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Signal Timing Optimization for Mixed Connected Automated Traffic Based on A Markov Delay Approximation

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Pith's one-line read This paper derives closed-form expressions for expected average delay and optimal cycle length at an isolated signalized intersection with mixed connected automated and human-driven traffic, using a Markov-chain platoon model and…

desk verdict The paper's central claim—a closed-form optimal cycle length for mixed CAV/HDV traffic—is invalid: the derivation contradicts its own monotonicity result and produces negative values, so the main contribution collapses. read the letter →

arxiv 2505.11522 v1 pith:CMDRYOVN submitted 2025-05-09 eess.SY cs.SY

Ximin Yue , Yunlong Zhang , Zihao Li , Yang Zhou This is my paper · ORCID
classification eess.SYcs.SY
keywords signaltimingmixedtrafficconnectedautomatedvehiclesdelayapproximationMarkovchainqueuingtheoryoptimalcyclelengthintersectioncapacity
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

The paper sets out to give traffic engineers a simple analytical replacement for simulation when estimating delay at a signalized intersection where some vehicles are connected automated vehicles (CAVs) and the rest are human-driven. It models the alternating sequence of CAV and HDV platoons as a discrete Markov chain, converts the resulting steady-state platoon probabilities into an expected capacity, and then integrates queuing-theory delay formulas for the two departure behaviors. The output is a closed-form expected average delay and a closed-form optimal cycle length. This matters because signal timing is currently tuned with simulation-heavy or HDV-only formulas, and a closed form makes mixed-traffic signal planning fast and transparent.

What carries the argument

The machinery is a discrete-time Markov chain whose state is the number of consecutive CAVs in a platoon: each following vehicle is a CAV with probability $p$ and an HDV with probability $1-p$, returning the state to 0. Its steady-state distribution, Eq. (5), feeds an expected time-gap $\mathbb{E}[\tau]$, Eq. (6), in which successive CAV time gaps follow the string-stability condition of Eq. (7), giving the mixed-capacity expression Eq. (9). On top of that, delay is computed as the area between the cumulative arrival curve and a piecewise departure curve, Eq. (12), whose HDV branch is a reaction-time delay $T_r$ plus a quadratic acceleration phase of length $T_a$; the closed-form integrals (18) and (24), mixed by penetration probability, yield the final delay and cycle-length formulas.

What would settle it

Measure cumulative departures from an isolated approach over many signal cycles and compare CAV-led and HDV-led platoons. The model predicts zero start-up lost time for CAV-led platoons and an optimal cycle near 15 seconds as the CAV penetration approaches 1; if real CAV-led queues show measurable reaction delay, or if the measured optimum is far above 15 seconds, the central claim fails.

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Extended reading notes

Core claim

The central claim is that for an isolated approach with arrival rate $\bar{q}$, capacity $c$ derived from the Markov chain, red time $R$, green time $G$, CAV penetration $p$, HDV reaction time $T_r$, and HDV acceleration duration $T_a$, the expected average delay is given by Eq. (27), formed as the penetration-weighted combination of a CAV-led delay $D_{\mathrm{CAV}}$ and an HDV-led delay $D_{\mathrm{HDV}}$, and the optimal cycle length is Eq. (30), obtained by differentiating the summed approach delay with respect to cycle length. The paper further proves Proposition 1: total delay is monotonically increasing in cycle length, so the shortest safe cycle is the best cycle. Under the model's assumptions, increasing CAV penetration lowers both delay and optimal cycle length, while green ratio changes affect delay more than penetration changes.

Load-bearing premise

The formulas stand or fall on the departure-behavior assumption: CAV-led platoons discharge at full saturation flow the instant green begins, with zero reaction or start-up lost time, while HDV-led platoons lose a fixed reaction time followed by a quadratic acceleration phase; if either behavior differs in the field, the closed-form delay and the optimal cycle length both change.

Editorial extensions

If this is right

  • At zero CAV penetration the model reproduces HDV-only delay behavior; at full penetration the optimal cycle falls to about 15 seconds and average delay to roughly 4.5 seconds per cycle in the numerical settings.
  • Green ratio has a larger effect on expected delay than penetration rate, so signal timing optimization remains a high-leverage policy even before CAVs are widespread.
  • Because total delay is monotonically increasing in cycle length, the minimum permissible cycle set by clearance and lost-time constraints is also the delay-minimizing cycle under the model.
  • Higher CAV penetration is more beneficial at longer cycle lengths; shorter cycles compress the delay gap between penetration levels.
  • Closed-form expressions allow signal retiming to be evaluated instantaneously without simulation, making the model suitable for real-time or planning applications.

