rapidly drivers adjust acceleration in response to deviations from equilibrium. Therefore, $γ$ strongly influences the amplification or attenuation of small disturbances in a vehicle platoon. In contrast, the parameter $δ$ operates at a different dynamical level. Rather than controlling perturbation damping directly, $δ$ determines the tolerance threshold for task saturation exceedance, the onset of the BAR, and the equilibrium speed–spacing relationship under overload conditions. Therefore, while $γ$ governs local oscillation propagation around a given operating point, $δ$ shifts the operating point by modifying equilibrium speed, flow, and regime occupancy. This distinction explains why $γ$ dominates linear string stability conditions, and $δ$ significantly affects macroscopic characteristics, such as capacity and equilibrium flow (as shown in Figure 2).

Figure 5.

String stability region of the IDMTS model in the behavioral adaptation regime (BAR) for two speeds: (a) 50 km/h; and (b) 120 km/h.

The solid red line separates stable and unstable regions. Each point denotes a parameter combination that yields string instability in simulation, while the empty region corresponds to string-stable outcomes as a function of the smoothing factor and risk sensitivity.

Calibration and Validation

Models are simplified approximations of the real-world, which implies that they are subject to errors and limitations ( 36 ). The primary objective is to narrow down these discrepancies and ensure that the model’s outcomes align closely with observed values via calibration.

Data

Three complementary data sets were considered for model calibration and validation: (1) controlled driving simulator experiments; (2) naturalistic trajectory data from the NGSIM program ( 37 ); and the MiTra data set ( 38 ). The motivation for combining these data sets lies in their distinct advantages and limitations. Driving simulator data enables the simultaneous capture of trajectory information and HF inputs such as distraction, which are rarely observable in real-world trajectory data sets. They also allow systematic manipulation of behavioral conditions within a consistent driver population. However, simulator environments may not fully replicate naturalistic driving, particularly for car-following distances and risk-taking tendencies. In contrast, naturalistic data provides high-resolution, real-world trajectory data that reflects authentic traffic conditions and heterogeneous driver behaviors. However, it lacks direct HF measurements, such as distraction or risk sensitivity. By integrating both sources, the proposed IDMTS model is calibrated and validated against the behavioral richness of simulator data and against the empirical realism of freeway traffic, ensuring both behavioral soundness and practical applicability.

Driving Simulator Data

Calibration of the proposed model needs trajectory data and HF inputs. For this calibration, trajectory data derived from simulations in which participants were subject to distractions from mobile phone conversations during their drive were utilized. This experiment used the CARRS-Q Advanced Driving Simulator, as detailed in the study by Saifuzzaman et al. ( 21 ). The participant group consisted of 32 drivers. Each participant was subjected to two distinct driving scenarios within the simulator: (1) normal driving without any phone conversation; and (2) driving while engaged in a conversation through a handheld mobile phone.

The focus was on car-following, which took place on city roads with a designated speed limit of 40 km/h. A 245-m-long section of the road was chosen to monitor the car-following behaviors of every participant. This particular road stretch consisted of four lanes moving in a single direction. Vehicles were parked in outer lanes, restricting driving to the two middle lanes.

During this car-following event, the participant’s vehicle stopped at signals. No intervening vehicles were between the participant’s car and the leading ones. As the gap between the participant’s vehicle and the foremost vehicle diminished to 60 m, both leading cars increased their speed to 20 km/h. When the gap narrowed to 30 m or less, these vehicles surged to 35 km/h, maintaining that velocity until the car-following event was completed. The signal at the second intersection remained green to guarantee smooth flow. Essential data observation points, such as the position and velocity of the vehicles, were recorded every 0.05 s ( 21 ). From the driver population, 22 participants were randomly selected for model calibration, and the remaining participants were used for validation.

NGSIM I-80 Data Set

To complement the controlled experiment data, empirical trajectory data from the NGSIM data set were also utilized ( 39 ). The data set captures detailed vehicle movements along a 500-m segment of Interstate 80 (I-80) in Emeryville, California (Figure 6). Vehicle trajectories were recorded at 10 Hz, providing high-resolution measurements of vehicle positions, velocities, accelerations, and headways under varying traffic conditions. Because raw NGSIM trajectories are known to contain measurement noise and occasional inconsistencies, preprocessing was conducted before model calibration. Trajectory denoising and reconstruction were performed following the methodology proposed by Montanino and Punzo ( 40 ), which has been widely adopted in studies using NGSIM data to reduce high-frequency measurement noise and improve kinematic consistency.

In this study, Lane 2 was the focus, which exhibited relatively uninterrupted flow with fewer lane changes compared with the adjacent merge and weaving lanes. The full 45 min of data were considered, subject to stringent filtering criteria to ensure valid leader–follower pairs. Specifically: (1) only passenger cars were retained; (2) each car-following episode was required to last at least

60 s; (3) leader and follower were required to remain in the same lane throughout the observation window; and (4) a minimum initial speed differential of 5 m/s was imposed to exclude passive or nonreactive sequences ( 41 ). These filtering steps ensured that the retained trajectories represented active longitudinal interactions suitable for car-following analysis. After preprocessing and filtering, 396 valid car-following pairs were identified. These were randomly divided into 277 pairs (70%) for calibration and 119 pairs (30%) for validation. The resulting data set covers a broad range of traffic states, including free-flow and congested conditions, with corresponding variations in vehicle speeds and spacing.

MiTra Data Set

To complement the I-80 data set and mitigate its limitations for spatial coverage and observation duration, the MiTra data set was also utilized ( 38 ). While I-80 covers mostly congested traffic, MiTra has freeway traffic with varied traffic conditions. The data set was collected using six unmanned aerial vehicles (drones) flying in formation over a 900-m section of the A50 urban freeway in Milan, Italy (see Figure 7). Trajectory data were captured across nine flight campaigns totaling 135 min, spanning all traffic states from free-flow to heavily congested conditions, and include detailed vehicle positions, speeds, accelerations, and surrounding vehicle identifiers. Data were extracted at a sampling frequency of 30 frames per second. For this study, analysis was restricted to the mainline lanes (Lanes 1–7), which carry uninterrupted directional flow, while ramp and auxiliary lanes (0, 10, 11, 20, and 21) were excluded to avoid the confounding effects of merging and diverging maneuvers on car-following behavior. The same trajectory selection criteria used for the NGSIM data were applied. However, the MiTra data set has already been denoised. After preprocessing and filtering, 101 valid car-following pairs were identified. These were randomly divided into 70 pairs (70%) for calibration and 31 pairs (30%) for validation.

