Pith. sign in

REVIEW 3 major objections 3 minor 42 references

A multi-class non-local macroscopic model with time delay for mixed autonomous / human-driven traffic

T0 review · 3 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper extends a scalar delayed non-local traffic model to multiple vehicle classes and proves global existence, uniqueness, and L1 stability of entropy weak solutions, with numerical evidence that autonomous vehicles dampen…

desk verdict Solid new well-posedness theorem for a multi-class delayed non-local traffic model, but the proved model does not enforce total road capacity, and the paper's own numerics show it. read the letter →

arxiv 2501.09440 v1 pith:5TTE3J4H submitted 2025-01-16 math.AP cs.NAmath.NA

Ilaria Ciaramaglia (ACUMES , LJAD , UniCA) , Paola Goatin (ACUMES , UniCA , LJAD) , Gabriella Puppo (UNIROMA) This is my paper · ORCID
classification math.APcs.NAmath.NA MSC 35L6535L0365M1276A30
keywords Non-localconservationlawsTimedelayMulti-classtrafficflowAutonomousvehiclesEntropyweaksolutionsBVestimatesL1stabilityHilliges-Weidlichscheme
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper extends a scalar delayed non-local traffic model to multiple vehicle classes, letting each class have its own reaction time, look-ahead distance, speed law, and saturation. For bounded-variation initial data, it proves global existence of a unique entropy weak solution on any time horizon, together with L1 stability with respect to initial data and delay parameters. A corollary gives convergence of delayed solutions to the delay-free multi-class model, and also supplies a global existence result for that non-delayed model, improving earlier local-in-time results. Numerical experiments on a ring road indicate that replacing human-driven vehicles with autonomous vehicles reduces oscillatory stop-and-go patterns, with the smoothest flow near a 70 percent autonomous share.

What carries the argument

The Hilliges-Weidlich finite volume flux $F_{i,j+1/2} = \rho_{i,j} f_i(\rho_{i,j+1}) v_i\big(\Delta x \sum_k \omega_i^k r_{j+k}\big)(t-\tau_i)$ produces approximate solutions whose positivity, weak maximum principle, discrete entropy inequality, and spatial and temporal BV bounds pass to the limit. The saturation function $f_i$ is load-bearing: it enforces the maximum principle and makes the L∞ and BV estimates global. The L1 stability inequality (4.6) is obtained by adapting Kru\v{z}kov's doubling-of-variables technique to the delayed non-local velocities, using uniform BV bounds on each approximate component.

What would settle it

Run the Hilliges-Weidlich scheme for the total-density saturation model (2.7) on a ring road with a small BV perturbation of a constant state and track the total variation: if some bounded-variation initial datum produces total variation that grows without bound as the mesh refines, the no-BV-estimate limitation is real and the well-posedness theorem cannot extend to hard capacity constraints. Alternatively, evaluate the functional $J(p)$ with the triangular speed law (5.13) for a non-uniform autonomous-vehicle distribution; if for some positive $p$ the value $J(p)$ exceeds $J(0)$, the numerical claim that autonomous vehicles improve stability fails in that regime.

Watch

Extended reading notes

Core claim

The central discovery is that adding a class-specific saturation function $f_i(\rho_i)$ to each delayed non-local conservation law makes the mixed-traffic system globally well-posed. Each class density stays in $[0,R_i]$, L1 mass is conserved, total variation grows at most exponentially with a rate that increases with delays and decreases with look-ahead distance, and a Kru\v{z}kov-type doubling-of-variables argument yields L1 stability with respect to initial data and delay parameters. This provides the first global existence result for the multi-class non-local model with delay, and it improves the earlier no-saturation multi-class model, for which existence was only known locally in time. A separate numerical comparison shows that the class-specific saturation model can let the total density exceed the road's maximal capacity, while a model with saturation depending on total density preserves the capacity simplex but lacks the BV estimates needed for the well-posedness proof.