Reading between the lines

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

  • If the constant-arrival assumption is relaxed to stochastic arrivals, extra queueing variance terms should appear in the delay expression, and the monotone-in-cycle result in Proposition 1 would likely need an oversaturation caveat.
  • Extending the Markov state space to include a 'CAV leader with lost communication' state would yield a testable prediction: the optimal cycle length as a function of communication reliability, interpolating between the penetration-driven cycle and the HDV-only cycle.
  • The model suggests a feedback loop not analyzed in the paper: if signal controllers adopt shorter cycles as CAV share grows, platoon formation itself changes, which would alter the very headway distribution the Markov chain computes.
  • A direct empirical test of the mechanism would compare cumulative departure curves of CAV-led versus HDV-led platoons; the predicted difference is a start-up lost time measurable from trajectory data.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a stochastic analytical framework for estimating delay and optimizing cycle length at an isolated signalized intersection with mixed connected automated vehicle (CAV) and human-driven vehicle (HDV) traffic. A discrete-time Markov chain models the distribution of CAV platoon lengths, from which an expected capacity is derived using car-following headway relations. Queuing-theoretic delay expressions are developed separately for CAV-led and HDV-led platoons, combined through the CAV penetration rate, and then differentiated with respect to the cycle length to obtain a closed-form optimal cycle length. Numerical experiments illustrate the sensitivity of average delay to arrival rate, green ratio, CAV penetration rate, and cycle length, and report an optimal cycle length that decreases with CAV penetration.

Significance. If correct, the framework would give traffic engineers a simple closed-form tool for signal timing in mixed traffic, which is currently lacking. The paper has the virtue of attempting to connect stochastic platoon composition, car-following headways, and queuing delay in one analytical model, and it provides explicit formulas that could be tested against simulation. However, the central deliverables—the capacity formula and especially the optimal cycle length—are undermined by internal mathematical inconsistencies that make the main claims untenable as stated.

major comments (3)
  1. [Eq. (5)] The steady-state probabilities in Eq. (5) do not sum to 1. Summing π_i = p^i / (p^n/(1-p) + Σ_{m=0}^{n-1} p^m) over i=0..n gives (1-p^{n+1})/(1-p) / (1/(1-p)) = 1 - p^{n+1}, not 1. For the DTMC defined by Eq. (1) with the self-loop at state n, the correct steady state is π_i=(1-p)p^i for i=0,...,n-1 and π_n=p^n. This normalization error propagates directly into the capacity formula Eq. (9) and hence into every delay expression derived from it, so it must be corrected before the model can be evaluated.
  2. [Eq. (30) and Proposition 1] The optimal-cycle-length formula Eq. (30) is internally contradictory with Proposition 1. Proposition 1 proves that D_total(C) is monotonically increasing for all C>0, so no interior stationary point exists and the minimum occurs at the boundary C_min. Yet Eq. (30) is presented as the result of setting the derivative to zero. Furthermore, for under-saturated approaches (0<λ_i<1, 0<p<1), the numerator of Eq. (30) contains (T_a+2T_r)(λ_i−λ_i p+p−1) = (T_a+2T_r)(1-p)(λ_i−1) < 0, while the denominator (λ_i−1)^2 > 0, so Eq. (30) returns a negative C* for every parameter combination. Figure 9 reports positive optimal cycle lengths between about 220 s and 15 s, which cannot be produced by Eq. (30). This is a load-bearing flaw in the paper's central claim.
  3. [Eq. (32) and the definition of λ_i] The derivation equivocates on the meaning of λ_i. In Eq. (29), λ_i is defined as the green split G/C, so that red time is R_i=(1−λ_i)C. In the proof of Proposition 1, the text states that λ_i = q_i/c, the demand-to-capacity ratio. The derivative in Eq. (32) uses the latter definition, but the expression R_i=(1−λ_i)C only holds when λ_i is the green split. These two definitions are not generally equal, so the derivative computation leading to both Eq. (32) and Eq. (30) is not valid. The contradiction between Proposition 1 and Eq. (30) is a direct consequence of this equivocation, and it cannot be resolved without re-deriving the optimality condition.
minor comments (5)
  1. [Throughout] The manuscript contains several typos that should be fixed: 'DMTC' should be 'DTMC', 'Illustative' should be 'Illustrative', and the summation index in Eq. (36) is written as 'n' rather than a phase index.
  2. [Eq. (9)] The presentation of Eq. (9) is difficult to parse because the denominator appears to combine terms in a way that does not match the standard formula c = 1/(E[τ]+L/v_free). After correcting the steady-state probabilities, the capacity expression should be re-derived and written as a single unambiguous fraction.
  3. [Figure 7] Figure 7 contains a dark blue region described as over-saturation, but the model explicitly excludes over-saturated conditions. The paper should clarify why this region is shown and whether the plotted delay values in that region are produced by the model or by an extrapolation.
  4. [Section 4.3 and Figure 9] The text claims that the optimal cycle length decreases with CAV penetration, but it does not state whether Figure 9 was generated using Eq. (30), the boundary minimum C_min from Eq. (34), or a numerical search. Since Eq. (30) yields negative values, the figure must come from some other procedure, and the authors should specify it explicitly.
  5. [References] The car-following headway relation in Eq. (7) is adopted from Chen et al. [27] without derivation. Given that this relation is a key input to the capacity and delay formulas, the paper should at least state its origin prominently and discuss conditions under which it applies to mixed CAV/HDV platoons.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: delay and optimal cycle length follow from the stated queuing and car-following assumptions, with only minor self-referentiality from a self-cited headway model and self-exercise numerical experiments.

