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Decision Timescales in Max-Pressure Traffic Signal Control: Theory, Methodology, and Validation

Chaopeng Tan

arXiv:2610.32280v1eess.SYcs.SY

TL;DR

MP control commonly prescribes when to reconsider a phase decision, although that interval should reflect the traffic load and switching losses. This paper derives a load-dependent timescale and proposes local, asynchronous VT-MP; in a store-and-forward network, it reduces total queueing time by 31.0% versus the best fixed interval.

  • Problem

    MP’s decision interval is commonly prescribed rather than derived from the operating load, despite its trade-off between switching efficiency and responsiveness to changing traffic.

    Source: §1 · pp. 2–3

  • Method

    The paper derives switching-adjusted timescale bounds and proposes VT-MP, which selects local, time-varying intervals while preserving MP phase selection and establishing stability under uniformly admissible local loads and bounded intervals.

    Source: §1 · p. 3, §6 · p. 26

  • Results

    31.0% less total queueing time than the best fixed interval was achieved by heterogeneous VT-MP in the store-and-forward network.

    Source: §6 · p. 26

  • Takeaways & Limitations

    The analysis identifies the scaling T ∝ I/(1 − ρ) and a performance-oriented range Tc < T ≤ 2Tc.

    Source: §6 · p. 26

    Near unit load, small load-estimation errors can greatly change the recommended interval, and deeper oversaturation requires an explicit finite-storage and recovery model in the theory.

    Source: §5.4 · p. 25

Abstract

from arXiv · show

Max-pressure (MP) control adapts signal phases to prevailing traffic states, but its decision timescale is typically fixed a priori. This paper treats the decision timescale as an explicit dimension of MP control. We first characterize the trade-off between responsiveness and switching efficiency by quantifying how an equivalent switching loss reduces service capacity over a finite decision interval. This yields a load-dependent critical timescale determined by switching loss and normalized traffic load, together with a performance-oriented operating range above the stability boundary. Building on this result, we propose Variable-Timescale Max-Pressure (VT-MP), in which each intersection selects its decision interval according to local operating conditions, allowing the control timescale to vary across time and space. We establish strong stability of the resulting asynchronous heterogeneous controller under uniformly admissible local loads and bounded decision intervals, and introduce a downstream receiving-supply correction for finite-storage implementation. Three experiments provide mechanism validation and network-level evaluation. A single-intersection experiment confirms that the preferred decision interval increases systematically with traffic load and follows the derived characteristic timescales. In a store-and-forward network, heterogeneous VT-MP reduces total queueing time by 31.0% relative to the best fixed-interval benchmark. In a microscopic 5x5 SUMO network, it reduces mean trip delay by 24.3%, from 213.3 s to 161.5 s, and peak network accumulation by 12.3%. These findings show that the MP decision timescale should be treated as a traffic-state-dependent control variable rather than a fixed implementation parameter.

1 Introduction

MP phase selection is established, but the interval for reconsidering a decision is usually prescribed rather than derived from traffic load. The paper relates that interval to switching loss and load, proposes local variable-timescale control, and evaluates its performance.

1 Introduction

  • The decision interval governs a trade-off: shorter frames refresh pressure more often but incur proportionally greater switching loss, while longer frames retain decisions as traffic evolves.A network-wide interval cannot simultaneously match intersections operating at different saturation levels.Source: §1 · pp. 2–3
  • The analysis gives a load-dependent critical timescale, Tc = I/(1 − ρ), and a performance-oriented range Tc < T ≤ 2Tc by accounting for switching loss within finite decision intervals.The paper connects decision resolution, traffic load, and effective switching loss.Source: §1 · p. 3
  • VT-MP makes each intersection’s decision interval local and time-varying, enabling asynchronous updates while retaining strong stability under uniformly admissible local loads and bounded intervals.The stability result also extends to sufficiently persistent piecewise-stationary demand.Source: §1 · p. 3
  • 31.0% less total queueing time than the best fixed interval was achieved by heterogeneous VT-MP in the store-and-forward network.In the microscopic SUMO network, it reduced mean trip delay by 24.3% and peak network accumulation by 12.3% relative to the seed-specific best fixed benchmark.Source: §1 · p. 3

2 Max-Pressure Control and the Decision-Timescale Problem

This section establishes the conventional movement-level Max-Pressure model and stability concepts as the baseline for studying finite decision frames. It then frames the timescale choice as balancing switching efficiency against responsiveness while preserving positive service margin.