Figure 6.

Layout of the NGSIM I-80 study site in Emeryville, California.

Figure 7.

Aerial view of the Milan Trajectories (MiTra) data set study site, showing the monitored freeway segment and the coverage areas of the six drones used for trajectory extraction ( 38 ).

Measures of Performance and Goodness-of-Fit

The calibration problem was formulated by defining the model’s measures of performance (MoP) as a function of the parameter vector $β$ , as expressed in Equation 39. The objective is to minimize the deviation between observed and simulated MoP using a predefined goodness-of-fit (GoF) function, subject to parameter bounds, as shown in Equations 39–41.

{MoP}^{sim} = F (β)

(39)

\min_{β} f ({MoP}^{obs}, {MoP}^{sim})

(40)

s . t . {LB}_{β} \leq β \leq {UB}_{β}

(41)

where $β$ denotes the vector of model parameters to be estimated, with lower and upper bounds ${UB}_{β}$ and ${LB}_{β}$ , respectively, $F (\cdot)$ represents the car-following model under calibration, ${MoP}^{obs}$ and ${MoP}^{sim}$ represent the observed and simulated performance measures (e.g., spacing), and $f (\cdot)$ represents the GoF criterion, which is minimized to obtain the parameter set that best reproduces observed trajectories ( 42 ).

To evaluate model performance, various GoF metrics and MoP are employed in the literature. Punzo et al. ( 36 ) suggested that when calibrating car-following models, focusing on spacing yields optimal optimization for spacing and suboptimal results for speed. In contrast, calibrating on speed would result in an optimal speed trajectory but would leave spacing indeterminate. Therefore, spacing is preferred as MoP and was adopted in this study. For the GoF, the preferred root mean square error (RMSE) ( 36 ) was adopted, which is defined in Equation 42. Table 2 gives the range of parameters for the calibration of models.

RMSE (s) = \sqrt{\frac{1}{T} \sum_{t = 1}^{T} (s^{obs} (t) - s^{sim} (t))^{2}}

(42)

where $s^{obs} (t)$ represents the observed spacing at time t, $s^{sim} (t)$ represents the simulated spacing at time t, and $T$ represents total simulation time.

Table 2.

Parameters Range for Calibration of Models

Parameter description	Parameter (unit)	Minimum	Maximum
Maximum acceleration	$a$ (m/s²)	0.50	4.00
Desired deceleration	$b$ (m/s²)	0.50	4.50
Minimum spacing	$s_{0}$ (m)	1.00	10.00
Desired speed	$v_{0}$ (km/h)	36.00	120.00
Desired headway	$T$ (s)	0.20	3.00
Risk sensitivity	$δ$	0.00	0.90
Smoothness factor	$γ$	1.00	4.00

Optimization Algorithm

The calibration process seeks to determine the model parameters that reduce the difference between the simulated values and observed data. In general, this is conducted using optimization techniques ( 43 , 44 ). In this study, the Whale Optimization Algorithm (WOA) was used to identify the optimum model parameters ( 45 ). The WOA is a population-based metaheuristic inspired by the bubble-net hunting behavior of humpback whales ( 45 ). The algorithm alternates between exploration, which searches broadly across the solution space, and exploitation, which refines candidate solutions near promising regions. These processes are implemented using mechanisms that mimic whale behaviors such as encircling prey, spiral bubble-net attacking, and random search ( 45 ). Such a balance between exploration and exploitation helps prevent premature convergence while maintaining efficient local search capability. Therefore, the WOA has demonstrated strong performance in solving complex nonlinear optimization problems and has been successfully applied across a wide range of engineering optimization tasks ( 46 ).

Car-following model calibration typically involves highly nonlinear objective functions with multiple local minima, where analytical gradients are unavailable and derivative-based optimization techniques are unsuitable ( 47 , 48 ). Population-based metaheuristics such as WOA are particularly appropriate for these problems because they perform global search without requiring gradient information and can effectively explore complex parameter spaces ( 46 ). To ensure robustness and reproducibility, identical optimization settings were applied across all experiments. The WOA was executed with a population size of 200 over 100 iterations. Multiple calibration runs with different initial populations produced consistent objective function values and parameter estimates, indicating that the optimization results were stable and not excessively sensitive to initialization. Convergence of the objective function was also monitored across iterations, and stable convergence behavior was observed.

Results and Discussions

This section evaluates the proposed model from two complementary perspectives: numerical soundness and behavioral soundness. Numerical soundness concerns the model’s quantitative validity, which is its ability to reproduce observed trajectories with low error and maintain plausible parameters. Behavioral soundness, in contrast, reflects the model’s plausible behavioral credibility, assessing whether simulated responses and regime transitions accord with established human behavior ( 2 ). At the microscopic level, behavioral soundness evaluates whether simulated speed, spacing, acceleration patterns, and endogenous regime transitions are consistent with established HF theories such as task difficulty homeostasis. This microscopic assessment focuses on trajectory-level behavioral adaptation under varying task demands. At the macroscopic level, behavioral soundness has already been examined analytically in earlier sections using equilibrium analysis and string stability properties. Considering both microscopic simulation behavior and macroscopic analytical properties is essential: a model that is numerically accurate but behaviorally implausible offers limited explanatory value ( 2 , 3 ). Evaluating the IDMTS using these complementary lenses provides a balanced assessment of its robustness, realism, and theoretical consistency.

Driving Simulator Experiment

Numerical Soundness

The calibration results obtained from the driving simulator experiments provide an initial assessment of the numerical soundness of the proposed IDMTS relative to the baseline IDM+. Both models were calibrated separately under normal and distracted driving conditions. The results presented in Table 3 indicate the calibrated parameters for the IDM+ and IDMTS models derived from normal driving trajectory data. The parameters obtained are plausible for the IDM+ and IDMTS models. The close correspondence between the IDM+ and IDMTS parameters demonstrates an important property of the proposed formulation: the addition of the cognitive layer enhances model responsiveness to changing task demands without disturbing the fundamental dynamics or parameter interpretability of the base model. The small reduction in calibration and validation RMSE confirms that the IDMTS improves numerical accuracy while preserving parsimony and numerical stability.

Table 3.