Load-bearing premise

The well-posedness proof covers only saturation functions that depend on each class's own density $\rho_i$, not on the total density $r$; the more physically natural total-density saturation is shown numerically to preserve the capacity simplex, but for it no BV estimates are available, and the class-specific model can let total density exceed the road's capacity.

Editorial extensions

If this is right

  • For any time horizon $T>0$ and bounded-variation initial data, the delayed multi-class system (1.1)-(1.2) has a global entropy weak solution whose components remain in $[0,R_i]$ and conserve L1 mass.
  • Entropy solutions are unique and depend continuously on both initial data and the delay vector $\tau$, with the explicit bound $\|\rho(t,\cdot)-\sigma(t,\cdot)\|_1 \le e^{K_1 T}\big(K_3\|\rho_0-\sigma_0\|_1 + K_2\|\tau-\nu\|_1\big)$.
  • As all delays tend to zero, solutions converge in L1 to the solution of the non-delayed multi-class model, which now inherits global existence from the delayed analysis.
  • In the numerical AV-human scenarios, the total variation of the total density decreases as the autonomous-vehicle penetration rate increases, with the minimum of the functional $J(p)$ occurring near $p=0.7$.
  • Delay increases the total variation bounds and produces more oscillatory density profiles, while larger look-ahead distances shrink the relevant constants and stabilize the solution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If delay is the main destabilizing mechanism, the stability estimate suggests that even small communication latencies among autonomous vehicles could measurably increase density oscillations in real mixed-traffic flows, a prediction testable in car-following experiments.
  • The non-monotonic dependence of $J(p)$ on the penetration rate indicates that an intermediate autonomous share, around 70 percent in this setup, is preferable; the paper's model uses no external control, so adding active controllers could shift that optimum in either direction.
  • The missing BV theory for the total-density saturation model (2.7) is the main obstacle to applying the well-posedness result under a hard road-capacity constraint, so proving compactness for (2.7) would be a natural next step with direct traffic-engineering consequences.
  • The constant $K_2\|\tau-\nu\|_1$ in (4.6) offers a quantitative way to compare human reaction times with autonomous-vehicle latencies: calibrating $\tau_H$ from empirical reaction-time data would let the model predict the minimum autonomous penetration needed to stabilize a given flow.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper introduces a multi-class non-local conservation-law model with time delays for mixed autonomous/human-driven traffic. Each class has its own saturation function, speed function, and convolution kernel, and the classes are coupled through the non-local velocity that depends on total density. The authors construct Hilliges-Weidlich finite-volume approximations, prove positivity, L1 and L∞ bounds, spatial and spatio-temporal BV estimates, and a discrete entropy inequality. From these they derive global existence of entropy weak solutions for BV initial data (Theorem 1), L1 stability with respect to initial data and delay parameters with uniqueness as a consequence (Theorem 2), and convergence to the non-delayed model as delays vanish (Corollary 3). Numerical experiments study the effect of saturation, convergence to the non-delayed model, and the stabilizing influence of autonomous vehicles at different penetration rates.

Significance. If the proofs are correct, the paper makes a useful mathematical contribution: it extends the scalar delayed non-local traffic model of [10] to a multi-class system, permits zero delays for some classes (as needed for autonomous vehicles), and improves on [6] by obtaining global existence through saturation terms. The explicit CFL conditions, uniform estimates, discrete entropy inequality, and L1-stability-with-delay result provide a solid quantitative framework. The numerical section offers a plausible and clearly described exploration of AV penetration rates. The main limitation, which the authors themselves acknowledge, is that the proved well-posedness covers only the class-specific saturation f_i(ρ_i); the physically more natural total-density saturation f_i(r), which enforces the road-capacity constraint, is outside the theory. This scope gap is not an internal inconsistency, but it does mean the advertised application to mixed traffic with a hard capacity constraint is only partially underwritten.