2.1.1 Traffic dynamics

  • Movement queues track vehicles by their current link and intended next link, with turning proportions routing upstream departures into downstream movements.The store-and-forward baseline assumes sufficient link storage, so downstream receiving capacity does not directly constrain the model.Source: §2.1.1 · p. 4

2.1.2 Max-Pressure control and classical stability

  • Conventional MP assigns service to the feasible phase with the largest pressure-weighted benefit, and its classical stability result applies to strictly admissible demand under bounded-arrival, bounded-service, and open-routing assumptions.This established stability result serves as the baseline for the paper’s timescale analysis.Source: §2.1.2 · pp. 4–5

2.2 Decision frames and switching loss

  • A decision interval T is the number of basic slots between MP evaluations; a phase decision is held for that frame, except for any required transition interval.Equivalent switching loss I represents yellow, all-red, and startup effects in service-time units; reserving one loss leaves T − I guaranteed productive slots.Source: §2.2 · pp. 5–6

2.3 The decision-timescale problem

  • Short intervals refresh pressure frequently but devote a larger share of service time to switching, whereas long intervals reduce relative switching loss but hold decisions based on older traffic states.The analysis relates T to switching loss I and the nominal service reserve 1 − ρ, which shrinks as normalized load approaches one.Source: §2.2 · p. 6, §2.3 · p. 6

3 Decision-Timescale Theory for Max-Pressure Control

The theory makes MP decision intervals explicit: switching loss reduces guaranteed service, while longer frames make decisions less responsive to evolving queues. It derives a load-dependent stability boundary and a performance-oriented interval above it.

3.1 Switching-adjusted service and the critical timescale

  • The theory first converts transition time into a conservative inner capacity region, then compares normalized demand with that reduced region to obtain a minimum frame length.The guarantee reserves one equivalent switching loss in every frame, even when consecutive decisions might retain the same phase.Source: §3.1 · p. 6, §3.1.1 · pp. 6–7

3.1.1 Switching-adjusted guaranteed inner region

  • Reserving one equivalent switching loss per frame scales the guaranteed capacity region by 1 − I/T; every arrival vector in its interior is supportable under that convention.The exact two-phase special case changes the boundary from x + y ≤ 1 to x + y ≤ 1 − I/T.Source: §3.1.1 · pp. 6–7, Figure 1 · p. 7

3.1.2 Critical decision timescale

  • For normalized load ρ and switching loss I, T_c = I/(1 − ρ) is a sufficient lower boundary for positive conservative service slack.The boundary increases with load and diverges as ρ approaches 1.Source: §3.1.2 · pp. 7–8

3.2 Performance-oriented decision timescale

  • Under bounded arrivals and service, fixed-timescale MP is strongly stable when T > T_c(ρ), yielding a finite long-run all-slot average expected total movement queue.The result assumes the mean arrival vector has normalized load ρ < 1 and the selected phase is retained during the productive part of each frame.Source: §3.2 · pp. 8–9
  • Holding a phase selected from the frame-initial queue creates an O(T²) pressure-mismatch term as the traffic state evolves during the frame.This is the responsiveness cost that competes with the service slack gained by lengthening the frame.Source: §3.2 · pp. 8–9
  • The queue-bound envelope scales with T² divided by switching-adjusted frame slack, so it grows near T_c and for sufficiently long frames.Moving above T_c initially restores service slack faster than the frame-length penalty grows.Source: §3.2 · pp. 9–10
  • The performance-oriented range is T_c(ρ) < T ≤ 2T_c(ρ), with 2T_c minimizing the derived queue-bound envelope.The lower boundary ensures positive switching-adjusted service slack; the upper characteristic timescale marks where the quadratic frame penalty dominates.Source: §3.2 · p. 10

3.3 Normalized decision-timescale policy

  • The normalized policy sets T(ρ) = κT_c(ρ), with 1 < κ ≤ 2, placing the interval between the stability boundary and the queue-bound minimizer.This scales intervals with switching loss and the available service reserve rather than prescribing one physical interval for every traffic condition.Source: §3.3 · p. 10

4 Variable-Timescale Max-Pressure Control

VT-MP keeps conventional MP phase selection but assigns each intersection a load-dependent decision interval, producing asynchronous heterogeneous updates. The section establishes stability under bounded intervals and uniformly admissible local loads, explains the potential performance benefit of heterogeneous timing, and adds a receiving-supply correction for finite storage.