Calibrated Parameters of IDM+ and IDMTS with Normal Driving Data

Parameter	IDM+ (normal driving)				IDMTS (normal driving)
Parameter	Mean	SD	Min.	Max.	Mean	SD	Min.	Max.
$a$ (m/s²)	3.45	0.90	0.88	4.00	3.55	0.68	1.06	4.00
$b$ (m/s²)	3.67	1.25	0.85	4.50	3.72	1.15	0.87	4.50
$s_{0}$ (m)	6.94	3.30	1.00	10.00	6.20	3.10	1.00	10.00
$v_{0}$ (km/h)	79.49	30.37	32.45	108.00	79.06	27.20	33.93	108.00
$T$ (s)	1.29	0.80	0.20	3.00	1.29	0.78	0.20	3.00
$δ$	na	na	na	na	0.59	0.30	0.04	0.90
$γ$	na	na	na	na	3.00*	0.92	1.00	4.00
RMSE (calibration)	2.77	0.92	0.82	4.68	2.53	0.92	0.72	4.12
RMSE (validation)	3.63	0.98	0.93	6.32	2.63	0.94	0.76	4.72

Note: IDMTS = Intelligent Driver Model with Task Saturation; IDM+ = Intelligent Driver Model Plus; RMSE = root mean square error; a = maximum acceleration; b = desired acceleration; s₀ = minimum spacing; v₀ = desired speed; T = desired headway; γ = smoothness factor; δ = risk sensitivity; SD = standard deviation; Min. = minimum; Max. = maximum across all calibrated trajectories; * = parameter $γ$ values show mode as this is a categorical variable (values 1, 2, 3, and 4); na = not applicable.

When distracted driving data were used (Table 4), systematic parameter shifts were observed that are consistent with behavioral expectations and reinforce the model’s numerical soundness. Both the maximum acceleration $a$ and comfortable deceleration $b$ decreased relative to normal driving, reflecting the higher task demand and cognitive load induced by phone use. This reduction implies that distracted drivers adopt smoother, less aggressive control inputs, which is an outcome empirically observed in earlier HF studies (e.g., Saifuzzaman et al., [ 21 ]). The desired speed $v_{0}$ also declined modestly, while the desired time headway $T$ increased, signaling compensatory behavior aimed at maintaining safety margins. These shifts emerged endogenously from calibration rather than being imposed ex ante, illustrating the model’s ability to capture adaptive tendencies through its parameter structure.

Table 4.

Calibrated Parameters of IDM+ and IDMTS with Distracted Driving Data

Parameters	IDM+ (distracted driving)				IDMTS (distracted driving)
Parameters	Mean	SD	Min.	Max.	Mean	SD	Min.	Max.
$a$ (m/s²)	3.01	1.12	0.73	4.00	3.09	1.05	0.69	4.00
$b$ (m/s²)	3.24	1.44	0.50	4.50	3.73	1.08	0.90	4.50
$s_{0}$ (m)	5.97	3.20	1.00	10.00	6.13	2.77	1.00	10.00
$v_{0}$ (km/h)	78.74	30.44	27.37	108.00	77.81	26.04	27.55	108.00
$T$ (s)	1.68	0.93	0.20	3.00	1.55	0.85	0.20	3.00
$δ$	na	na	na	na	0.60	0.26	0.05	0.90
$γ$	na	na	na	na	3.00*	0.98	1.00	4.00
RMSE (calibration)	2.25	1.03	0.31	4.47	1.92	1.01	0.32	4.32
RMSE (validation)	3.21	1.02	0.39	5.32	2.32	1.03	0.36	4.68

For numerical fit, the IDMTS achieved lower calibration and Validation RMSEs than the IDM, demonstrating enhanced trajectory-reproduction capability under cognitively demanding conditions. The inclusion of the risk sensitivity parameter $δ$ and the adaptation smoothness factor $γ$ did not introduce instability or over-parameterization; rather, they improved the model’s flexibility in representing heterogeneous driver responses while maintaining consistent convergence across drivers. The modest dispersion of $δ$ (mean ≈ 0.6, standard deviation [SD] ≈ 0.26) indicates realistic variability in risk sensitivity within the experimental sample, aligning with the notion that many participants behaved as moderately risk-averse drivers.

These results establish the numerical soundness of the IDMTS. The model reproduces observed trajectories with reduced error, yields plausible and interpretable parameters, and remains numerically stable across contrasting behavioral contexts. Its ability to retain the structural elegance of IDM+ while delivering measurable gains in calibration and validation accuracy confirms that the cognitive extensions, task saturation and risk sensitivity are mathematically consistent, computationally efficient, and empirically justified.

Behavioral Soundness

Beyond numerical soundness, a car-following model must reproduce psychologically plausible behavioral patterns. The driving simulator experiments offer a controlled setting in which the behavioral soundness of the IDMTS can be assessed against established HF theories. In particular, the scenarios explicitly manipulate task demand via mobile phone conversations, operationalizing the core constructs of the TCI framework and task difficulty homeostasis ( 5 , 12 , 21 ). The key question is whether the IDMTS can endogenously reproduce the observed adaptations in spacing and speed between normal and distracted driving and whether the calibrated risk sensitivity and adaptation parameters provide a psychologically meaningful representation of intra-driver heterogeneity.

Figures 8 and 9 show two representative drivers spanning risk-averse and risk-taking profiles. For Driver 8, the model captures a case of risk-averse adaptation under distraction. Under the normal condition, the IDMTS remains almost entirely in the CFR, and its spacing and speed trajectories closely mirror IDM+, consistent with low task saturation: the driver experiences the task as manageable, and no additional adaptation is required. Under distraction, however, the observed trajectory exhibits larger headways and reduced speeds, in line with previous empirical findings that drivers often increase their safety margins when engaged in mobile phone conversations ( 21 , 26 ). The IDM+ is unable to represent this context-specific change in behavior and underestimates spacing during the distracted segment, and shows high speed. In contrast, the IDMTS transitions from CFR to the BAR as task saturation rises, and the simulated spacing profile aligns closely with the observed adjustment. This regime shift is not imposed exogenously but arises from the interaction between the task saturation function and the driver’s calibrated risk sensitivity $δ$ (risk sensitivity $δ$ of 0.63), providing a direct operationalization of task difficulty homeostasis ( 5 ).

Figure 8.

Comparison of IDM+ and IDMTS performance under: (a) normal; and (b) distracted driving conditions for a risk-averse driver.

Figure 9.

