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arxiv: 2604.24771 · v1 · submitted 2026-04-15 · 📡 eess.SY · cs.SY

Stochastic and Dynamic Fundamental Diagram for Mixed Traffic

Jiwan Jiang , Soyoung Ahn This is my paper

Pith reviewed 2026-05-10 12:31 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords mixed trafficautomated vehiclesfundamental diagramtraffic hysteresisstochastic modelingvehicle sequencingdynamic fundamental diagramcar-following models
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The pith

Sequencing of AVs and HDVs in platoons determines the magnitude of traffic hysteresis in mixed traffic.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a dynamic fundamental diagram framework for mixed traffic of automated vehicles and human-driven vehicles. It applies describing function analysis to approximate the nonlinear behaviors of HDV car-following models and builds a sequence-based stochastic model for platoons. Monte Carlo simulations then evaluate how different vehicle orders and AV shares affect the size and variability of hysteresis loops in flow-density relations. A sympathetic reader would care because the findings indicate that traffic flow instabilities depend on both the proportion of AVs and their specific placement within groups of vehicles, with implications for managing congestion as automation increases. The work shows that hysteresis is shaped by the unfolding interactions through sequencing rather than composition alone.

Core claim

The paper formulates a sequence-based stochastic dynamic fundamental diagram for mixed platoons by deriving approximate linear transfer functions via describing function analysis of nonlinear HDV car-following models, then uses Monte Carlo simulations to show that AV-HDV sequencing significantly alters the size of traffic hysteresis loops while higher AV shares generally dampen hysteresis magnitude and variability depending on how AVs are distributed within the platoon.

What carries the argument

The sequence-based stochastic dynamic fundamental diagram, which models mixed platoons to evaluate hysteresis across sequencing scenarios and AV penetration levels.

If this is right

  • Differences in AV-HDV sequencing significantly alter the size of traffic hysteresis loops.
  • Higher AV shares generally dampen hysteresis magnitude and variability.
  • The net impact of AV penetration on hysteresis depends on how AVs are distributed within the platoon.
  • Traffic hysteresis in mixed environments is governed by both the composition of AVs and HDVs and how their interactions unfold through sequencing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Traffic control strategies could prioritize specific AV insertion points in human-driven streams to minimize flow instabilities.
  • The framework could be tested in field experiments by deploying AVs in controlled platoon orders and observing real hysteresis patterns.
  • Extensions might include optimizing platoon formation rules in mixed traffic to reduce overall congestion variability.
  • Similar sequencing effects could appear in other flow systems, such as pedestrian or supply chain models with mixed agent types.

Load-bearing premise

The describing function analysis accurately approximates the nonlinear HDV car-following models for use in the mixed platoon dynamic FD, and the stochastic elements reflect real traffic variability without needing empirical calibration.

What would settle it

Collect empirical flow-density data from real mixed traffic platoons with known AV-HDV sequences and measure whether the observed hysteresis loop sizes match the model's Monte Carlo predictions for different sequencing patterns.

Figures

Figures reproduced from arXiv: 2604.24771 by Jiwan Jiang, Soyoung Ahn.

Figure 1
Figure 1. Figure 1: Dynamic FD vs. Stochastic Dynamic FD [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of Nonlinear Acceleration Behaviors of (a-d) HDV CF Laws; (e-f) Linear Feedback Controllers of CAV. Jiang & Ahn: Preprint submitted to Elsevier Page 14 of 21 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Hybrid Model Distribution Results (𝑁 = 3) [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Frequency Response of G Using I-80 Dataset for (a) FVDM (b) GFM (c) OVM (d) IDM. Jiang & Ahn: Preprint submitted to Elsevier Page 15 of 21 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Probability Density Distribution of the Mixed Dynamic FD (Sequence) (a) AV first, (b) HDV first, (c) Alternating, (d) Random sequencing. Jiang & Ahn: Preprint submitted to Elsevier Page 16 of 21 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Probability Density Distribution of the Mixed Dynamic FD - AV first (Penetration Rate): (a) 0%AV, (b) 33.3%AV, (c) 66.7%AV, (d) 100%AV) Jiang & Ahn: Preprint submitted to Elsevier Page 17 of 21 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Probability Density Distribution of the Mixed Dynamic FD - HDV first (Penetration Rate): (a) 0% AV, (b) 33.3% AV, (c) 66.7% AV, (d) 100%AV) Jiang & Ahn: Preprint submitted to Elsevier Page 19 of 21 [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Probability Density Distribution of the Mixed Dynamic FD - Random (Penetration Rate): (a) 0% AV, (b) 33.3% AV, (c) 66.7% AV, (d) 100%AV) Jiang & Ahn: Preprint submitted to Elsevier Page 20 of 21 [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
read the original abstract

This study develops a dynamic fundamental diagram (FD) framework tailored to mixed traffic environments comprising automated vehicles (AVs) and human-driven vehicles (HDVs). Describing function analysis is employed to derive approximate linear transfer functions for nonlinear HDV car-following models. A sequence-based stochastic dynamic FD is then formulated for mixed platoons, enabling the evaluation of hysteresis in the evolution of flow-density relations across different vehicle sequencing scenarios and AV penetration levels. Monte Carlo simulation results reveal that (i) differences in AV-HDV sequencing significantly alter the size of traffic hysteresis loops; and (ii) higher AV shares generally dampen hysteresis magnitude and variability, yet the net impact depends on how AVs are distributed within the platoon. The results suggest that traffic hysteresis in mixed environments is governed not only by the composition of AVs and HDVs, but also by how their interactions unfold through sequencing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper develops a dynamic fundamental diagram (FD) for mixed AV-HDV traffic by applying describing function analysis to obtain amplitude-dependent linear approximations of nonlinear HDV car-following models. It then constructs a sequence-based stochastic dynamic FD for mixed platoons and uses Monte Carlo simulations to evaluate how AV-HDV sequencing and penetration levels affect the size and variability of traffic hysteresis loops in flow-density relations.