4.1 Local formulation of VT-MP

  • VT-MP maps each intersection’s local load to its own decision interval while leaving the conventional MP pressure definition and phase-ranking rule unchanged.Local load reflects offered movement flow relative to the local service region and can vary across intersections and time.Source: §4.1 · pp. 11–12
  • With uniformly bounded switching loss and local load below one, the rule preserves uniformly positive absolute and fractional service margins at every node.The fractional reserve is ((κ−1)/κ)(1−ρ_n,k).Source: §4.1 · p. 12

4.2.1 Stability under asynchronous heterogeneous updates

  • Under uniformly admissible local loads and bounded decision intervals, asynchronous heterogeneous VT-MP is strongly stable.The proof relies on positive service reserve and bounded stale-pressure error over a finite common analysis horizon; that horizon is analytical, not a shared control interval.Source: §4.2.1 · pp. 13–14
  • Under piecewise-stationary demand, strong stability holds when local loads remain uniformly below one and operating periods exceed a finite duration threshold.After local intervals adjust to a demand change, the same drift bounds apply across periods.Source: §4.2.1 · p. 14

4.2.2 Performance implication of heterogeneous timescales

  • Heterogeneous intervals can improve service utilization when a common interval holds a depleted phase, and they cannot worsen the minimum analytical queue-bound surrogate.A lightly loaded intersection may otherwise retain an unproductive phase, while each node can use its own load-dependent timescale.Source: §4.2.2 · pp. 15–16

4.3 Finite-storage implementation

  • For finite-storage networks, a downstream receiving-supply check shortens the planned interval when available storage cannot accommodate its predicted discharge.The executed interval matches the planned interval when sufficient receiving space is available; the analytical stability results use the infinite-storage model.Source: §4.3 · p. 16

5 Validation by Numerical and Simulation Experiments

The three experiments progress from validating the load–timescale mechanism at one intersection to evaluating network performance. They use a store-and-forward network aligned with the analysis and a microscopic SUMO network with finite storage and spillback.

5 Validation by Numerical and Simulation Experiments

  • The experiments progress from single-intersection mechanism validation to network-level evaluation under increasingly realistic traffic dynamics.The MP phase-selection rule remains unchanged across all three experiments.Source: §5 · p. 16
  • Experiment 1 isolates the load–timescale relationship and supplies the dimensionless timing factor used in the network experiments.It tests the theoretical prediction against a microscopic decision-interval sweep.Source: §5 · p. 16
  • Experiments 2 and 3 evaluate network effectiveness in a 5 × 5 store-and-forward network and microscopic SUMO traffic with finite storage and spillback.The microscopic experiment includes detector-based state estimation.Source: §5 · p. 16

5.1 Experiment 1: Single-intersection mechanism

The single-intersection experiment tests whether the preferred MP decision interval varies with traffic load as predicted by the theory. Its measured interval trend follows the characteristic scales over subcritical loads, while near capacity finite storage limits feasible intervals.

5.1 Experiment 1: Single-intersection mechanism

  • Preferred feasible intervals rise from roughly 5–10 s under light traffic to about 20 s at ρ=0.8 and approximately 40 s at ρ=0.90.This upward trend supports the predicted increase in decision timescale as service reserve shrinks.Source: Figure 3 · p. 17, §5.1 · p. 17
  • The experiment varies deterministic uniform arrivals across ρ=0.30–1.20 and tests fixed intervals primarily from 5–60 s.Pretrials estimate an average saturation flow of approximately 2339 veh/h and effective switching loss I_eff=2.534 s.Source: §5.1 · p. 17
  • For subcritical cases through ρ=0.90, the fitted normalized timing factor is κ≈1.6, and preferred intervals lie broadly between Tc and 2Tc.The calibrated factor is retained without further tuning in subsequent network experiments.Source: §5.1 · p. 17, Figure 3 · p. 17
  • At ρ=0.95, only three tested intervals remain feasible; at ρ=0.98, none prevents spillback.The passage attributes this near-capacity deviation from the point-queue prediction to finite storage and queue propagation.Source: §5.1 · p. 17, Figure 3 · p. 17

5.2 Experiment 2: Store-and-forward network

In the store-and-forward network, heterogeneous VT-MP outperforms the best fixed interval, while temporal and spatial results show that preferred intervals vary with load and location. The larger gains arise when intersections can use different local timescales.