Comparison of IDM+ and IDMTS performance under normal and distracted driving conditions for risk-taking driver.

In contrast, Driver 17 exemplifies a risk-taking strategy (Figure 9). The empirical data show shorter headways and higher speeds during distraction compared with normal driving, consistent with drivers who underestimate task difficulty or accept elevated risk levels ( 6 , 20 ). The calibrated risk sensitivity for this driver is relatively low ( $δ \approx 0.51$ ), and simulation results indicate that the BAR is rarely activated: the model remains almost exclusively in CFR for normal and distracted conditions. Therefore, the IDMTS and IDM+ produce very similar trajectories for this driver, with the cognitive layer effectively “switching off” because task saturation rarely exceeds the driver’s high personal tolerance. This behavior is theoretically desirable. It shows that the IDMTS does not spuriously generate adaptation when the empirical data do not support it and that the cognitive extension is selective rather than universally active. In the TCI, Driver 17 accepts a narrower safety margin and operates closer to the overload boundary without engaging compensatory behavior ( 12 , 21 ).

These three cases demonstrate that the IDMTS achieves behavioral soundness along several dimensions. First, regime selection is endogenous: transitions between free driving, car-following, and behavioral adaptation are governed by the evolving task saturation level and individual risk sensitivity. Second, the distribution of calibrated $δ$ values across drivers is consistent with a spectrum from risk-averse to risk-taking behavior observed in experimental and naturalistic studies of headway choice and perceived risk ( 20 , 27 , 28 ). Risk-averse drivers (higher $δ$ ) show frequent and early BAR activation and larger headways under distraction; risk-taking drivers (lower $δ$ ) rarely enter the BAR and maintain small headways even when distracted. Third, the smoothness parameter $γ$ governs how quickly drivers return to their preferred task saturation level, aligning with the notion that some drivers adjust sharply while others adapt more gradually ( 6 ).

The simulator analysis provides strong evidence that the IDMTS is numerically and behaviorally sound. It replicates key stylized facts about distracted driving, such as increased headways and reduced speeds for risk-averse drivers, and persistent short headways for risk-taking drivers, and maps these patterns onto interpretable HF parameters within a unified car-following law. This dual alignment with data and theory distinguishes the IDMTS from many existing engineering models and supports its use as a behavioral testbed for studying distraction, task demand and intra-driver heterogeneity in microscopic traffic flow simulations.

However, there are some important limitations. Behavioral interpretation of $δ$ and $γ$ is partly constrained by the finite duration and structure of the simulator scenarios. Single car-following episodes with prescribed leader trajectories and discrete distraction events limit the range of observable adaptation patterns, potentially inflating uncertainty in individual parameter estimates. In addition, the current task saturation formulation aggregates speed, spacing and desired headway into a single scalar, whereas the TCI distinguishes between different components of workload (perceptual and cognitive) and between primary and secondary tasks ( 5 , 24 ). Despite this simplification, the simulator results indicate that the IDMTS can reproduce the direction and timing of behavioral adaptation observed empirically and that its parameters can be interpreted in a manner consistent with the broader HF literature.

NGSIM Experiment

Numerical Soundness

The evaluation of the IDMTS on the I-80 trajectory data set provides a test of its numerical stability and generalizability under naturalistic traffic conditions characterized by noise, heterogeneity, and complex interactions. Table 5 summarizes the calibration and validation statistics for the IDM+ and IDMTS across 396 leader–follower pairs. As in the driving simulator results, the IDMTS consistently achieves lower RMSEs in both calibration and validation phases, confirming that the proposed cognitive extension enhances numerical soundness without compromizing model stability. The mean calibration and validation RMSE for IDMTS decreases, indicating an average improvement of roughly 20%–25% in spacing accuracy relative to IDM+. These gains are meaningful in microscopic trajectory modeling, where sub-meter accuracy differences often translate into significant improvements in simulated flow stability and macroscopic realism ( 36 ).

Table 5.

Calibrated parameters of IDM+ and IDMTS on I-80 data

Parameters	IDM+				IDMTS
Parameters	Mean	SD	Min.	Max.	Mean	SD	Min.	Max.
$a$ (m/s²)	2.16	1.23	0.73	4.00	2.31	1.15	0.72	4.00
$b$ (m/s²)	2.92	1.39	0.50	4.50	2.52	1.24	0.50	4.50
$s_{0}$ (m)	2.23	1.57	0.5	4.30	2.26	1.28	0.50	5.23
$v_{0}$ (km/h)	96.76	25.17	54.16	113.99	99.46	23.39	54.87	115.54
$T$ (s)	2.06	0.54	0.50	2.50	1.65	0.58	0.51	2.50
$δ$	na	na	na	na	0.60	0.29	0	0.9
$γ$	na	na	na	na	3.00*	1.06	1.00	4.00
RMSE (calibration)	4.72	1.90	1.08	12.15	3.98	2.08	0.64	12.23
RMSE (validation)	5.32	1.80	2.45	12.09	4.81	1.72	1.71	11.45

Of note, the IDMTS retains the numerical soundness of the IDM family. No instances of divergence, oscillatory instability, or negative spacing were observed during simulation, even under high-density, stop-and-go conditions. This stability can be attributed to the model’s design: the BAR modifies acceleration smoothly via the task saturation function, avoiding abrupt deceleration jumps that could cause numerical blow-ups. This feature contrasts with some hybrid or switching-logic models in which discontinuities in regime boundaries induce instability ( 49 , 50 ).

From a parameter estimation standpoint, the IDMTS demonstrates convergence comparable to IDM+, with parameter ranges remaining physically plausible. The mean desired time headway $T$ (∼1.6 s), desired speed $v_{0}$ (∼100 km/h), and minimum spacing $s_{0}$ (∼2.3 m) align with empirical headway and spacing distributions reported in freeway studies ( 33 , 36 ). The newly introduced cognitive parameters exhibit interpretable variation: the average risk sensitivity parameter $δ = 0.6$ (SD = 0.3) suggests moderate risk aversion across the data set, while the adaptation smoothness parameter $γ$ clusters around three, indicating that most drivers adjust gradually rather than abruptly. The consistency of these estimates across hundreds of trajectories evidences the model’s capacity to learn stable driver-specific characteristics from real data without overfitting. Furthermore, the superior validation performance of the IDMTS demonstrates that the added cognitive mechanism improves transferability, an aspect critical for practical traffic simulation and control applications ( 36 ). By embedding task saturation, the model implicitly regularizes driver responses under diverse conditions, mitigating over-calibration to idiosyncratic leader–follower patterns that can plague traditional deterministic models. Therefore, the IDMTS achieves a more balanced trade-off between data fit and behavioral generality: it explains variance through interpretable behavioral parameters rather than arbitrary parameter distortion.