major comments (3)
  1. [Section 5.2, Remark 1, Eq. (2.7)] The well-posedness results (Theorems 1 and 2, Eqs. (4.5)-(4.6)) are proved only for the saturation f_i(ρ_i) in (1.1). The physically more natural model (2.7)/(5.6), where saturation depends on the total density r and the simplex S is invariant, is explicitly outside the theory: the text states that 'BV estimates are not available in general' for (2.7) and that 'well-posedness results for (5.6) are currently missing.' Since the abstract advertises the model for mixed autonomous/human traffic, where road capacity is a hard constraint, the central claim does not cover the capacity-respecting variant. I recommend either extending the analysis to (2.7), at least under additional structural assumptions, or substantially revising the abstract and Section 5 to present (1.1) as the analyzed model and (2.7) as a numerically investigated alternative.
  2. [Section 3, Proposition 2] Proposition 2, which provides the uniform space-time BV estimate needed for Helly compactness in the proof of Theorem 1, is not proved in the manuscript; its proof is replaced by 'See proof of [10, Proposition 3.6].' The scalar delayed case of [10] does not automatically cover the multi-class coupled system with possibly zero delays for some classes, and the BV recursion in Proposition 1 already shows that the delayed multi-class case requires a separate argument. The paper should include a complete proof, or at least a detailed statement of the modifications needed for the multi-class setting.
  3. [Abstract, Section 5.4] The abstract's claim that 'the presence of autonomous vehicles improves overall traffic flow and stability' is supported numerically by the penetration-rate experiments of Section 5.4, but those experiments use the class-specific saturation model (5.1) with fixed parameters (τ_H=2.5, τ_A=0, L_H=0.1, L_A=0.2, and the speed laws (5.3) or (5.13)). Section 5.2 shows that this model can produce total density r>1, violating the road's maximal capacity, whereas the capacity-respecting model (5.6) is outside the proved well-posedness theory. The numerical conclusion is therefore an extrapolation from a model whose capacity constraint is violated; the paper should either repeat the stabilization study for the capacity-respecting model or clearly qualify the claim as a numerical observation for (5.1) only.
minor comments (3)
  1. [Section 2, Lemma 3] In the proof of Lemma 3, the last term in the expression for ∂Φ/∂ρ_i,j is non-positive because v'_i≤0; the displayed inequality '≥ 0' is therefore incorrect as written. Monotonicity follows by bounding the absolute value of this term with the last part of the CFL condition (2.6); please correct the display accordingly.
  2. [Section 5.5] There is a typo: 'Not that the initial total density is constant' should read 'Note that the initial total density is constant.'
  3. [Section 5.4, Eq. (5.13)] The triangular speed law (5.13) is only piecewise linear and does not satisfy the C1 regularity required by Assumption 1. The authors mention that smoothing would recover the assumption, but the numerical experiments appear to use the unsmoothed law; please state explicitly whether the reported simulations use the non-smooth version and treat the results as formal, or use a smoothed approximation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the well-posedness theorem is proved in this paper from model assumptions; self-citations to the scalar delayed model are independent support, and the AV-stabilization claim is a qualitative simulation consequence, not a fitted prediction.

full rationale

The paper's central derivation chain is self-contained. Theorem 1 is proved via the discrete entropy inequality, Helly compactness, and a Lax–Wendroff argument completed in Appendix B; Theorem 2 is proved by a Kruzhkov doubling-of-variables argument in Section 4. The assumptions in Assumption 1 are model hypotheses, not fitted parameters, and no target quantity is inserted into the construction. The self-citations to [10] are used as proof templates and as a scalar predecessor; [10] is a published, parameter-free result with stated assumptions that do not include the multi-class target, so under the review rules it counts as independent evidence rather than circularity. The convergence-to-the-non-delayed model (Corollary 3) follows directly from the L1 stability estimate (4.6), not from an imported conclusion. The numerical statement that AVs improve traffic flow is an illustration with hand-set parameters (tau_H=2.5, tau_A=0, L_H=0.1, L_A=0.2, kernels (5.10)); nothing is fitted from the output, so no fitted parameter is renamed as a prediction. The paper explicitly flags the scope limitation that well-posedness is proved only for class-specific saturation f_i(rho_i), while the total-density saturation model (2.7)/(5.6) lacks BV estimates; Section 5.2 even shows that (1.1) can violate total road capacity. This is a correctness/scope gap, not a circular step: no equation is equivalent by construction to an input. The honest finding is therefore no significant circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central theorem has no fitted parameters: it holds for all model coefficients satisfying Assumption 1. The hand-chosen numerical parameters listed above shape the AV-stabilization conclusion, not the well-posedness proof. The main unprovided input is the class-specific saturation coupling, which the authors acknowledge is less physical than total-density saturation, plus the constant backward extension of initial data. No new entities such as particles or forces are introduced.

