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Modeling Cascading Driver Interventions in Partially Automated Traffic: A Semi-Markov Chain Approach

Zihao Li, Fan Pu, Soyoung Ahn, Yang Zhou

TL;DR

This work develops a leader-dependent semi-Markov framework to model cascading driver interventions in partially automated traffic, capturing lockout delays and mode-switch cascades between AV and HDV states. By approximating deterministic lockout with a phase-type (specifically Erlang-$k$) distribution, the model becomes nonlinear yet tractable as a piecewise affine bilinear system, enabling analysis of equilibrium existence and global stability via Brouwer’s theorem and a common quadratic Lyapunov function. Through numerical experiments, the authors demonstrate how cascading transitions can substantially degrade throughput, particularly under downward-dominant conditions, and show that real-speed profiles (NGSIM data) amplify these effects. The framework provides actionable insights into how the share of permanent HDVs, initial mode mixes, and leader-dependent decisions influence system performance, with validation and sensitivity analyses guiding potential policy or design interventions for partially automated traffic. Future work includes validation with more real-world data and extending driver decision models to capture richer behavioral dynamics.

Abstract

This paper presents an analytical modeling framework for partially automated traffic, incorporating cascading driver intervention behaviors. In this framework, drivers of partially automated vehicles have the flexibility to switch driving modes (either AV or HDV) under lockout constraints. The cascading impact is captured by making the switching probability leader-dependent, highlighting the influence of the leading vehicle on mode choice and the potential propagation of mode changes throughout traffic. Due to the complexity of this system, traditional Markov-based methods are insufficient. To address this, the paper introduces an innovative semi-Markov chain framework with lockout constraints, ideally suited for modeling the system dynamics. This framework reformulates the system as a nonlinear model whose solution can be efficiently approximated using numerical methods from control theory, such as the Runge-Kutta algorithm. Moreover, the system is proven to be a piecewise affine bilinear system, with the existence of solutions and both local and global stability established via Brouwer's Fixed Point Theorem and the 1D Uncertainty Polytopes Theorem. Numerical experiments corroborate these theoretical findings, confirming the presence of cascading impacts and elucidating the influence of modeling parameters on traffic throughput, thereby deepening our understanding of the system's properties.

Modeling Cascading Driver Interventions in Partially Automated Traffic: A Semi-Markov Chain Approach

TL;DR

This work develops a leader-dependent semi-Markov framework to model cascading driver interventions in partially automated traffic, capturing lockout delays and mode-switch cascades between AV and HDV states. By approximating deterministic lockout with a phase-type (specifically Erlang-) distribution, the model becomes nonlinear yet tractable as a piecewise affine bilinear system, enabling analysis of equilibrium existence and global stability via Brouwer’s theorem and a common quadratic Lyapunov function. Through numerical experiments, the authors demonstrate how cascading transitions can substantially degrade throughput, particularly under downward-dominant conditions, and show that real-speed profiles (NGSIM data) amplify these effects. The framework provides actionable insights into how the share of permanent HDVs, initial mode mixes, and leader-dependent decisions influence system performance, with validation and sensitivity analyses guiding potential policy or design interventions for partially automated traffic. Future work includes validation with more real-world data and extending driver decision models to capture richer behavioral dynamics.

Abstract

This paper presents an analytical modeling framework for partially automated traffic, incorporating cascading driver intervention behaviors. In this framework, drivers of partially automated vehicles have the flexibility to switch driving modes (either AV or HDV) under lockout constraints. The cascading impact is captured by making the switching probability leader-dependent, highlighting the influence of the leading vehicle on mode choice and the potential propagation of mode changes throughout traffic. Due to the complexity of this system, traditional Markov-based methods are insufficient. To address this, the paper introduces an innovative semi-Markov chain framework with lockout constraints, ideally suited for modeling the system dynamics. This framework reformulates the system as a nonlinear model whose solution can be efficiently approximated using numerical methods from control theory, such as the Runge-Kutta algorithm. Moreover, the system is proven to be a piecewise affine bilinear system, with the existence of solutions and both local and global stability established via Brouwer's Fixed Point Theorem and the 1D Uncertainty Polytopes Theorem. Numerical experiments corroborate these theoretical findings, confirming the presence of cascading impacts and elucidating the influence of modeling parameters on traffic throughput, thereby deepening our understanding of the system's properties.
Paper Structure (18 sections, 9 theorems, 47 equations, 12 figures, 1 table)

This paper contains 18 sections, 9 theorems, 47 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Statement and Assumptions
  3. Leader-Dependent Semi-Markov Mixed Traffic System
  4. State Dynamics
  5. System Property Analysis
  6. Steady State Analysis
  7. Existence of Equilibrium
  8. Stability Analysis
  9. Traffic Throughput Analysis
  10. Numerical Experiments
  11. Erlang-$k$ Approximation for Deterministic Lockout Time
  12. Global Stability of the System
  13. Leader-Dependent Dynamics
  14. Throughput Analysis Using NGSIM Data
  15. Sensitivity Analysis
  16. ...and 3 more sections

Key Result

Theorem 1

Let $F$ be any cumulative distribution function on $[0,\infty)$ with finite mean. Then, for every $\varepsilon > 0$, there exists a phase-type distribution $F_{\mathrm{PH}}$ such that

Figures (12)

  • Figure 1: Downward Mode Transitions (AV mode to HDV mode) and cascading effects in mixed traffic
  • Figure 2: PAV state transition diagram
  • Figure 3: PAV state transition diagram without downward transition lockout period
  • Figure 4: Example of piecewise constant approximation for sigmoid headway transition function
  • Figure 5: Approximation of deterministic lockout time using Erlang-$k$ distribution.
  • ...and 7 more figures

Theorems & Definitions (19)

  • Theorem 1: Phase-Type Distributions, david1987least
  • Proposition 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2: Brouwer’s Fixed-Point Theorem, brouwer1911abbildung
  • Remark 4
  • Proposition 2
  • proof
  • ...and 9 more