Although the IDMTS introduces two additional parameters relative to IDM+, the consistent reduction in validation RMSE across both the simulator and the I-80 data sets indicates that performance gains are not confined to calibration data. Moreover, the BAR is conditionally activated; during large portions of trajectories where task saturation remains below tolerance, the IDMTS effectively collapses to IDM+. This conditional structure limits unnecessary parameter exploitation and reduces the risk of overfitting. The bounded parameter domains for $δ$ and $γ$ further constrain flexibility, ensuring that improvements arise from structured behavioral representation rather than unrestricted mathematical degrees of freedom.

Behavioral Soundness

While numerical accuracy establishes a model’s quantitative soundness, behavioral soundness assesses whether its simulated dynamics remain psychologically and behaviorally plausible when exposed to the complexity of real-world traffic. The NGSIM I-80 data, which capture heterogeneous car-following interactions across varying densities and traffic regimes, provide an ideal testbed for this purpose. Under such uncontrolled conditions, drivers continuously balance task demand, situational awareness, and risk acceptance, an interplay central to TCI ( 5 , 12 ) and risk-homeostasis theories ( 27 ). Therefore, the key question is whether the IDMTS can replicate these adaptive responses in naturalistic car-following behavior without explicit regime labeling or driver-type classification.

Figure 10 shows the simulated trajectories for two representative leader–follower pairs extracted from the NGSIM I-80 data set. For the IDMTS simulations, the follower dynamics are further decomposed into regime-specific segments corresponding to the FDR, CFR, and BAR. This decomposition emphasizes how the proposed model dynamically transitions between regimes as task saturation evolves. In particular, periods of reduced spacing and elevated traffic demand trigger the activation of the BAR regime, during which the IDMTS produces adaptive deceleration and spacing adjustments that more closely follow the observed trajectory compared with IDM+. Figure 10 shows how the task saturation mechanism enables the model to capture localized behavioral adaptation in naturalistic traffic conditions.

Figure 10.

Simulation for two representative leader–follower pairs calibrated on I-80 data set comparing IDM+ and IDMTS. In the IDMTS model, speed and spacing profiles are decomposed into regime-specific segments: (a) Leader-follower pair (Vehicle 145-Vehicle 152) and (b) Leader-follower pair (Vehicle 397-Vehicle 401).

The calibrated IDMTS parameters reveal a behaviorally interpretable spectrum of intra-driver heterogeneity. The risk sensitivity parameter $δ$ is from 0.1 to 0.9 (mean ≈ 0.57), spanning drivers who tolerate high task demand (risk-taking) to those who exhibit strong avoidance of overload (risk-averse). When mapped against the distribution of average time headways and speed variability, behavioral patterns emerge: higher- $δ$ drivers maintain longer headways and exhibit smoother accelerations, and lower- $δ$ drivers follow more closely and display sharper accelerations and decelerations. This relationship mirrors empirical observations from field and simulator studies showing that risk-averse drivers compensate for uncertainty by enlarging temporal gaps ( 28 ).

Crucially, these behavioral tendencies arise endogenously in the IDMTS through the activation of the BAR. Under congested or unstable conditions, where spacing shortens and task saturation increases, the model predicts localized deceleration waves for high- $δ$ drivers, consistent with proactive adaptation to elevated workload. In contrast, low- $δ$ drivers remain predominantly in the CFR, tolerating higher task saturation and sustaining smaller gaps, behavior that aligns with aggressive driving types identified in empirical heterogeneity analyses ( 20 , 41 ). The selective activation of BAR is indicative of behavioral soundness: the cognitive mechanism engages only when task demand exceeds the driver’s individual tolerance.

The calibrated smoothness parameter $γ$ , concentrated around three, further supports the plausibility of driver response dynamics. Values in this range correspond to gradual, continuous adaptation rather than abrupt regime switching, matching empirical evidence that drivers regulate speed and spacing smoothly even in stop-and-go traffic ( 36 , 49 ). This continuous adjustment contrasts with the discrete perception thresholds of psycho-physical models such as Wiedemann ( 18 ), but remains behaviorally consistent with observed incremental braking and acceleration in dense flows. It is important to clarify that $δ$ and $γ$ are inferred behavioral parameters rather than directly observed cognitive measurements. NGSIM data does not provide physiological or subjective workload indicators; therefore, these parameters should be interpreted as latent behavioral traits reflected in longitudinal control patterns. Their cognitive interpretation is theoretically grounded in the TCI framework but remains indirectly inferred from trajectory dynamics. Future research integrating multimodal behavioral data sets (e.g., eye tracking or workload indices) would enable stronger causal validation of these constructs.

The NGSIM experiment results confirm that the IDMTS is not merely a numerically improved variant of IDM+ but a behaviorally sound model capable of reproducing intra-driver adaptation and risk heterogeneity under naturalistic traffic conditions. By coupling task saturation with risk sensitivity in a single acceleration law, the IDMTS bridges the gap between physics-based and cognition-based modeling, providing a theoretically grounded and empirically verifiable framework for human-centered traffic flow analysis.

MiTra Experiment

Numerical Soundness

To further assess the robustness of the proposed model under naturalistic traffic conditions, the calibration and validation analyses were conducted using the MiTra trajectory data set. Table 6 presents the calibrated parameters for the baseline IDM+ and the proposed IDMTS model across all MiTra car-following trajectories. The optimization results indicate stable and physically plausible parameter estimates for both models. The calibrated maximum acceleration parameter $a$ shows a slightly higher mean value for IDMTS (2.04 m/s²) compared with IDM+ (1.64 m/s²), suggesting that the task saturation framework allows the model to better capture more responsive driver behavior observed in the empirical trajectories. Similarly, the deceleration parameter $b$ remains within realistic ranges for both models, with slightly lower mean values observed for IDMTS.

Table 6.

