arXiv subjects

Martin Treiber

Publications and source records attributed to Martin Treiber.

At least 19 recordsLinked to original sources

arXiv2026

Second-Order Continuum Model for Disordered Traffic

In this study, a two-dimensional second-order macroscopic traffic flow model is proposed to capture the complex dynamics of disordered traffic by explicitly incorporating coupled longitudinal and lateral interactions. The framework extends the classical continuity equation to two spatial dimensions and introduces acceleration equations consisting of self-driven, interaction-induced, and road-boundary-related components. The macroscopic longitudinal acceleration is formulated based on the Full Velocity Difference Model (FVDM), while lateral dynamics are governed by interaction principles consistent with the Optimal Velocity Model (OVM). The explicit inclusion of road-boundary effects enables the representation of vehicle confinement and space-sharing behavior that are essential in lane-free and weakly lane-disciplined traffic systems. The model is examined through a series of numerical experiments in which longitudinal and lateral dynamics are analysed individually as well as simultaneously within a coupled two-dimensional setting under a range of initial conditions, including transitions between free-flow, medium congestion, and heavy congestion, and different lateral density configurations. The simulations demonstrate the model's ability to reproduce key traffic features such as shockwave propagation, lateral dispersion, vehicle rearrangement, and stable density evolution across the road width. The numerical scheme, based on upwind and Lax-Friedrichs discretisations, ensures stable solutions of the coupled partial differential equations. Overall, the proposed framework provides a robust macroscopic description of disordered traffic and offers a consistent basis for analysing two-dimensional vehicular flow dynamics.

physics.soc-ph↗

arXiv2026

Beyond Lanes: Traffic Flow Dynamics in Disordered Conditions Based on High-Resolution Trajectory Data

Disordered traffic flow is characterized by weak or non-existent lane discipline in the presence of strong vehicle heterogeneity and continuous lateral interactions, challenging traditional lane-based modeling assumptions. This study presents an empirical study of macroscopic and microscopic aspects of disordered traffic using high-resolution UAV trajectory data collected on an urban arterial. A two-dimensional extension of Edie's framework is applied to quantify aggregate traffic variables and produce a two-dimensional fundamental diagram, revealing that traffic states cannot be adequately represented using one-dimensional formulations and highlighting the persistent role of lateral redistribution. The propagation of congestion is estimated directly from the spatiotemporal speed fields, demonstrating the emergence of coherent stop-and-go waves and showing a similar dynamics as conventional lane-based flow, in spite of the heterogeneous vehicle interactions. At the microscopic level, steady-state follower-leader identification is used to examine desired time gaps and minimum lateral spacing, vehicle dimension distributions, and kinematic characteristics, revealing pronounced inter-class heterogeneity that explains disordered traffic behavior. The study provides an empirical framework linking vehicle-level interactions and aggregate traffic dynamics and establishes a data-driven basis for the calibration and validation of traffic models for disordered mixed traffic systems.

physics.soc-ph↗

arXiv2024

Leader-Follower Identification with Vehicle-Following Calibration for Non-Lane-Based Traffic

Most car-following models were originally developed for lane-based traffic. Over the past two decades, efforts have been made to calibrate car-following models for non-lane-based traffic. However, traffic conditions with varying vehicle dimensions, intermittent following, and multiple leaders often occur and make subjective Leader-Follower (LF) pair identification challenging. In this study, we analyze Vehicle Following (VF) behavior in traffic with a lack of lane discipline using high-resolution microscopic trajectory data collected in Chennai, India. The paper's main contributions are threefold. Firstly, three criteria are used to identify LF pairs from the driver's perspective, taking into account the intermittent following, lack of lane discipline due to consideration of lateral separation, and the presence of in-between vehicles. Second, the psycho-physical concept of the regime in the Wiedemann-99 model is leveraged to determine the traffic-dependent "influence zone" for LF identification. Third, a joint and consistent framework is proposed for identifying LF pairs and estimating VF parameters. The proposed methodology outperforms other heuristic-based LF identification methods from the literature in terms of quantitative and qualitative performance measures. The proposed approach can enable robust and more realistic LF identification and VF parameter calibration with practical applications such as LOS analysis, capacity, and travel time estimation.