free parameters (4)
  • reaction delays tau_H=2.5, tau_A=0 = 2.5, 0
    Hand-chosen in Section 5.4 (Eq. 5.8); the AV stabilization result depends on this asymmetry.
  • look-ahead distances L_H=0.1, L_A=0.2 = 0.1, 0.2
    Chosen in Eq. (5.9); larger AV look-ahead is a key driver of the stabilizing effect.
  • exponential saturation rate 50 = 50
    Used in Eq. (5.5) to approximate the indicator of [0,R_i); affects the estimates through |f'_i|.
  • critical densities rho_c,H=0.4, rho_c,A=0.6 = 0.4, 0.6
    Used in the triangular speed law (5.13) and Figure 7 to give AVs a higher free-flow density threshold.
assumptions (5)
  • domain assumption Assumption 1: v_i in C^2 nonincreasing, f_i in C^1 nonincreasing, omega_i in C^1 nonincreasing with positive integral.
    Stated in Section 2 and used in all proofs for positivity, L-infinity, BV, and entropy estimates.
  • domain assumption Initial data on [-||tau||,0] is a constant backward extension of rho_i(0,x) (Eq. 1.2).
    This reduces the delay problem to a classical Cauchy problem; nonconstant past data would require a different well-posedness analysis.
  • domain assumption Solutions are entropy weak solutions in the sense of Definition 2.
    The nonstandard entropy inequality includes a velocity-derivative term; it is assumed to be the correct admissibility criterion for uniqueness.
  • ad hoc to paper Saturation functions depend on the individual density rho_i, not on total density r.
    Needed for the BV estimates; the authors acknowledge the physically more natural f_i(r) model (2.7) has no general BV estimates (Remark 1).
  • standard math Standard results: Helly's theorem, Lax-Wendroff consistency, Kruzhkov doubling of variables, Gronwall lemma.
    Invoked in Section 4 and Appendix B without proof, as common background in conservation law theory.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A multi-class non-local macroscopic model with time delay for mixed autonomous / human-driven traffic." pith.science (2026). https://pith.science/paper/5TTE3J4H

@misc{pith2026250109440,
  author       = {Pith},
  title        = {Pith review of: A multi-class non-local macroscopic model with time delay for mixed autonomous / human-driven traffic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5TTE3J4H}},
  note         = {Machine review of arXiv:2501.09440}
}
abstract

In this paper, we present a class of systems of non-local conservation laws in one space-dimension incorporating time delay, which can be used to investigate the interaction between autonomous and human-driven vehicles, each characterized by a different reaction time and interaction range. We construct approximate solutions using a Hilliges-Weidlich scheme and we provide uniform L $\infty$ and BV estimates which ensure the convergence of the scheme, thus obtaining existence of entropy weak solutions of bounded variation. Uniqueness follows from an L 1 stability result derived from the entropy condition. Additionally, we provide numerical simulations to illustrate applications to mixed autonomous / human-driven traffic flow modeling. In particular, we show that the presence of autonomous vehicles improves overall traffic flow and stability.

Figures

Figures reproduced from arXiv: 2501.09440 by the authors.

