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Three-phase traffic theory and two-phase models with a fundamental diagram in the light of empirical stylized facts
Martin Treiber, Arne Kesting, Dirk Helbing
TL;DR
Freeway congestion theory remains divided over models with or without a fundamental diagram and over how empirical congestion patterns should be interpreted. The paper compares three-phase theory with fundamental-diagram phase diagrams while incorporating model parameters, traffic heterogeneity, freeway design, and measurement processing. It finds that both model classes can generate similar empirical-style patterns, although several plausible mechanisms remain and further observations are needed.
Problem
The paper addresses whether freeway congestion is better described by models with or without a unique equilibrium flow-density or speed-distance relationship.
Method
The paper compares Kerner’s three-phase theory with fundamental-diagram models while considering empirical stylized facts, realistic traffic factors, and the measurement process.
Results
Features associated with three-phase theory can be reproduced by conventional two-phase models with suitably specified parameters, while three-phase models produce similar spatiotemporal states and phase diagrams.
Takeaways & Limitations
The controversy may be addressed through more precise terminology and quantitative comparisons of competing models using representative measurements and calibrated parameters.
Takeaways & Limitations
Several plausible, mutually compatible or incompatible mechanisms explain the observations, so further observations and experiments are necessary to confirm or reject them.
Abstract
from arXiv · showhide
Despite the availability of large empirical data sets and the long history of traffic modeling, the theory of traffic congestion on freeways is still highly controversial. In this contribution, we compare Kerner's three-phase traffic theory with the phase diagram approach for traffic models with a fundamental diagram. We discuss the inconsistent use of the term "traffic phase" and show that patterns demanded by three-phase traffic theory can be reproduced with simple two-phase models, if the model parameters are suitably specified and factors characteristic for real traffic flows are considered, such as effects of noise or heterogeneity or the actual freeway design (e.g. combinations of off- and on-ramps). Conversely, we demonstrate that models created to reproduce three-phase traffic theory create similar spatiotemporal traffic states and associated phase diagrams, no matter whether the parameters imply a fundamental diagram in equilibrium or non-unique flow- density relationships. In conclusion, there are different ways of reproducing the empirical stylized facts of spatiotemporal congestion patterns summarized in this contribution, and it appears possible to overcome the controversy by a more precise definition of the scientific terms and a more careful comparison of models and data, considering effects of the measurement process and the right level of detail in the traffic model used.
1 Introduction
The paper addresses whether freeway congestion is better described by models with or without a fundamental diagram. It compares three-phase theory with the phase-diagram approach and proposes explaining their apparent differences through terminology, model assumptions, and empirical details.
- 1 Introduction: The central controversy concerns whether traffic models should have a unique equilibrium flow-density or speed-distance relationship.The paper focuses on this question rather than the separate debate over faster-than-vehicle characteristic propagation speeds.
- 1 Introduction: Kerner’s three-phase theory distinguishes free traffic, wide moving jams, and synchronized flow.Synchronized flow is characterized by widely scattered flow-density data.
- 1 Introduction: The phase-diagram approach represents conditions for traffic states and can derive them semi-quantitatively from model instabilities and congested-traffic outflow.The approach concerns models with a fundamental diagram.
- 1 Introduction: Both approaches use inconsistent terminology, simplify real traffic, and are criticized for mismatches involving empirical patterns such as the general and widening synchronized patterns.Criticisms also concern model complexity, accuracy, parameter counts, and the placement of homogeneous traffic states.
- 1 Introduction: The paper compares the approaches and reports that three-phase patterns can arise in fundamental-diagram models with suitably chosen parameters and realistic traffic factors.It also examines the converse similarity between patterns generated by two-phase and three-phase models.
2 Overview of empirical observations
The paper adopts a data-oriented approach to freeway congestion, emphasizing measurement, smoothing, interpolation, and interpretation choices. It uses detector-based velocity data and treats acquisition and processing as part of comparisons between observations and simulations.
- 2 Overview of empirical observations: The empirical overview uses data from several German freeways and avoids technical terminology from competing traffic theories where possible.The goal is to establish observations as generally valid rather than relying only on the extensively studied German A5.
- 2.1 Data issues: Direct comparison requires simulating data acquisition and subsequent processing or interpretation steps.The analysis focuses on aggregated stationary detector data, with velocity preferred because it is measured directly.
- 2.1 Data issues: Traffic flow is computed from vehicle counts over typically 60-second intervals, while velocity is the arithmetic mean of passing vehicles’ individual velocities.Virtual detectors record each vehicle’s passage time and velocity before aggregation.