physics.soc-ph↗

arXiv2023

The intelligent agent model -- a fully two-dimensional microscopic traffic flow model

Recently, a fully two-dimensional microscopic traffic flow model for lane-free vehicular traffic flow has been proposed [Physica A, 509, pp. 1-11 (2018)]. In this contribution, we generalize this model to describe any kind of human-driven directed flow including lane-based vehicular flow, lane-free mixed traffic, bicycle traffic, and pedestrian flow. The proposed intelligent-agent model (IAM) has the same philosophy as the well-known social-force model (SFM) for pedestrians but the interaction and boundary forces are based on car-following models making this model suitable for higher speeds. Depending on the underlying car-following model, the IAM includes anticipation, response to relative velocities, and accident-free driving. When adding a suitable floor field, the IAM reverts to an integrated car-following and lane-changing model with continuous lane changes. We simulate this model in several lane-based and lane-free environments in various geometries with and without obstacles. We observe that the model produces accident-free traffic flow reproducing the observed self-organisation phenomena.

physics.soc-ph↗

arXiv2023

Two-dimensional LWR model for lane-free traffic

While macroscopic models for single or multi-lane traffic flow are well established, these models are not applicable to the dynamics and characteristics of disordered traffic which is characterized by widely different types of vehicles and no lane discipline. We propose a first-order two-dimensional Lighthill-Whitham-Richards (LWR) model for the continuous macroscopic longitudinal and lateral dynamics of this type of traffic flow. The continuity equation is extended into two dimensions and the equation is closed by assuming a longitudinal flow-density relationship as in traditional one-dimensional models while the lateral dynamics is based on boundary repulsion and a desire of a majority of the drivers to go to less dense regions. This is equivalent to Fick's law giving rise to a lateral diffusion term. Using the proposed model, several numerical tests were conducted under different traffic scenarios representing a wide range of traffic conditions. Even for extreme initial conditions, the model's outcome turned out to be plausible and consistent with observed traffic flow dynamics. Moreover, the numerical convergence test is performed using an analytical solution for lateral steady-state conditions. The model was applied for bicycle simulation and reproduced the evolution of lateral density profile with asymmetric behavior.

physics.soc-ph↗

arXiv2023

On the calibration of stochastic car following models

Recent experimental and empirical observations have demonstrated that stochasticity plays a critical role in car following (CF) dynamics. To reproduce the observations, quite a few stochastic CF models have been proposed. However, while calibrating the deterministic CF models is well investigated, studies on how to calibrate the stochastic models are lacking. Motivated by this fact, this paper aims to address this fundamental research gap. Firstly, the CF experiment under the same driving environment is conducted and analyzed. Based on the experimental results, we test two previous calibration methods, i.e., the method to minimize the Multiple Runs Mean (MRMean) error and the method of maximum likelihood estimation (MLE). Deficiencies of the two methods have been identified. Next, we propose a new method to minimize the Multiple Runs Minimum (MRMin) error. Calibration based on the experimental data and the synthetic data demonstrates that the new method outperforms the two previous methods. Furthermore, the mechanisms of different methods are explored from the perspective of error analysis. The analysis indicates that the new method can be regarded as a nested optimization model. The method separates the aleatoric errors caused by stochasticity from the epistemic error caused by parameters, and it is able to deal with the two kinds of errors effectively. Finally, we find that under the calibration framework of stochastic CF models, the calibrated parameter set using spacing as MoP may not always outperform that using velocity as MoP. These findings are expected to enhance the understanding of the role of stochasticity in CF dynamics where the new calibration framework for stochastic CF models is established.