Figure 1
Figure 1. Comparison between the solution of the model (5.1) with no saturation ( [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Comparison between the model (5.1) considered in this work, and the modified [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Convergence of the delayed model (5.1) with initial data [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Comparison between the total density of the solution of (5.1) with initial data [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]
Figure 5
Figure 5. Figure 5: Density profiles of each class taken individually, corresponding to the total [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]
Figure 6
Figure 6. Figure 6: Functional J defined in (5.12) for T = 30 associated to the initial datum (5.7)- (5.11a)-(5.11b) and to the delay τH ∈ {2, 2.1, 2.2, 2.3, 2.4, 2.5}. Left: Greenshields’ speed law (5.3). Right: Triangular speed law (5.13). p = 0, 0.2, 0.4, 0.6, 0.8, 1. We can see that i…
Figure 7
Figure 7. Figure 7: Speed-density relation described in (5.13) with [PITH_FULL_IMAGE:figures/full_fig_p028_7.png]
Figure 8
Figure 8. Figure 8: Solution at the final time T = 30 of the model (5.1) with initial condition given by (5.14) and penetration rate respectively equal to p = 0.2, 0.4, 0.6, 0.8 [PITH_FULL_IMAGE:figures/full_fig_p030_8.png]
Figure 9
Figure 9. Figure 9: Total variation of the total density r = ρH + ρA with respect to time associated to the tests in [PITH_FULL_IMAGE:figures/full_fig_p030_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 26 canonical work pages

  1. [10]

    Ciaramaglia, P

    I. Ciaramaglia, P. Goatin, and G. Puppo. Non-local traffic flow models with time delay: Well-posedness and numerical approximation. Discrete and Continuous Dynamical Systems - B, 2024. https://doi.org/10.3934/dcdsb.2024113

    doi:10.3934/dcdsb.2024113 2024

  2. [6]

    F. A. Chiarello and P. Goatin. Non-local multi-class traffic flow models. Netw. Heterog. Media, 14(2):371–387, 2019. https://doi.org/10.3934/nhm.2019015

    work page doi:10.3934/nhm.2019015 2019

  3. [1]

    Aggarwal, R

    A. Aggarwal, R. M. Colombo, and P. Goatin. Nonlocal systems of conservation laws in several space dimensions. SIAM J. Numer. Anal., 53(2):963–983, 2015. https://doi.org/ 10.1137/140975255

    doi:10.1137/140975255 2015

  4. [2]

    S. S. Avedisov, G. Bansal, and G. Orosz. Impacts of connected automated vehicles on freeway traffic patterns at different penetration levels. IEEE Transactions on Intelligent Transportation Systems, 23(5):4305–4318, 2022. https://doi.org/10.1109/TITS.2020. 3043323

    doi:10.1109/tits.2020 2022

  5. [3]

    Benzoni-Gavage and R

    S. Benzoni-Gavage and R. M. Colombo. An n-populations model for traffic flow. European J. Appl. Math., 14(5):587–612, 2003. https://doi.org/10.1017/S0956792503005266

    work page doi:10.1017/s0956792503005266 2003

  6. [4]

    Blandin and P

    S. Blandin and P. Goatin. Well-posedness of a conservation law with non-local flux arising in traffic flow modeling. Numer. Math., 132(2):217–241, 2016. http://doi.org/10.1007/ s00211-015-0717-6

    work page 2016

  7. [5]

    F. A. Chiarello and P. Goatin. Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel. ESAIM Math. Model. Numer. Anal., 52(1):163–180,

    work page

  8. [7]

    F. A. Chiarello, P. Goatin, and E. Rossi. Stability estimates for non-local scalar conservation laws. Nonlinear Anal. Real World Appl., 45:668–687, 2019. https://doi.org/10.1016/j. nonrwa.2018.07.027

    doi:10.1016/j 2019

Show all 42 references
  1. [8]

    F. A. Chiarello, B. Piccoli, and A. Tosin. Multiscale control of generic second order traffic models by driver-assist vehicles. Multiscale Modeling & Simulation, 19(2):589–611, 2021. https://doi.org/10.1137/20M1360128

    2021 doi

  2. [9]

    Chitturi and R

    M. Chitturi and R. Benekohal. Passenger-car equivalents for heavy vehicles in work zones. December 2007. 37

    2007

  3. [11]

    D. J. Fagnant and K. Kockelman. Preparing a nation for autonomous vehicles: opportunities, barriers and policy recommendations. Transportation Research Part A: Policy and Practice, 77:167–181, 2015. https://doi.org/10.1016/j.tra.2015.04.003