- 2.1 Data issues: Detector spacing of 1–3 km can be comparable to wavelengths of non-homogeneous congestion patterns, creating ambiguities in reconstructed dynamics.Heterogeneous traffic and measurement noise can further obscure underlying patterns.
- 2.1 Data issues: Adaptive smoothing is applied consistently to real and virtual-detector data because alternative post-processing methods may not provide continuous estimates or may rely explicitly on traffic models.The method had been validated with high-spatial-resolution empirical data.
- 2 Overview of empirical observations: The overview treats recurring spatiotemporal congestion findings as stylized facts and assembles them as a testbed for traffic models and theories.The list is intended to include all relevant findings, including previously published observations.
(1) Congestion patterns on real (non-circular) freeways are typically caused
Congestion on real freeways is commonly localized or spatially extended, with fronts and internal structures governed by bottlenecks, demand, and characteristic propagation speeds. The controversial pinch or general pattern combines stationary congestion, growing upstream perturbations, and wide jams.
- (1) Congestion patterns on real (non-circular) freeways are typically caused: Congestion is caused by bottlenecks including ramps, junctions, lane reductions, accidents, and gradients.The A8-East example combines gradients around Irschenberg with an accident at x = 43.5 km between 17:40 h and 18:15 h.
- (1) Congestion patterns on real (non-circular) freeways are typically caused: About 200 out of 400 observed breakdowns corresponded to spatially extended patterns, while localized patterns typically had widths of about 1 km.Localized patterns either remain stationary at bottlenecks or move upstream.
- (1) Congestion patterns on real (non-circular) freeways are typically caused: The characteristic congestion-front speed c_cong is typically between −20 km/h and −15 km/h and does not depend on congestion type.This speed describes upstream motion of localized patterns and can also describe moving downstream fronts of extended congestion.
- (1) Congestion patterns on real (non-circular) freeways are typically caused: Extended-pattern upstream fronts have no characteristic speed: they move upstream when demand exceeds capacity and downstream when demand is below capacity.Downstream movement toward the bottleneck is the most frequent way congestion dissolves.
- (1) Congestion patterns on real (non-circular) freeways are typically caused: Internal structures in extended congestion generally propagate upstream in parallel at approximately c_cong.These structures are also described as oscillations, stop-and-go traffic, or small jams.
- (1) Congestion patterns on real (non-circular) freeways are typically caused: Oscillations may emerge upstream or appear at a downstream boundary, and neighboring perturbations may merge as they grow.For strong bottlenecks, oscillations can be difficult to distinguish from noise because their periods approach the smoothing-window duration.
- (1) Congestion patterns on real (non-circular) freeways are typically caused: The pinch effect or general pattern combines a stationary pinch region, growing upstream oscillatory structures, and some structures developing into wide jams.The paper leaves open whether this is a separate elementary pattern or a composition of several elementary patterns.
3 The meaning of traffic phases
Traffic research uses “phase” in both thermodynamic and nonequilibrium senses, creating inconsistent classifications of traffic states. For observable freeway behavior, the paper emphasizes dynamic spatiotemporal patterns and their occurrence conditions over the number of physical phases.
- Thermodynamic and nonequilibrium meanings: Thermodynamic phases describe aggregate states of large, homogeneous, closed systems in equilibrium, whereas traffic systems are open, driven, and inhomogeneous.Traffic models also use phase terminology for states distinguished by flow stability and instability mechanisms.
- Model classifications: One-phase models produce homogeneous equilibrium traffic, two-phase models add oscillatory states, and three-phase models add synchronized flow with scattered traffic variables.The classification is tied mainly to the instability properties of the modeled traffic flows.
- Model classifications: Whether Kerner’s car-following models possess three thermodynamic phases remains unresolved because an appropriate order parameter and thermodynamic interpretation have not been established.The paper distinguishes this question from reproducing observed nonequilibrium traffic patterns.
- Dynamic traffic phases: Boundary conditions, bottlenecks, freeway inhomogeneities, and perturbation history shape traffic patterns, favoring a dynamic-phase concept for observable freeway phenomena.Metastability makes the resulting state depend on perturbation size and traffic history.
- Dynamic traffic phases: Model quality therefore depends less on having two or three physical phases than on predicting observed spatiotemporal patterns and their occurrence conditions.The phase diagram approach classifies models by the qualitatively similar or different traffic states produced under specified conditions.