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arXiv2022

Vessel-following model for inland waterways based on deep reinforcement learning

While deep reinforcement learning (RL) has been increasingly applied in designing car-following models in the last years, this study aims at investigating the feasibility of RL-based vehicle-following for complex vehicle dynamics and strong environmental disturbances. As a use case, we developed an inland waterways vessel-following model based on realistic vessel dynamics, which considers environmental influences, such as varying stream velocity and river profile. We extracted natural vessel behavior from anonymized AIS data to formulate a reward function that reflects a realistic driving style next to comfortable and safe navigation. Aiming at high generalization capabilities, we propose an RL training environment that uses stochastic processes to model leading trajectory and river dynamics. To validate the trained model, we defined different scenarios that have not been seen in training, including realistic vessel-following on the Middle Rhine. Our model demonstrated safe and comfortable driving in all scenarios, proving excellent generalization abilities. Furthermore, traffic oscillations could effectively be dampened by deploying the trained model on a sequence of following vessels.

cs.CE↗

arXiv2021

Formulation and validation of a car-following model based on deep reinforcement learning

We propose and validate a novel car following model based on deep reinforcement learning. Our model is trained to maximize externally given reward functions for the free and car-following regimes rather than reproducing existing follower trajectories. The parameters of these reward functions such as desired speed, time gap, or accelerations resemble that of traditional models such as the Intelligent Driver Model (IDM) and allow for explicitly implementing different driving styles. Moreover, they partially lift the black-box nature of conventional neural network models. The model is trained on leading speed profiles governed by a truncated Ornstein-Uhlenbeck process reflecting a realistic leader's kinematics. This allows for arbitrary driving situations and an infinite supply of training data. For various parameterizations of the reward functions, and for a wide variety of artificial and real leader data, the model turned out to be unconditionally string stable, comfortable, and crash-free. String stability has been tested with a platoon of five followers following an artificial and a real leading trajectory. A cross-comparison with the IDM calibrated to the goodness-of-fit of the relative gaps showed a higher reward compared to the traditional model and a better goodness-of-fit.

cs.LG↗

arXiv2018

A Microscopic Decision Model for Route Choice and Event-Driven Revisions

We propose a microscopic decision model for route choice based on discrete choice theory. The correlation of overlapping routes is included in the random portions of the utility explicitly. For computational efficiency, we restrict the choice set to the turning possibilities at the next intersection, assuming shortest paths to the destination afterwards. The proposed decision model also regards traffic conditions (e.g. traffic lights, long queues) such that drivers may revise their previously taken decision. Due to its compatibility to already existing microscopic traffic flow models, the proposed route choice model can be readily simulated with available software. Combined, the proposed decision model features realistic behavior, i.e., adaptive choice based on incomplete information and simultaneously allows for a straightforward implementation.

physics.soc-ph↗

arXiv2018

Self-Healing Road Networks: A Self-Organized Management Strategy for Traffic Incidents in Urban Road Networks

We propose a novel self-organized traffic management strategy for incidents in traffic-light-controlled road networks. During incidents, it regulates the inflow into already congested road segments by restricting or skipping green times. Simultaneously, the remaining green times are used for the yet free turning directions. In this way, drivers can utilize unused road capacities and circumvent congested areas by revising their originally chosen routes. Consequently, the network has "healed itself" as soon as all affected traffic flows have been redistributed along remaining road capacities. Driver's route choice is modeled by means of discrete choice theory and regards the signaling at the next observable intersection. To this end, we propose a microscopic route choice model for event-oriented decisions. Subsequently, we examine four distinct incident scenarios in a grid and real-size road network and compare fixed-time and traffic-dependent traffic light controls with and without the self-healing strategy employed. As result, the self-healing strategy accounts for significantly lower vehicle accumulations during and after the incident as it effectively antagonizes gridlock effects in the disturbed network.

physics.soc-ph↗

arXiv2018

Theoretical investigation, simulation and empirical analysis of the growth pattern of traffic oscillations in the Euler coordinates