    2015 doi

  4. [12]

    Fan and D

    S. Fan and D. B. Work. A heterogeneous multiclass traffic flow model with creeping. SIAM Journal on Applied Mathematics, 75(2):813–835, 2015. https://doi.org/10.1137/ 140977977

    2015

  5. [13]

    Friedrich, O

    J. Friedrich, O. Kolb, and S. G¨ ottlich. A Godunov type scheme for a class of L WR traffic flow models with non-local flux. Networks and Heterogeneous Media, 13(4):531–547, 2018. https://doi.org/10.3934/nhm.2018024

    2018 doi

  6. [14]

    Ghiasi, O

    A. Ghiasi, O. Hussain, Z. S. Qian, and X. Li. A mixed traffic capacity analysis and lane man- agement model for connected automated vehicles: A markov chain method. Transportation Research Part B: Methodological, 106:266–292, 2017. https://doi.org/10.1016/j.trb. 2017.09.022

    2017 doi

  7. [15]

    X. Gong, B. Piccoli, and G. Visconti. Mean-field of optimal control problems for hybrid model of multilane traffic. IEEE Control Systems Letters, 5(6):1964–1969, 2021. https: //doi.org/10.1109/LCSYS.2020.3046540

    1964

  8. [16]

    Greenshields

    B. Greenshields. A study of traffic capacity. Proceedings of the Highway Research Board, 14:448–477, 1935

    1935

  9. [17]

    Gu´ eriau, R

    M. Gu´ eriau, R. Billot, N.-E. El Faouzi, J. Monteil, F. Armetta, and S. Hassas. How to assess the benefits of connected vehicles? a simulation framework for the design of cooperative traffic management strategies. Transportation Research Part C Emerging Technologies, 67, 04 2...

    2016 doi

  10. [18]

    Herty, G

    M. Herty, G. Puppo, and G. Visconti. Model of vehicle interactions with autonomous cars and its properties. Discrete and Continuous Dynamical Systems - B, 28(2):833–853, 2023. https://doi.org/10.3934/dcdsb.2022100

    2023 doi

  11. [19]

    Hilliges and W

    M. Hilliges and W. Weidlich. A phenomenological model for dynamic traffic flow in networks. Transportation Research Part B: Methodological, 29(6):407–431, 1995. https://doi.org/ 10.1016/0191-2615(95)00018-9

    1995 doi

  12. [20]

    Huang and Q

    K. Huang and Q. Du. Stability of a nonlocal traffic flow model for connected vehicles. SIAM J. Appl. Math., 82(1):221–243, 2022

    2022

  13. [21]

    Hussain and S

    R. Hussain and S. Zeadally. Autonomous cars: Research results, issues, and future challenges. IEEE Communications Surveys & Tutorials, 21:1275–1313, 2019. https://doi.org/10. 1109/COMST.2018.2869360

    2019

  14. [22]

    Keimer and L

    A. Keimer and L. Pflug. Nonlocal conservation laws with time delay. NoDEA Nonlinear Differential Equations Appl., 26(6):Paper No. 54, 34, 2019. https://doi.org/10.1007/ s00030-019-0597-z . 38

    2019

  15. [23]

    S. N. Kruˇ zkov. First order quasilinear equations with several independent variables. Mat. Sb. (N.S.), 81 (123):228–255, 1970

    1970

  16. [24]

    LaFrance

    A. LaFrance. Our grandmother’s driverless car. https://www.theatlantic.com/ technology/archive/2016/06/beep-beep/489029/, 2016

    2016

  17. [25]

    M. W. Levin and S. D. Boyles. A multiclass cell transmission model for shared human and autonomous vehicle roads. Transportation Research Part C: Emerging Technologies, 62:103–116, 2016. https://doi.org/10.1016/j.trc.2015.10.005

    2016 doi

  18. [26]

    M. J. Lighthill and G. B. Whitham. On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London. Ser. A., 229:317–345, 1955. https://doi.org/10. 1098/rspa.1955.0089