4 Simulating the spatiotemporal traffic dynamics
The paper simulates realistic congestion as an open freeway system with bottlenecks and tests how traffic patterns depend on model choice, parameters, and perturbations. It reports that suitably chosen two-phase models can reproduce the stylized patterns associated with three-phase theory, including the pinch effect and WSP.
- Model capability: Two-phase models can reproduce all listed empirical stylized facts, including the pinch effect and widening synchronized pattern, when their parameters are suitably chosen.These patterns had been attributed exclusively to three-phase traffic theory.
- Bottleneck simulations: Realistic congestion patterns are simulated in an open system with a bottleneck, using main inflow, ramp flow, and lane count to characterize flow conditions.Downstream free and congested flows are then determined from these quantities.
- Bottleneck simulations: Because models may exhibit hysteresis, simulations test both slowly increasing inflows with minimum perturbations and large perturbations generated by temporarily blocking outflow.The pre-congestion time-dependent conditions matter for discontinuous, history-dependent transitions.
4.1 Two-phase models
The paper shows that two-phase models with a fundamental diagram can reproduce diverse congestion patterns, including patterns associated with three-phase theory, when parameters, perturbations, and bottleneck settings are appropriately specified.
- Model classification: Two-phase models are classified by dynamic instabilities within a density range, while some fundamental-diagram models are instead one-phase.A microscopic two-phase model requires dynamic acceleration or delays; a macroscopic model requires a dynamical velocity equation.
- GKT model: The GKT model produces localized, oscillatory, triggered, and homogeneous congested patterns, with localized clusters requiring sufficiently strong temporary perturbations.Its parameterization distinguishes driver time gaps, acceleration times, look-ahead distance, and velocity variance effects.
- GKT model: Changing GKT parameters replaces triggered stop-and-go traffic with widening synchronized and pinch patterns, while stability diagrams identify metastability and convective instability as relevant conditions.The widening synchronized pattern requires critical density ρ2 on the congested side of the fundamental diagram.
- Microscopic models: The IDM generates phase diagrams qualitatively similar to GKT results, but its high-density restabilization and homogeneous congested state depend on parameter choices.For the reported parameters, homogeneous congested traffic is missing; varying s0 and a produces qualitatively different phase diagrams.
- Microscopic models: In the IDM, heterogeneous driver-vehicle units yield quasi-periodic rather than perfectly periodic oscillations, while nonstationary perturbations favor pinch effects at on-ramp bottlenecks.Replacing an on-ramp with a flow-conserving bottleneck changes many pinch-effect cases into widening synchronized states.
- Model comparison: Other car-following models can produce different phase-diagram types through parameter variation, supporting the conclusion that observed patterns are not unique to three-phase models.The Gipps model consistently produces two diagram types, whereas the OVM and VDM can produce both depending on parameters.
4.2 Three-phase models
The KK micro-model reproduces several congestion patterns also generated by two-phase models, but differs in oscillation frequencies and some propagation behavior. Setting its synchronization distance factor to k = 1 yields essentially the same patterns as a conventional two-phase model.
- The KK micro-model reproduces WSP, pinch-effect, OCT, and TSG patterns that are essentially equivalent to patterns from IDM or GKT simulations.
- The KK micro-model produces lower oscillation frequencies than IDM, often closer to empirical values.Including next-nearest-neighbor reactions can increase IDM frequencies to realistic values.
- Moving synchronized patterns differ because downstream fronts and internal congested structures can propagate upstream at different velocities.The KK model can produce congested-traffic structure speeds exceeding 40 km/h, for which the authors report no empirical evidence.
- In stationary car-following, the KK time gap satisfies τ ≤ T ≤ kτ; setting k = 1 converts the model into a conventional two-phase model.The on-ramp scenario with k = 1 produced essentially the same patterns.
4.3 Different mechanisms producing the pinch effect
The paper identifies three mechanisms that can produce pinch effects and related congestion patterns: metastability with depletion, convective instability, and locally increased stability from off-ramp/on-ramp combinations. Simulations show that these mechanisms can generate stationary pinch regions, growing upstream disturbances, and wide jams.
- Mechanism I: metastability and depletion effect: Two-phase models can reproduce the pinch effect when parameters are suitably chosen, and nonstationary bottleneck perturbations may support its occurrence.Frequent lane changes caused by weaving flows are offered as one possible source of such perturbations.
