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A Second Order Model of Capacity Drop at Expressway Lane-Drop Bottlenecks

Fuminori Hattori, Kentaro Wada

TL;DR

This work extends Jin's first-order capacity-drop model for lane-drop bottlenecks to a second-order, bounded-acceleration framework. By formulating a BA-LWR-based continuum model and a one-dimensional iterated function system, the authors characterize not only the stationary capacity-drop state but also the transition dynamics from congestion onset to that state, proving global stability via contraction. The reduced model enables analytical insights and an efficient sensitivity analysis, while calibration and validation with Zen Traffic Data from the Hanshin Expressway demonstrate the model's ability to reproduce speed recovery and estimate bottleneck properties. The results suggest that capacity drop stabilizes immediately once congestion arises at lane-drop bottlenecks and offer a potential foundation for traffic-control and roadway-design strategies to mitigate congestion effects.

Abstract

This paper presents a second-order model of capacity drop at expressway lane-drop bottlenecks. The model is an extension of Jin's model (Jin, 2017). This model captures not only the stationary state associated with the capacity drop but also the transitional dynamics leading from the onset of congestion to that state. The characteristics of the proposed model are examined theoretically and umerically. The results show that the capacity drop stationary state is stable and is reached immediately once congestion occurs. Furthermore, we validate the model using empirical data. The results suggest that the model has the potential to provide new insights into congestion phenomena at expressway lane-drop bottlenecks.

A Second Order Model of Capacity Drop at Expressway Lane-Drop Bottlenecks

TL;DR

This work extends Jin's first-order capacity-drop model for lane-drop bottlenecks to a second-order, bounded-acceleration framework. By formulating a BA-LWR-based continuum model and a one-dimensional iterated function system, the authors characterize not only the stationary capacity-drop state but also the transition dynamics from congestion onset to that state, proving global stability via contraction. The reduced model enables analytical insights and an efficient sensitivity analysis, while calibration and validation with Zen Traffic Data from the Hanshin Expressway demonstrate the model's ability to reproduce speed recovery and estimate bottleneck properties. The results suggest that capacity drop stabilizes immediately once congestion arises at lane-drop bottlenecks and offer a potential foundation for traffic-control and roadway-design strategies to mitigate congestion effects.

Abstract

This paper presents a second-order model of capacity drop at expressway lane-drop bottlenecks. The model is an extension of Jin's model (Jin, 2017). This model captures not only the stationary state associated with the capacity drop but also the transitional dynamics leading from the onset of congestion to that state. The characteristics of the proposed model are examined theoretically and umerically. The results show that the capacity drop stationary state is stable and is reached immediately once congestion occurs. Furthermore, we validate the model using empirical data. The results suggest that the model has the potential to provide new insights into congestion phenomena at expressway lane-drop bottlenecks.

Paper Structure

This paper contains 13 sections, 1 theorem, 20 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Formulation
  4. Continuum Model in Lagrangian Coordinate
  5. Theoretical Analysis
  6. Reduced Model
  7. Stability of the capacity drop phenomenon
  8. Sensitivity Analysis
  9. Calibration and Validation
  10. Validation Framework
  11. Validation Results
  12. Conclusion
  13. Derivation of Model \ref{['overallCF']}

Key Result

theorem 1

The mapping $f: [0, \bar{v}] \to [0, \bar{v}]$, defined by IFS, has a unique fixed point and its fixed point is globally stable.

Figures (15)

  • Figure 1: Lane drop zone
  • Figure 2: Flow-density fundamental diagram
  • Figure 3: Evolution of queue discharge flow rate at $x=L$ under the second order model
  • Figure 4: The location of the $n$th vehicle at time $t$ (where color indicates traffic states)
  • Figure 5: Speed recovery profile in the stationary state of capacity drop (where color indicates traffic states)
  • ...and 10 more figures

Theorems & Definitions (2)

  • theorem 1
  • proof