Significance. If the describing-function approximations preserve the relevant phase and amplitude sensitivities, the work demonstrates that hysteresis in mixed traffic depends on both AV share and the specific ordering of vehicles within platoons, rather than composition alone. This could inform platoon management strategies. The Monte Carlo framework is a strength for exploring stochastic variability under the formulated model.

major comments (3)
  1. [§3.2] §3.2 (describing function derivation): The central hysteresis claims rest on the describing-function linearization of the nonlinear HDV models, yet no residual analysis, amplitude-dependent error bounds, or direct comparison of the approximated transfer functions against the original nonlinear ODEs is provided for the mixed-platoon sequences used in the Monte Carlo runs. Without this, it is unclear whether the reported differences in loop size are preserved or distorted by the approximation.
  2. [§5] §5 (stochastic dynamic FD formulation): The stochastic elements are introduced without calibration to field data or sensitivity analysis showing that the modeled variability matches observed traffic fluctuations; this directly affects the claim that higher AV shares dampen hysteresis variability in a manner representative of real mixed traffic.
  3. [§6] §6 (Monte Carlo results, Tables 2-4): The reported effects of sequencing on hysteresis magnitude are derived solely from the approximated model; no side-by-side validation runs using the original nonlinear HDV dynamics for identical sequences are shown, leaving the load-bearing claim that sequencing alters loop size without quantitative support for approximation fidelity.
minor comments (2)
  1. Notation for the sequence-dependent parameters in the stochastic FD could be clarified with an explicit table mapping symbols to physical meanings.
  2. Figure captions for the hysteresis plots should explicitly state the number of Monte Carlo realizations and the perturbation amplitude used.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments that highlight opportunities to strengthen the validation of our framework. We address each major comment point by point below and commit to specific revisions.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (describing function derivation): The central hysteresis claims rest on the describing-function linearization of the nonlinear HDV models, yet no residual analysis, amplitude-dependent error bounds, or direct comparison of the approximated transfer functions against the original nonlinear ODEs is provided for the mixed-platoon sequences used in the Monte Carlo runs. Without this, it is unclear whether the reported differences in loop size are preserved or distorted by the approximation.

    Authors: We agree that explicit validation of the describing-function approximation is needed to support the hysteresis claims. In the revised manuscript we will add residual analysis, amplitude-dependent error bounds, and direct comparisons of the approximated transfer functions against the original nonlinear ODEs for representative mixed-platoon sequences from the Monte Carlo runs. These additions will quantify approximation fidelity and confirm that observed differences in loop size are not distorted by the linearization. revision: yes

  2. Referee: [§5] §5 (stochastic dynamic FD formulation): The stochastic elements are introduced without calibration to field data or sensitivity analysis showing that the modeled variability matches observed traffic fluctuations; this directly affects the claim that higher AV shares dampen hysteresis variability in a manner representative of real mixed traffic.

    Authors: The stochastic dynamic FD is formulated as a theoretical model derived from sequence-dependent variability rather than an empirical fit. We will add a sensitivity analysis on the stochastic parameters in the revision and relate the modeled variability to reported traffic fluctuations in the literature. We will also explicitly state the limitations regarding direct calibration to field data and the representative nature of the results. revision: partial

  3. Referee: [§6] §6 (Monte Carlo results, Tables 2-4): The reported effects of sequencing on hysteresis magnitude are derived solely from the approximated model; no side-by-side validation runs using the original nonlinear HDV dynamics for identical sequences are shown, leaving the load-bearing claim that sequencing alters loop size without quantitative support for approximation fidelity.

    Authors: We recognize the value of direct validation against the nonlinear dynamics. In the revised manuscript we will include side-by-side Monte Carlo runs using the original nonlinear HDV car-following models for selected sequences and report quantitative comparisons of hysteresis loop sizes. This will supply the missing support for approximation fidelity and the sequencing effects shown in Tables 2-4. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is model-based simulation without self-referential reduction.

full rationale

The paper constructs a dynamic FD by applying describing-function linearization to nonlinear HDV car-following models, then assembles a sequence-based stochastic FD for mixed platoons and evaluates hysteresis via Monte Carlo simulation. These steps produce simulation outputs rather than tautological identities or fitted parameters renamed as predictions. No load-bearing self-citations, self-definitional equations, or ansatz smuggling appear in the provided derivation chain. The reported effects on hysteresis loop size and variability are direct consequences of the forward simulation under varying sequences and AV shares, not reductions to the inputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The framework builds on standard traffic modeling assumptions and introduces the sequencing as a key variable without new physical entities.

free parameters (2)
  • AV penetration levels
    Varied as input parameters in the Monte Carlo simulations to assess impact.
  • Sequencing scenarios
    Different orders of AVs and HDVs chosen for evaluation.
axioms (1)
  • domain assumption Describing function analysis provides a valid linear approximation for the nonlinear dynamics of HDV car-following models.
    Employed to derive approximate linear transfer functions for the HDV models in the mixed traffic framework.

pith-pipeline@v0.9.0 · 5442 in / 1405 out tokens · 47738 ms · 2026-05-10T12:31:01.208797+00:00 · methodology

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Reference graph

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