Calibrated Parameters of IDM+ and IDMTS on MiTra Data

Parameters	IDM+				IDMTS
Parameters	Mean	SD	Min.	Max.	Mean	SD	Min.	Max.
$a$ (m/s²)	1.64	1.30	0.50	4.00	2.04	1.44	0.5	4.00
$b$ (m/s²)	3.13	1.19	0.50	4.50	2.78	1.32	0.50	4.50
$s_{0}$ (m)	1.80	1.40	0.5	4.20	1.24	1.03	0.50	5.23
$v_{0}$ (km/h)	122.80	25.17	70.67	154.26	105.55	18.50	70.74	154.40
$T$ (s)	0.71	0.30	0.50	1.85	0.65	0.23	0.51	1.57
$δ$	na	na	na	na	0.42	0.30	0	0.9
$γ$	na	na	na	na	3.00*	1.21	1.00	4.00
RMSE (calibration)	3.28	3.25	1.02	8.18	2.88	3.15	1.04	6.20
RMSE (validation)	3.63	14.48	1.21	8.68	3.14	3.53	1.02	6.49

Note: IDMTS = Intelligent Driver Model with Task Saturation; IDM+ = Intelligent Driver Model Plus; RMSE = root mean square error; MiTra = Milan Trajectories; a = maximum acceleration; b = desired acceleration; s₀ = minimum spacing; v₀ = desired speed; T = desired headway; γ = smoothness factor; δ = risk sensitivity; SD = standard deviation; Min. = minimum; Max. = maximum across all calibrated trajectories; * = parameter $γ$ values show mode as this is a categorical variable (values 1, 2, 3, and 4); na = not applicable.

For model performance, the proposed IDMTS consistently outperforms IDM+. The average calibration RMSE decreases from 3.28 m (IDM+) to 2.88 m (IDMTS), while the validation RMSE reduces from 3.63 to 3.14 m, representing an improvement in predictive accuracy. Of note, this improvement persists in the validation stage, indicating that the proposed model does not simply overfit the calibration trajectories but generalizes well to unseen data. The dispersion of the RMSE values across trajectories also indicates that the proposed model achieves more stable performance across diverse driving situations captured in the MiTra data set. These results demonstrate that incorporating task saturation mechanisms improves the model’s ability to reproduce longitudinal vehicle dynamics observed in naturalistic traffic environments.

Behavioral Soundness

Figure 11 shows representative car-following episodes from the MiTra data set, comparing the observed trajectories with simulations generated by IDM+ and IDMTS. The position profiles indicate that both models broadly capture the general movement of the follower vehicle relative to the leader. However, noticeable differences emerge when examining spacing dynamics. The IDM+ model tends to overestimate spacing, producing trajectories that deviate from the observed follower behavior. In contrast, the IDMTS model reproduces the observed spacing dynamics more closely across different behavioral regimes.

Figure 11.

Simulation for two representative leader–follower pairs calibrated on Milan Trajectories data set comparing IDM+ and IDMTS. In the IDMTS model, spacing profiles are decomposed into regime-specific segments

The proposed model distinguishes between three behavioral regimes (FDR, BAR, and CFR). These regimes enable the model to adapt the follower’s response according to the level of task demand. As shown in the spacing profiles, the IDMTS transitions smoothly between these regimes, allowing it to capture moderate adjustments and stronger responses when the leader’s behavior changes. The speed trajectories further confirm the behavioral realism of the proposed model. The IDMTS closely tracks the observed follower speed while maintaining consistency with the leader’s speed profile. In comparison, IDM+ occasionally produces smoother but less responsive speed adjustments, which leads to deviations in spacing over time. The results demonstrate that the IDMTS framework provides behaviorally plausible responses while maintaining numerical stability. By explicitly accounting for task saturation and behavioral adaptation, the model is able to better reproduce the longitudinal interaction patterns observed in the MiTra data set. This finding further supports the general applicability of the proposed modeling framework across different empirical trajectory data sets.

The simulator and naturalistic data analyses demonstrate that the proposed IDMTS framework achieves a balance between numerical reliability and behavioral interpretability. Across both controlled and naturalistic conditions, the model reproduces empirical trajectories with lower errors while retaining plausible, stable parameters, confirming its numerical soundness. Simultaneously, the risk sensitivity and task saturation mechanisms yield behaviorally coherent adaptations that align with established theories of driver workload regulation and risk-homeostasis ( 5 , 28 ). This dual consistency indicates that the IDMTS extends beyond a mere data-fitting improvement: it embeds a cognitively meaningful decision structure within a mathematically tractable acceleration law. From a methodological standpoint, this integration bridges the long-standing gap between HF soundness and numerical rigor in microscopic traffic modeling. Practically, the framework enables more faithful simulations of driver heterogeneity, distraction effects, and adaptive behavior, providing a foundation for next-generation traffic flow models and human-centered automated-vehicle controllers that can dynamically account for task demand and risk sensitivity.

Conclusion

This study introduced the IDMTS, a car-following framework that embeds a cognitive layer in a mathematically tractable structure. By integrating task saturation and risk sensitivity into the acceleration law, the IDMTS bridges the long-standing divide between behaviorally interpretable and numerically reliable traffic flow models. Analytical derivations confirmed that the extended formulation preserves equilibrium, rational driving, and stability properties of the base IDM+, ensuring theoretical soundness.

Using complementary data sets, controlled driving simulator experiments and naturalistic (NGSIM and MiTra) trajectories, the model was evaluated along two complementary dimensions: numerical and behavioral soundness. Numerically, the IDMTS achieved lower calibration and validation errors, stable convergence, and physically plausible parameter values across heterogeneous drivers, confirming its computational robustness. Behaviorally, the model reproduced key empirical features of driver adaptation under distraction and congestion, including risk-averse headway enlargement, risk-taking persistence, and endogenous regime transitions governed by task demand. Of note, the model distinguishes between equilibrium adaptation driven by risk sensitivity and perturbation damping governed by adaptation smoothness, highlighting the multilayered nature of behavioral influence on macroscopic traffic dynamics. Therefore, the macroscopic robustness of the traffic system is influenced by both the stability properties governed by $γ$ and the equilibrium condition governed by $δ$ . This separation of roles clarifies that risk sensitivity remains central to behavioral adaptation even if it does not dominate the linear string stability condition. These findings substantiate the cognitive validity of the proposed mechanism and its consistency with the TCI and risk-homeostasis theories.