5.2 Experiment 2: Store-and-forward network

  • The 5 × 5 store-and-forward network uses 28 fixed paths, changing demand and OD composition, and a 3-s switching loss; five timing strategies include asynchronous, node-specific VT-MP.Intervals are quantized to 5-s steps, with κ=1.6 calibrated in Experiment 1; the fixed-interval sensitivity spans 5–60 s and uses ten arrival realizations.Source: §5.2 · p. 18

5.2.1 Overall network performance

  • 31.0% lower total queueing time (2443.6 to 1687.1 veh·h) and 31.0% lower mean network accumulation; peak accumulation falls from 465.8 to 310.6 vehicles versus best fixed.All five controllers serve the complete demand and clear the network, so the comparison reflects congestion and queueing efficiency rather than differences in served demand.Source: Figure 5 · p. 19, §5.2.1 · p. 19

5.2.2 Temporal adaptation

  • The best full-day fixed interval is 15 s, while the offline hourly-best benchmark varies from 5 to 20 s; local recommendations lengthen at peak loads and contract as demand subsides.Q85 tracks increasing network loading without being dictated by the most extreme local condition, whereas the maximum rule applies long intervals even to lightly loaded locations; intersections with different local reserves still share one timescale.Source: §5.2.2 · p. 19

5.2.3 Spatial heterogeneity

  • At 08:00, local intervals span 10–45 s, and at 17:30 they span 10–60 s; longer intervals concentrate at a few peak-period hotspots.The maximum local load is 0.886 at 08:00 and 0.913 at 17:30, while the network is more homogeneous at 12:00, when almost every intersection operates at 10 s.Source: §5.2.3 · pp. 19–20

5.3 Experiment 3: Microscopic SUMO network

In the microscopic finite-storage network, heterogeneous VT-MP improves system-level performance over the best fixed interval. Its recommended timescales rise with load and concentrate along heavily loaded central routes rather than changing uniformly across the network.

5.3 Experiment 3: Microscopic SUMO network

  • The 5 × 5 SUMO grid tests finite links and downstream congestion with four strengthened routes through C2; detector loads use a 300-s rolling window and intervals are capped at 40 s.The experiment uses three demand/route realizations; planned intervals are rounded upward to 5-s steps and shortened when downstream receiving supply constrains storage.Source: §5.3 · pp. 20–21

5.3.1 System-level traffic performance

  • 24.3% lower mean trip delay (213.3 to 161.5 s), with mean and peak network vehicles reduced by 14.3% and 12.3%, respectively, versus the seed-specific best fixed interval.Heterogeneous VT-MP achieves the best performance across all reported measures; the results average three demand/route realizations.Source: Table 1 · p. 22, §5.3.1 · p. 22

5.3.2 Temporal evolution of decision timescales

  • The best full-horizon fixed interval is 25–35 s, while the hourly benchmark rises from about 15 s early to 30–40 s during the heaviest demand; online mean recommendations rise from about 10 to 19 s near the peak.The upper tail reaches the 40-s operational cap, then recommendations contract as demand subsides; representative nodes follow different trajectories.Source: §5.3.2 · p. 23, Figure 10 · p. 24

5.3.3 Spatial distribution of decision timescales

  • Long intervals concentrate at C2 and along strengthened OD corridors, while peripheral intersections remain shorter; peak-period mean recommendations correlate 0.97 with detector-based load.The spatial timing pattern follows local traffic load rather than a network-wide change in operating scale, and expands only where local service reserve becomes small.Source: §5.3.3 · pp. 23–24

5.4 Discussion and limitations

The experiments validate the load-dependent timing mechanism from a single intersection through store-and-forward and microscopic finite-storage networks. The study’s validation is scoped to structured grids and assumes traffic states and model parameters can be observed or estimated with sufficient accuracy.

Discussion and validation scope

  • The validation sequence tests characteristic load–timescale scaling at one intersection, network effectiveness under store-and-forward assumptions, and then finite-storage traffic with online load estimates and spillback.In the microscopic setting, the timing rule plans the interval while a receiving-supply correction limits execution under current downstream storage conditions.Source: §5.4 · pp. 24–25

Limitations

  • Near capacity, the critical interval diverges as load approaches one, so small load-estimation errors can cause large recommendation changes; extending deeper into oversaturation requires finite-storage and recovery theory.Experiment 1 found spillback can eliminate the microscopic feasible interval before the ideal point-queue boundary; Experiment 3 therefore caps load at 0.90.Source: §5.4 · p. 25
  • Both network experiments use structured 5 × 5 grids and assume required traffic states and parameters can be observed or estimated accurately enough to isolate the timing mechanism.The study identifies irregular urban networks and measurement or parameter uncertainty as extensions, rather than testing them exhaustively here.Source: §5.4 · pp. 25–26

6 Conclusions

The paper makes MP decision timescale load-dependent, deriving a critical range and proposing asynchronous, heterogeneous VT-MP with strong-stability guarantees. Experiments report gains over fixed-interval benchmarks, while the framework’s practical boundary is near capacity.