full rationale

The paper's derivation chain is not circular in the prohibited sense. Capacity c (Eq. 9) is obtained from the DTMC stationary distribution and a car-following time-gap model; the delay expressions (Eqs. 18, 24, 26, 27) are obtained by integrating between assumed arrival and departure curves under under-saturation; the optimal cycle length (Eq. 30) is obtained by differentiating the resulting total delay following Webster's methodology. No output quantity (delay or C*) is used as an input, and no parameter is fitted to reproduce the target delay or cycle length. The one self-referential element is the CAV time-gap formula (Eq. 7), cited to Chen et al. (2023), which includes a co-author of the present paper; this is load-bearing for capacity but is an externally published car-following model, not a conclusion of this paper, so it constitutes independent support rather than circularity. The numerical experiments are sensitivity analyses that exercise the model's own formulas; the conclusion explicitly states that validation or calibration using simulation or field-test datasets is future work, which is a missing external check but not a circular step. A separate mathematical inconsistency is present in the optimal-cycle-length section (Eq. 30 as printed appears to give a negative numerator for 0<λ<1, and Proposition 1 states the delay derivative is nonnegative for all C), but this is an internal correctness/reproducibility defect, not a circular reduction of the derivation to its inputs.

Assumptions & free parameters 9 free parameters · 5 assumptions · 0 invented entities

The derivation rests on several externally chosen parameters and behavior assumptions. The most significant axiom is the assumed departure behavior contrast between CAV-led and HDV-led platoons, which is the main driver of the delay formulas. The imported car-following headway from Chen et al. (2023) also carries a self-citation component.

free parameters (9)
  • HDV desired time gap tau_HDV = 1.5 s
    Chosen from literature (ref 32) for numerical experiments; directly scales capacity in Eq. (9).
  • Safe time gap tau_safe = 0.3 s
    Chosen from literature; lower bound in CAV headway Eq. (7).
  • Acceleration process time T_a = 3 s
    Chosen from literature; defines HDV acceleration phase in delay model.
  • Reaction time T_r = 2 s
    Chosen from literature for HDV reaction time.
  • Maximum communication capacity n = 5 vehicles
    Chosen from literature; caps CAV platoon state space in Markov chain.
  • Spacing feedback gain omega_e = 1.2 s^-2
    Car-following parameter from Chen et al. (2023), used in Eq. (7).
  • Speed difference feedback gain omega_v = 0.5 s^-1
    Car-following parameter from Chen et al. (2023), used in Eq. (7).
  • Vehicle length L = 5 m
    Average vehicle length from literature; enters capacity Eq. (8).
  • Free-flow speed v_free = 15 m/s
    Chosen from literature; enters capacity Eq. (8).
assumptions (5)
  • domain assumption Vehicle-type sequence is a Bernoulli process with probability p that the next vehicle is a CAV, independent of earlier types.
    This underlies the DTMC transition matrix in Eq. (1); real platoon formation may be correlated.
  • domain assumption The Markov chain reaches stationarity, and the steady-state probabilities are used as long-run platoon fractions.
    Eqs. (3)-(5) assume a unique stationary distribution; not demonstrated for the specific finite chain.
  • domain assumption CAV desired time gap for a platoon of i consecutive CAVs is tau_CAV,i = max(tau_safe, 4*omega_v/(omega_e*(1+i))) from Chen et al. (2023).
    Equation (7) is imported from a cited prior work without derivation in this paper; a load-bearing input to capacity.
  • ad hoc to paper CAV-led platoons discharge at saturation flow rate immediately at green onset with no start-up lost time; HDV-led platoons have a reaction time T_r followed by a quadratic acceleration profile of duration T_a.
    Section 3 and Figures 2-3; this behavioral distinction is assumed, not measured, and directly shapes both delay expressions.
  • domain assumption Traffic is under-saturated (arrival rate q < capacity c) and arrivals are uniform at rate q.
    Stated before Eq. (16); the model explicitly excludes oversaturation and stochastic arrival patterns.