Beyond its empirical performance, the IDMTS offers methodological implications for the broader traffic flow modeling community. It demonstrates that cognitive constructs can be incorporated without sacrificing analytical elegance or numerical stability, paving the way for hybrid frameworks that unify physics-based dynamics with HF principles. Future research will extend the model to incorporate explicit reaction time variability, multitask demand components, and lane-changing behavior, as well as explore its application to adaptive cruise control and cooperative automated-vehicle systems. By quantitatively linking workload, risk sensitivity, and longitudinal control, the IDMTS contributes a step toward behaviorally sound and human-aware microscopic traffic modeling.

Footnotes

Acknowledgements

The author would like to express gratitude to Prof. Zuduo Zheng for generously providing the driving simulator data, which was instrumental in calibrating the model in this research.

Author Contributions

The author confirms sole responsibility for the following: study conception and design, data preparation and analysis, and manuscript preparation.

Declaration of Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Mohammad Tamim Kashifi

References

Saifuzzaman

Zheng

Incorporating Human-Factors in Car-Following Models: A Review of Recent Developments and Research Needs. Transportation Research Part C: Emerging Technologies, Vol. 48, 2014, pp. 379–403. https://doi.org/10.1016/j.trc.2014.09.008

Kashifi

M. T.

Sharma

Ali

Benchmarking Intra-Driver Heterogeneity Car-Following Models Using a Behavioural and Numerical Evaluation Framework. Transportation Research Part B: Methodologies, Vol. 205, 2026, p. 103399.

Kashifi

M. T.

Sharma

Ali

Modelling Intra-Driver Heterogeneity in Car-Following: A Neural Stochastic Differential Equations Approach. Proceedings of the Transportation Research Board 105th Annual Meeting, Washington, D.C., 2026.

Zhu

Wang

Tarko

Fang

Modeling Car-Following Behavior on Urban Expressways in Shanghai: A Naturalistic Driving Study. Transportation Research Part C: Emerging Technologies, Vol. 93, 2018, pp. 425–445.

Fuller

Driver Control Theory: From Task Difficulty Homeostasis to Risk Allostasis. In Handbook of traffic psychology, (P.C.C. Boyle and M.A. Regan, eds.), Elsevier, pp. 13–26.

Hamdar

S. H.

Mahmassani

H. S.

Treiber

From Behavioral Psychology to Acceleration Modeling: Calibration, Validation, and Exploration of Drivers’ Cognitive and Safety Parameters in a Risk-Taking Environment. Transportation Research Part B: Methodological, Vol. 78, 2015, pp. 32–53.

Treiber

Helbing

Memory Effects in Microscopic Traffic Models and Wide Scattering in Flow-Density Data. Physical Review E, Vol. 68, No. 4, 2003, p. 046119.

Treiber

Kesting

Helbing

Understanding Widely Scattered Traffic Flows, the Capacity Drop, and Platoons as Effects of Variance-Driven Time Gaps. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, Vol. 74, No. 1, 2006, p. 016123.

Treiber

Kesting

Helbing

Delays, Inaccuracies and Anticipation in Microscopic Traffic Models. Physica A: Statistical Mechanics and its Applications, Vol. 360, No. 1, 2006, pp. 71–88.

10.

Treiber

Kesting

The Intelligent Driver Model with Stochasticity-New Insights into Traffic Flow Oscillations. Transportation research procedia, Vol. 23, 2017, pp. 174–187.

11.

Xiong

B. K.

Jiang

Tian

J. F.

Improving Two-Dimensional Intelligent Driver Models to Overcome Overly High Deceleration in Car-Following. Physica A: Statistical Mechanics and its Applications, Vol. 534, 2019, p. 122313. https://doi.org/10.1016/J.PHYSA.2019.122313

12.

Fuller

Toward a General Theory of Driver Behaviour. Accident analysis & prevention, Vol. 37, No. 3, 2005, pp. 461–472.

13.

Treiber

Hennecke

Helbing

Congested Traffic States in Empirical Observations and Microscopic Simulations. Physical review E, Vol. 62, No. 2, 2000, p. 1805.

14.

Pipes

L. A.

An Operational Analysis of Traffic Dynamics. Journal of applied physics, Vol. 24, No. 3, 1953, pp. 274–281.

15.

May

A. D.

Traffic Flow Fundamentals. Prentice Hall, Englewood Cliffs, NJ, 1990.

16.

Gipps

P. G.

A Behavioural Car-Following Model for Computer Simulation. Transportation Research Part B: Methodological, Vol. 15, No. 2, 1981, pp. 105–111.

17.

Bando

Hasebe

Nakayama

Shibata

Sugiyama

Dynamical Model of Traffic Congestion and Numerical Simulation. Physical review E, Vol. 51, No. 2, 1995, p. 1035.

18.

Wiedemann

Simulation Des Strassenverkehrsflusses. Institut für Verkehrswesen, Universität Karlsruhe, Karlsruhe, Germany, 1974.

19.

Yang

H.-H.

Peng

Development of an Errorable Car-Following Driver Model. Vehicle System Dynamics, Vol. 48, No. 6, 2010, pp. 751–773.

20.

Kochi

Saito

Uchida

Itoh

Task Difficulty, Risk Feeling, and Safety Margin in the Determination of Driver Behavior to Prepare for Traffic Conflicts. Accident Analysis & Prevention, Vol. 192, 2023, p. 107284. https://doi.org/https://doi.org/10.1016/j.aap.2023.107284

21.

Saifuzzaman

Zheng

Haque

M. M.

Washington

Revisiting the Task–Capability Interface Model for Incorporating Human Factors into Car-Following Models. Transportation research part B: methodological, Vol. 82, 2015, pp. 1–19.

22.

van Lint

J. W. C.

Calvert

S. C.

A Generic Multi-Level Framework for Microscopic Traffic Simulation—Theory and an Example Case in Modelling Driver Distraction. Transportation Research Part B: Methodological, Vol. 117, 2018, pp. 63–86. https://doi.org/10.1016/j.trb.2018.08.009

23.

Tian

Zhang

H. M.

Treiber

Jiang

Gao

Z.-Y.

Jia

On the Role of Speed Adaptation and Spacing Indifference in Traffic Instability: Evidence from Car-Following Experiments and Its Stochastic Model. Transportation Research Part B: Methodological, Vol. 129, 2019, pp. 334–350. https://doi.org/https://doi.org/10.1016/j.trb.2019.09.014

24.

Van Lint

J. W. C.

Calvert

S. C.

25.

Fuller

Psychology and the Highway Engineer. Fuller & Santos, Pergamon Press, Oxford, UK, 2002.