6 Conclusions

  • Switching loss yields a critical timescale T_c(ρ)=I/(1−ρ), with a performance-oriented range T_c<T≤2T_c and characteristic scaling T∝I/(1−ρ).The queue-bound analysis has a characteristic minimizer at 2T_c.Source: §6 · p. 26
  • VT-MP assigns each intersection a local interval T_n=κI_n/(1−ρ_n), retaining strong stability under uniformly admissible local loads and bounded intervals.For microscopic networks, a receiving-supply correction shortens the planned interval when downstream storage is restrictive.Source: §6 · p. 26
  • 31.0% lower total queueing time in the store-and-forward network and 24.3% lower mean trip delay plus 12.3% lower peak accumulation in microscopic SUMO are reported against fixed-interval benchmarks.The SUMO mean trip delay fell from 213.3 s to 161.5 s relative to the seed-specific best fixed benchmark.Source: §6 · p. 26
  • The single-intersection study found that preferred decision intervals increase with traffic load and follow the derived characteristic scales.The study selected κ≈1.6 for the subsequent network tests.Source: §6 · p. 26
  • Near capacity, timescale recommendations become more sensitive to load estimates, and finite-storage spillback can become critical before the ideal point-queue capacity boundary.The microscopic timing recommendation is limited to the load range validated in the single-intersection experiment.Source: §6 · p. 26

A Detailed derivation of the T-frame drift bound

The derivation bounds frame-level Lyapunov drift by combining a switching-adjusted service reserve with bounded errors from queue evolution, routing, and unused service.

A Detailed derivation of the T-frame drift bound

  • Unused potential service is bounded using the maximum service rate, since positive unused service implies the queue is below its service amount.The proof identifies this as the bounded unused-service correction for Lemma 1.Source: A Detailed derivation of the T-frame… · pp. 26–27
  • Summing one-slot drift over a frame introduces bounded terms because arrivals and services are bounded, while movement pressure is a fixed linear transform of queues under fixed turning proportions.Replacing evolving queues and pressures with frame-initial values produces a bounded discrepancy.Source: A Detailed derivation of the T-frame… · p. 27
  • The switching-adjusted fraction ϕ_T=1−I/T exceeds load ρ, providing a pressure-domain reserve proportional to ϕ_T−ρ.The reserve is converted to a queue reserve using the nonnegative inverse of the open-routing operator.Source: A Detailed derivation of the T-frame… · p. 27
  • The pressure-maximizing phase’s frame-initial productive service is no worse than a feasible comparison allocation, supporting the frame drift bound.The proof combines the stated intermediate equations with summed one-slot drift and absorbs bounded terms into BT^2 for T≥1.Source: A Detailed derivation of the T-frame… · p. 27

B Detailed proof of asynchronous VT-MP stability

The asynchronous VT-MP stability proof establishes a queue-dependent service reserve despite locally retained phases and bounded timing differences. It then combines complete-frame drift with bounded boundary errors to obtain finite time-average queues.

B Detailed proof of asynchronous VT-MP stability

  • Under fixed mean operating conditions, the effective movement flow and pressure-weighted offered flow are related through the stated flow and drift equations.This provides the starting drift conversion used in the stability proof.Source: B Detailed proof of asynchronous VT-MP… · p. 27
  • Uniform positive switching-adjusted reserve at each node yields a queue-dependent reserve proportional to total queue magnitude.The proof assumes uniformly positive slack and uses downward-closed service sets to construct a comparison service vector.Source: B Detailed proof of asynchronous VT-MP… · p. 28
  • Asynchronous phase retention creates an additive pressure error independent of queue magnitude when local intervals are bounded.Bounded queue increments and linear phase pressure underpin the stale-pressure bound.Source: B Detailed proof of asynchronous VT-MP… · p. 28
  • Complete local frames contribute negative drift over at least H−2T_max slots, while boundary fragments and bounded errors are absorbed into a positive drift condition.Telescoping the H-step drift establishes finite all-slot time-average queues and strong stability.Source: B Detailed proof of asynchronous VT-MP… · p. 28

B.1 Piecewise-stationary extension

The piecewise-stationary extension preserves strong stability by bounding adjustment after demand changes and aggregating complete windows to dominate boundary drift.

B.1 Piecewise-stationary extension

  • After a mean arrival-rate change, nodes adopt the new condition within at most T_max slots, and the resulting boundary drift is uniformly bounded.The argument uses uniform bounds on timing, slack, and stale-pressure error across operating periods.Source: §B.1 · p. 29
  • Choosing enough complete H-windows to dominate boundary drift yields a common positive block-drift coefficient and strong stability across operating periods.The block includes the adjustment interval and any remaining interval shorter than H.Source: §B.1 · p. 29
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