The formation and development of oscillations is an important traffic flow phenomenon. Recent studies found that along a vehicle platoon described in the Lagrangian specification, traffic oscillations grow in a concave way. Since stationary bottlenecks are more intuitively described in the Eulerian framework, this paper investigates whether the concave growth pattern of traffic oscillations in the Lagrangian coordinates can be transferred to the Euler coordinates (i.e. the concave increase in standard deviation is no longer measured across the vehicle indices but as a function of the road location). To this end, we theoretically unify these two ways of measuring oscillations by revealing their mapping relationship. We show that the growth pattern measured in the Lagrangian coordinates can be transferred to the Euler coordinates. We believe this finding is nontrivial since the scenarios are significantly different: while in vehicle platoons (Lagrangian view, non-penetrable moving bottleneck), the speed variance for a given vehicle is ideally constant, the drivers in the Eulerian setting (penetrable stationary bottleneck triggering the waves) experience all amplitudes, first the big ones and then the small ones. To test this proposition, we performed simulation using two different kinds of car-following models. Simulation results validate the theoretical analysis. Finally, we performed empirical analysis using the NGSIM data, which also validates the theoretical analysis.

nlin.PS↗

arXiv2018

Simulating bicycle traffic by the Intelligent-Driver Model -- reproducing the traffic-wave characteristics observed in a bicycle-following experiment

Bicycle traffic operations become increasingly important and yet are largely ignored in the traffic flow community, until recently. We hypothesize that there is no qualitative difference between vehicular and bicycle traffic flow dynamics, so the latter can be described by reparameterized car-following models. To test this proposition, we reproduce bicycle experiments on a ring with the Intelligent-Driver Model and compare its fit quality (calibration) and predictive power (validation) with that of the Necessary-Deceleration-Model which is specifically designed for bike traffic. We find similar quality metrics for both models, so the above hypothesis of a qualitative equivalence cannot be rejected.

physics.soc-ph↗

arXiv2018

Self-Driven Particle Model for Mixed Traffic and Other Disordered Flows

Vehicles in developing countries have widely varying dimensions and speeds, and drivers tend to not follow lane discipline. In this flow state called "mixed traffic", the interactions between drivers and the resulting maneuvers resemble more that of general disordered self-driven particle systems than that of the orderly lane-based traffic flow of industrialized countries. We propose a general multi particle model for such self-driven "high-speed particles" and show that it reproduces the observed characteristics of mixed traffic. The main idea is to generalize a conventional acceleration-based car-following model to a two-dimensional force field. For in-line following, the model reverts to the underlying car-following model, for very slow speeds, it reverts to an anisotropic social-force model for pedestrians. With additional floor fields at the position of lane markings, the model reverts to an integrated car-following and lane-changing model with continuous lateral dynamics including cooperative aspects such as zip merging. With an adaptive cruise control (ACC) system as underlying car-following model, it becomes a controller for the acceleration and steering of autonomous vehicles in mixed or lane-based traffic.

physics.soc-ph↗

arXiv2017

The Intelligent Driver Model with Stochasticity -- New Insights Into Traffic Flow Oscillations

Traffic flow oscillations, including traffic waves, are a common yet incompletely understood feature of congested traffic. Possible mechanisms include traffic flow instabilities, indifference regions or finite human perception thresholds (action points), and external acceleration noise. However, the relative importance of these factors in a given situation remains unclear. We bring light into this question by adding external noise and action points to the Intelligent Driver Model and other car-following models thereby obtaining a minimal model containing all three oscillation mechanisms. We show analytically that even in the subcritical regime of linearly stable flow (order parameter $\epsilon<0$), external white noise leads to spatiotemporal speed correlations "anticipating" the waves of the linearly unstable regime. Sufficiently far away from the threshold, the amplitude scales with $(-\epsilon)^{-0.5}$. By means of simulations and comparisons with experimental car platoons and bicycle traffic, we show that external noise and indifference regions with action points have essentially equivalent effects. Furthermore, flow instabilities dominate the oscillations on freeways while external noise or action points prevail at low desired speeds such as vehicular city or bicycle traffic. For bicycle traffic, noise can lead to fully developed waves even for single-file traffic in the subcritical regime.