26.

Hoogendoorn

Van Arem

Hoogendoorn

Incorporating Driver Distraction in Car-Following Models: Applying the TCI to the IDM. 2013 16th International IEEE Conference on Intelligent Transportation Systems - (ITSC 2013), IEEE, The Hague, Netherlands, 2013, pp. 2229–2234.

27.

Heino

van der Molen

H. H.

Wilde

G. J. S.

Differences in Risk Experience between Sensation Avoiders and Sensation Seekers. Personality and individual differences, Vol. 20, No. 1, 1996, pp. 71–79.

28.

Lewis-Evans

De Waard

Brookhuis

K. A.

That’s Close Enough—A Threshold Effect of Time Headway on the Experience of Risk, Task Difficulty, Effort, and Comfort. Accident Analysis & Prevention, Vol. 42, No. 6, 2010, pp. 1926–1933.

29.

Schakel

W. J.

Knoop

V. L.

Van Arem

Integrated Lane Change Model with Relaxation and Synchronization. Transportation Research Record, Vol. 2316, No. 1, 2012, pp. 47–57.

30.

Fuller

Human Factors and Driving. In Human factors for highway engineers (R. Fuller and J. A. Santos, eds.), Pergamon Press, Oxford, UK, 2002.

31.

Treiber

Kesting

Treiber

Kesting

Elementary Car-Following Models. In Traffic Flow Dynamics: Data, Models and Simulation, Springer-Verlag, Berlin, Heidelberg, Germany, 2013, pp. 157–180.

32.

Sun

Zheng

Sun

The Relationship between Car Following String Instability and Traffic Oscillations in Finite-Sized Platoons and Its Use in Easing Congestion via Connected and Automated Vehicles with IDM Based Controller. Transportation Research Part B: Methodological, Vol. 142, 2020, pp. 58–83. https://doi.org/https://doi.org/10.1016/j.trb.2020.10.004

33.

Treiber

Kesting

Traffic Flow Dynamics. Traffic Flow Dynamics: Data, Models and Simulation, Springer-Verlag Berlin Heidelberg, 2013, pp. 983–1000.

34.

Wilson

R. E.

Ward

J. A.

Car-Following Models: Fifty Years of Linear Stability Analysis–a Mathematical Perspective. Transportation Planning and Technology, Vol. 34, No. 1, 2011, pp. 3–18.

35.

Sun

Zheng

Sun

Stability Analysis Methods and Their Applicability to Car-Following Models in Conventional and Connected Environments. Transportation Research Part B: Methodological, Vol. 109, 2018, pp. 212–237. https://doi.org/10.1016/j.trb.2018.01.013

36.

Punzo

Zheng

Montanino

About Calibration of Car-Following Dynamics of Automated and Human-Driven Vehicles : Methodology, Guidelines and Codes. Transportation Research Part C, Vol. 128, No. December2020, 2021, p. 103165. https://doi.org/10.1016/j.trc.2021.103165

37.

Intelligent Transportation Systems Joint Program Office. Next Generation Simulation (NGSIM) Interstate 80 Freeway Dataset. FHWA-HRT-06-137. Federal Highway Administration, U.S. Department of Transportation, Washington, D.C., 2016.

38.

Chaudhari

A. A.

Treiber

Okhrin

Mitra: A Drone-Based Trajectory Data for an All-Traffic-State Inclusive Freeway with Ramps. Scientific Data, Vol. 12, No. 1, 2025, p. 1174.

39.

FHWA. The Next Generation Simulation (NGSIM). Federal Highway Administration, U.S. Department of Transportation, Washington, DC, 2007. http://www.ngsim.fhwa.dot.gov/. Accessed May 20, 2023

40.

Montanino

Punzo

Trajectory Data Reconstruction and Simulation-Based Validation against Macroscopic Traffic Patterns. Transportation Research Part B: Methodological, Vol. 80, 2015, pp. 82–106. https://doi.org/10.1016/j.trb.2015.06.010

41.

Ossen

Hoogendoorn

S. P.

Heterogeneity in Car-Following Behavior: Theory and Empirics. Transportation research part C: emerging technologies, Vol. 19, No. 2, 2011, pp. 182–195.

42.

Sharma

Zheng

Bhaskar

Is More Always Better? The Impact of Vehicular Trajectory Completeness on Car-Following Model Calibration and Validation. Transportation research part B: methodological, Vol. 120, 2019, pp. 49–75.

43.

Kashifi

M. T.

Robust Spatiotemporal Crash Risk Prediction with Gated Recurrent Convolution Network and Interpretable Insights from SHapley Additive Explanations. Engineering Applications of Artificial Intelligence, Vol. 127, 2024, p. 107379.

44.

Kashifi

M. T.

Deep Hybrid Attention Framework for Road Crash Emergency Response Management. IEEE Transactions on Intelligent Transportation Systems, 2024, pp. 1–12. https://doi.org/10.1109/TITS.2024.3376653

45.

Mirjalili

Lewis

The Whale Optimization Algorithm. Advances in Engineering Software, Vol. 95, 2016, pp. 51–67. https://doi.org/https://doi.org/10.1016/j.advengsoft.2016.01.008

46.

Mirjalili

SCA: A Sine Cosine Algorithm for Solving Optimization Problems. Knowledge-based systems, Vol. 96, 2016, pp. 120–133.

47.

Kashifi

M. T.

Investigating Two-Wheelers Risk Factors for Severe Crashes Using an Interpretable Machine Learning Approach and SHAP Analysis. IATSS Research, Vol. 47, No. 3, 2023, pp. 357–371. https://doi.org/10.1016/j.iatssr.2023.07.005

48.

Tamim Kashifi

Ahmad

Efficient Histogram-Based Gradient Boosting Approach for Accident Severity Prediction With Multisource Data. Transportation Research Record: Journal of the Transportation Research Board, 2022. 2676: 036119812210743.

49.

Thiemann

Treiber

Kesting

Estimating Acceleration and Lane-Changing Dynamics from next Generation Simulation Trajectory Data. Transportation Research Record, Vol. 2088, No. 1, 2008, pp. 90–101.

50.

Ossen

Hoogendoorn

S. P.

Heterogeneity in Car-Following Behavior: Theory and Empirics. Transportation Research Part C: Emerging Technologies, Vol. 19, No. 2, 2011, pp. 182–195. https://doi.org/https://doi.org/10.1016/j.trc.2010.05.006