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Pith. "Pith review of Signal Timing Optimization for Mixed Connected Automated Traffic Based on A Markov Delay Approximation." pith.science (2026). https://pith.science/paper/CMDRYOVN

@misc{pith2026250511522,
  author       = {Pith},
  title        = {Pith review of: Signal Timing Optimization for Mixed Connected Automated Traffic Based on A Markov Delay Approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CMDRYOVN}},
  note         = {Machine review of arXiv:2505.11522}
}
read the original abstract

Connected Automated Vehicles (CAVs) offer unparalleled opportunities to revolutionize existing transportation systems. In the near future, CAVs and human-driven vehicles (HDVs) are expected to coexist, forming a mixed traffic system. Although several prototype traffic signal systems leveraging CAVs have been developed, a simple yet realistic approximation of mixed traffic delay and optimal signal timing at intersections remains elusive. This paper presents an analytical approximation for delay and optimal cycle length at an isolated intersection of mixed traffic using a stochastic framework that combines Markov chain analysis, a car following model, and queuing theory. Given the intricate nature of mixed traffic delay, the proposed framework systematically incorporates the impacts of multiple factors, such as the distinct arrival and departure behaviors and headway characteristics of CAVs and HDVs, through mathematical derivations to ensure both realism and analytical tractability. Subsequently, closed-form expressions for intersection delay and optimal cycle length are derived. Numerical experiments are then conducted to validate the model and provide insights into the dynamics of mixed traffic delays at signalized intersections.

Figures

Figures reproduced from arXiv: 2505.11522 by the authors.

Figure 1
Figure 1. Illustative example of DTMC for mixed traffic platoon Pij =    p if j = i + 1, j = i = n 1 − p if j = 0 0 otherwise (1) Then, we can denote π as long-run (steady-state) probabilities to describe the probability of each state when the DMTC reaches equilibrium: π = [π0, π1, π2, . . . , πi , . . . , πn−1, πn] (2) where πi denotes the steady-state probabilities of each state. The long-run or steady-state properties… view at source ↗
Figure 3
Figure 3. Arrival and departure behavior of HDV-led platoon. To approximate the delay of these two departure behaviors, the HDV acceleration phase is modeled as a quadratic function (28), where HDV-led platoon can reach the capacity c (veh/s) after the acceleration process time Ta. The intersection parameters include R (red time), G (green time), Tr (HDV reaction time). According to queuing theory, the total delay at the inte… view at source ↗
Figure 2
Figure 2. Arrival and departure behavior of CAV-led platoon. In contrast, as shown in Figures 3 (a) and (b), HDVs rely solely on visual cues and react to the signal state in real￾time. When approaching a red light, HDVs gradually slow down and come to a complete stop at the intersection. When the signal turns green, human drivers always have time lost due to their reaction time and acceleration time to reach the free-flow spe… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Arrival and departure behavior of an HDV-led platoon. Specifically, the total delay of the CAV-led platoon is represented by the area DCAV in [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Illustration of start-up delay of the HDV-led platoon NUMERICAL EXPERIMENTS AND ANALYSIS In this section, we conducted several numerical experiments based on the previous derivation to better understand mixed traffic delays. Specifically, we performed a sensitivity ana…
Figure 6
Figure 6. Figure 6: Expected average delay results for CAV market penetration rate and cycle length with different arrival rate q impact on TFR, whereas the effect of CAV penetration rates is comparatively less pronounced. (34)’s study also emphasizes the immediate impact of signal timing…
Figure 8
Figure 8. Figure 8: Delay Reduction Rate across different market penetration rate and cycle length reveals that the green ratio has a more significant influence on delay reduction compared to the market penetration rate of CAVs, particularly at moderate to high arrival rates. Furthermore,…
Figure 7
Figure 7. Figure 7: The joint impact on delay time of CAV (p vs c) and (G/c ratio vs p) condition of the intersection, which is not accounted for in our model. Our results demonstrate that the expected average delay decreases as the market penetration rate p and the green ratio G/C increa…
Figure 9
Figure 9. Figure 9: Penetration Rate vs. Optimal Cycle Length and Average Delay CONCLUSION This paper presents a Markov-based analytical framework for approximating mixed traffic delay at an isolated signalized intersection and derives the optimal cycle length based on the proposed delay …

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