    1955

  19. [27]

    J. Lint, S. Hoogendoorn, and M. Schreuder. Fastlane: New multiclass first-order traffic flow model. Transportation Research Record: Journal of the Transportation Research Board, 2088, 12 2008. https://doi.org/10.3141/2088-19

    2008 doi

  20. [28]

    Y. Pan, Y. Wu, L. Xu, C. Xia, and D. L. Olson. The impacts of connected autonomous vehicles on mixed traffic flow: A comprehensive review. Physica A: Statistical Mechanics and its Applications, 635:129454, 2024. https://doi.org/10.1016/j.physa.2023.129454

    2024

  21. [29]

    Piccoli, N

    B. Piccoli, N. Pouradier Duteil, and E. Tr´ elat. Sparse control of Hegselmann-Krause models: black hole and declustering. SIAM J. Control Optim., 57(4):2628–2659, 2019

    2019

  22. [30]

    Rakha, A

    H. Rakha, A. Ingle, K. Hancock, and A. Al-Kaisy. Estimating truck equivalencies for freeway sections. Transportation Research Record, 2027(1):73–84, January 2007. https://doi.org/ 10.3141/2027-10

    2027 doi

  23. [31]

    P. I. Richards. Shock waves on the highway. Operations Res., 4:42–51, 1956. https: //doi.org/10.1287/opre.4.1.42

    1956 doi

  24. [32]

    S. Singh. Critical reasons for crashes investigated in the national motor vehicle crash cau- sation survey. Traffic Safety Facts Crash •Stats DOT HS 812 506, National Highway Traffic Safety Administration, Washington, DC, March 2018

    2018

  25. [33]

    R. E. Stern, S. Cui, M. L. Delle Monache, R. Bhadani, M. Bunting, M. Churchill, N. Hamil- ton, R. Haulcy, H. Pohlmann, F. Wu, B. Piccoli, B. Seibold, J. Sprinkle, and D. B. Work. Dissipation of stop-and-go waves via control of autonomous vehicles: Field ex- periments. Transpor...

    2018 doi

  26. [34]

    Talebpour and H

    A. Talebpour and H. S. Mahmassani. Influence of connected and autonomous vehicles on traf- fic flow stability and throughput. Transportation Research Part C: Emerging Technologies, 71:143–163, 2016. https://doi.org/10.1016/j.trc.2016.07.007

    2016 doi

  27. [35]

    Highway capacity manual: Special report 209

    Transportation research board. Highway capacity manual: Special report 209. National Research Council: Washington D.C., USA, 1985. 39

    1985

  28. [36]

    Webster and L

    N. Webster and L. Elefteriadou. A simulation study of truck passenger car equivalents (PCE) on basic freeway sections. Transportation Research Part B: Methodological, 33(5):323–336, June 1999. https://doi.org/10.1016/S0965-8564(98)00036-6

    1999 doi

  29. [37]

    Wong and S

    G. Wong and S. Wong. A multi-class traffic flow model – an extension of L WR model with heterogeneous drivers. Transportation Research Part A: Policy and Practice, 36(9):827–841,

  30. [38]

    Global status report on road safety 2023

    World Health Organization. Global status report on road safety 2023. Geneva 2023

    2023

  31. [39]

    Ye and T

    L. Ye and T. Yamamoto. Modeling connected and autonomous vehicles in heterogeneous traffic flow. Physica A: Statistical Mechanics and its Applications, 490:269–277, 2018. https: //doi.org/10.1016/j.physa.2017.08.015

    2018 doi

  32. [40]

    Zhou and F

    J. Zhou and F. Zhu. Modeling the fundamental diagram of mixed human-driven and connected automated vehicles. Transportation Research Part C: Emerging Technologies, 115:102614, 2020. https://doi.org/10.1016/j.trc.2020.102614. 40

    2020

  33. [2002]

    https://doi.org/10.1016/S0965-8564(01)00042-8

    doi

  34. [2018]

    https://doi.org/10.1051/m2an/2017066

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.