- Mechanism I: metastability and depletion effect: In the metastability mechanism, sufficiently large perturbations grow into jams whose reduced outflow is represented by slow-to-start behavior.Possible implementations include velocity-dependent stochastic deceleration, memory effects, velocity-variance-dependent driving, or suitable IDM parameters.
- Mechanism I: metastability and depletion effect: When large jams discharge more slowly than small jams, most small jams dissolve, leaving only a few wide moving jams through the depletion effect.
- Mechanism II: convective string instability: The pinch effect combines a stationary congested pinch region near the bottleneck with small perturbations that grow into fully developed moving jams upstream.
- Mechanism II: convective string instability: Gipps and IDM simulations show merging perturbations near an on-ramp growing upstream and transforming into wide jams several kilometers upstream.The Gipps simulation uses an on-ramp system with ∆t = 1.2 s.
- Mechanism III: locally increased stability: An off-ramp upstream of an on-ramp can create a stationary pinch region and upstream stop-and-go traffic through locally increased stability and effective ramp flows.The effective ramp flow is defined as Q_rmp,eff = Q_rmp,on − Q_rmp,off.
- Mechanism III: locally increased stability: With GKT parameter set 2, the on-ramp produces OCT while the effective ramp flow implies TSG or OCT with longer oscillation periods.Different oscillation periods upstream of the on-ramp and off-ramp produce merging phenomena resembling those caused by other mechanisms.
4.4 Summary of possible explanations
The paper argues that the pinch effect and other controversial spatiotemporal traffic phenomena are compatible with both three-phase and conventional two-phase models. Macroscopic, microscopic, and cellular-automaton models can reproduce them when models and parameters are suitably chosen.
- Table 2 summarizes controversial traffic phenomena and possible explanations across models consistent with Kerner’s theory and conventional models with traffic instabilities.
- The pinch effect or general pattern can be produced by both three-phase and conventional two-phase models with a fundamental diagram.This compatibility has been demonstrated using macroscopic, microscopic, and cellular-automaton models with suitable parameter choices.
5 Conclusions
The paper concludes that two-phase and three-phase approaches share more commonalities than differences, partly because terminology and model detail shape the comparison. It recommends quantitative out-of-sample evaluation while acknowledging that available evidence does not yet identify the mechanisms underlying all observations.
- Traffic phases have different meanings across the two approaches: three-phase theory emphasizes equilibrium concepts, whereas phase diagrams classify non-equilibrium patterns in non-homogeneous systems.Some three-phase distinctions additionally require a nonlocal criterion concerning propagation through the next bottleneck.
- Suitable fundamental-diagram model parameters can reproduce WSP when maximum-flow traffic is metastable and instability lies entirely on the congested side.
- Several plausible explanations fit many empirical observations, making the underlying mechanism difficult to determine.The authors view convective instability as likely for the pinch effect, ramp combinations as dominant at some intersections, and depletion as favored for transitions to wide jams.
- The authors judge that three-phase models do not explain more observations than simpler two-phase models, apart from fluctuations that can also arise from driver-vehicle heterogeneity.
- Competing models should be compared statistically on new representative measurements after calibration, using fit measures that account for model complexity.
- Model complexity should be matched to purpose: macroscopic models may suffice for travel-time forecasts, whereas driver-assistance studies require microscopic speeds, distances, and accelerations.
A Wide scattering of congested flow–density data
Wide flow–density scattering is not unique to synchronized traffic: empirical wide moving jams also show substantial scattering, challenging its use as a distinguishing feature.
- Empirical measurements show considerable flow–density scattering in wide moving jams, although theory expects a jam line.
- This suggests that wide scattering characterizes congested traffic more generally rather than synchronized flow specifically.
- The observation questions a basis of three-phase traffic theory because synchronized-flow scattering is claimed to distinguish it from wide moving jams.
B Discussion of homogeneous congested traffic
Evidence for homogeneous congested traffic under strong bottlenecks has been ambiguous, but high-resolution data suggest such states exist and model calibration can determine whether they appear.
- Evidence for homogeneous congested traffic at strong, typically accident-related bottlenecks has been ambiguous.
- Adaptive smoothing can make the spatiotemporal speed profile appear almost homogeneous, whereas measured flow may show oscillations.
- Small-wavelength oscillations may arise from driver–vehicle heterogeneity, difficulty maintaining low speeds, or perturbations from merging flows.
- High-resolution traffic data suggest homogeneous congested states exist but are very rare.
- Models with a fundamental diagram can be calibrated to generate homogeneous patterns for high bottleneck strengths or to suppress them.