physics.soc-ph↗

arXiv2016

On the role of speed adaptation and spacing indifference in traffic instability: evidence from car-following experiments and its stochastic modeling

Understanding the mechanisms responsible for the emergence and evolution of oscillations in traffic flow has been subject to intensive research by the traffic flow theory community. In our previous work, we proposed a new mechanism to explain the generation of traffic oscillations: traffic instability caused by the competition between speed adaptation and the cumulative effect of stochastic factors. In this paper, by conducting a closer examination of car following data obtained in a 25-car platoon experiment, we discovered that the speed difference plays a more important role on car-following dynamics than the spacing, and when its amplitude is small, the growth of oscillations is mainly determined by the stochastic factors that follow the mean reversion process; when its amplitude increases, the growth of the oscillations is determined by the competition between the stochastic factors and the speed difference. An explanation is then provided, based on the above findings, to why the speed variance in the oscillatory traffic grows in a concave way along the platoon. Finally, we proposed a mode-switching stochastic car-following model that incorporates the speed adaptation and spacing indifference behaviors of drivers, which captures the observed characteristics of oscillation and discharge rate. Sensitivity analysis shows that reaction delay only has slight effect but indifference region boundary has significant on oscillation growth rate and discharge rate.

physics.soc-ph↗

arXiv2016

Measuring and Modelling Crowd Flows - Fusing Stationary and Tracking Data

The two main data categories of vehicular traffic flow, stationary detector data and floating-car data, are also available for many Marathons and other mass-sports events: Loop detectors and other stationary data sources find their counterpart in the RFID tags of the athletes recording the split times at several stations during the race. Additionally, more and more athletes use smart-phone apps generating track data points that are the equivalent of floating-car data. We present a methodology to detect congestions and estimate the location of jam-fronts, the delay times, and the spatio-temporal speed and density distribution of the athlete's crowd flow by fusing these two data sources based on a first-order macroscopic model with triangular fundamental diagram. The method can be used in real-time or for analyzing past events. Using synthetic "ground truth" data generated by simulations with the Intelligent-Driver Model, we show that, in a real-time application, the proposed algorithm is robust and effective with minimal data requirements. Generally, two stationary data sources and about ten "floating-athlete" trajectories per hour are sufficient.

physics.soc-ph↗

arXiv2016

Calibrating the Local and Platoon Dynamics of Car-following Models on the Reconstructed NGSIM Data

The NGSIM trajectory data are used to calibrate two car-following models - the IDM and the FVDM. We used the I80 dataset which has already been reconstructed to eliminate outliers, unphysical data, and internal and platoon inconsistencies contained in the original data.We extract from the data leader-follower pairs and platoons of up to five consecutive vehicles thereby eliminating all trajectories that are too short or contain lane changes. Four error measures based on speed and gap deviations are considered. Furthermore, we apply three calibration methods: local or direct calibration, global calibration, and platoon calibration. The last approach means that a platoon of several vehicles following a data-driven leader is simulated and compared to the observed dynamics.

physics.soc-ph↗

arXiv2016

Improved 2D Intelligent Driver Model simulating synchronized flow and evolution concavity in traffic flow

This paper firstly show that 2 Dimensional Intelligent Driver Model (Jiang et al., PloS one, 9(4), e94351, 2014) is not able to replicate the synchronized traffic flow. Then we propose an improved model by considering the difference between the driving behaviors at high speeds and that at low speeds. Simulations show that the improved model can reproduce the phase transition from synchronized flow to wide moving jams, the spatiotemporal patterns of traffic flow induced by traffic bottleneck, and the evolution concavity of traffic oscillations (i.e. the standard deviation of the velocities of vehicles increases in a concave/linear way along the platoon). Validating results show that the empirical time series of traffic speed obtained from Floating Car Data can be well simulated as well.

nlin.PS↗