Traffic light optimization with low penetration rate vehicle trajectory data

The study addresses outdated or infrequently optimized fixed-time traffic signals, a major contributor to congestion-related costs at the over 320,000 signalized intersections in the United States. Detector-based systems (e.g., actuated control, ATCS) have limited deployment due to installation and maintenance costs, leaving many intersections without detection and retimed only every 3–5 years, leading to inefficiencies as demand changes. With the growth of connected vehicle trajectory data, the authors aim to leverage low-penetration-rate trajectory data to monitor and optimize signals detector-free. The key research question is how to estimate recurrent traffic states and optimize fixed-time signal parameters using sparse trajectory observations without roadside sensors, thereby achieving scalable and frequent retiming to reduce delay and stops.

The paper situates the work within: (1) increasing availability and advantages of vehicle trajectory data over fixed detectors (broader spatial-temporal coverage and richer performance measures); (2) limitations of existing signal control studies assuming high CAV penetration; (3) challenges in using current traffic flow models with trajectory data—Eulerian models (e.g., LWR and variants) lack direct trajectory observability; Lagrangian models are high-dimensional; hybrid stochastic models become complex. Prior work with trajectories has focused on monitoring (volume/queue estimation) or heuristic signal optimization with limited scalability. The authors point to a gap for a stochastic model aligned with trajectories that is tractable and supports prediction under different signal parameters for optimization. Supplementary Section 2 contains an extended review.

Modeling framework: The authors propose Newellian coordinates based on Newell's deterministic car-following, suitable for urban stop-and-go conditions. After a discrete approximation (time step Δt; unit flow Δu at saturation; jam space headway h), vehicle trajectories are mapped to a point-queue under Newellian coordinates with coordinates (t, n), where t is free-flow arrival time and n indexes unit flows. A deterministic mapping relates the point queue X(t) (number of stopped vehicles) to the spatial queue X′(t) (location of last stopped vehicle) accounting for elapsed green time. Stochastic point-queue model: Arrivals A(t) are Bernoulli with probability α(t); departures B(t)=1 occur when green and queue non-empty. Queue evolves as X(t)=X(t−1)+A(t)−B(t). Recursive equations provide queue-length pmf x(t,k) and departure probabilities b(t) given arrival profile a(t) and signal state S(t). The model focuses on undersaturated, recurrent stationary cycles under cyclic arrivals and signal timing. Probabilistic Time-Space (PTS) diagram: The stochastic point-queue is projected back to spatial-temporal space by assigning probabilities to edges of the Newellian grid: probabilities of stop (horizontal), free-flow (vertical), arrival, and departure edges derived from x(t,k), a(t), and b(t). Edge transparencies visualize the spatial-temporal distribution of trajectories, reconstructing recurrent traffic states from sparse observations. Estimation: Assuming random sampling of vehicles at penetration rate φ, historical trajectories are aggregated by time-of-day into an aggregated time-space (TS) diagram aligned by free-flow arrival within a cycle. Arrival and departure histograms (a_obs(t), b_obs(t)) are scaled by number of cycles N_e, unit flow A_u, and φ to produce a(t), b(t). Parameters are estimated via method of moments: choose φ such that model-estimated average delay d^(φ) matches observed average delay d^obs from trajectories. For networks, a centralized program estimates movement-specific φ with a dispersion regularizer across movements, and upstream arrivals are modeled by shifted/scaled upstream departures using turning ratios and offsets. OSaaS system: A closed-loop pipeline—monitoring (compute performance from trajectories), modeling (calibrate penetration and arrival rates; compute stationary PTS), diagnosis (identify optimality gaps for cycle lengths, green splits, and offsets using gradients or line search), and optimization. For isolated intersections, gradient-based updates to cycle/splits minimize a performance index I(s)=D(s)+α·L(s). For corridors, offset optimization uses relative offsets Δo and coordinate descent with line search to minimize delay and stops, reconstructing offset plans from relative offsets. Data and implementation: GM probe trajectories (GNSS/IMU; ~3 s sampling; 3–5 m accuracy), OpenStreetMap network, and SPaT from road agency records. Field test in Birmingham, Michigan (34 signals). Three weeks of pre-implementation data (03/07/2022–03/25/2022) used for modeling and optimization; new timings implemented 03/31–04/01/2022; three weeks post data (04/04/2022–04/22/2022) used for evaluation. Estimated citywide penetration rate ~7%.

• Accurate reconstruction of recurrent spatial-temporal traffic states from low-penetration trajectories using the PTS diagram; validation via queueing-area IoU across intersections averaging ~69.92% (intersection IoUs reported included 54.72%, 70.31%, 70.50%, 65.79%, 85.07%).
• Parameter estimation: Penetration rates inferred by matching model-estimated average delay to observed delay; example movement estimated φ≈7.3% (illustrative plot) with consistent arrival/departure profiles.
• Diagnosis capability: Gradient-based analysis revealed green split imbalances and excessive cycle lengths at isolated intersections; pair-wise coordination line searches identified offset improvements (example: adding 36 s relative offset between two intersections predicted −16% delay, −27% stops for the pair).
• Field implementation outcomes: • Adams Rd corridor: Overall average control delay −12.23% (13.89 s to 12.19 s), average number of stops −18.51% (0.44 to 0.36), space-mean speed +7.98% (36.51 to 39.43 km/h). Period-specific changes: AM delay −22.05%, MD −11.27%, PM −9.09%; stops AM −28.69%, MD −21.13%, PM −10.55%. • Old Woodward Ave corridor: Overall delay −3.78% (18.29 to 17.60 s), stops −10.77% (0.49 to 0.44), speed +1.19% (17.64 to 17.85 km/h). Notable: AM delay −15.66%, PM stops −14.63%; mid-day showed minimal optimality gap. • Example mid-day on Adams Rd: post-optimization, northbound average delay −18.7% and stops −42.4%; southbound delay −4.5% and stops −4.1%.
• Citywide statement: Delay and stops at signalized intersections decreased by up to 20% and 30%, respectively, demonstrating scalability without roadside detectors.

The proposed OSaaS leverages low-penetration trajectory data to continuously monitor and optimize fixed-time signals without detectors, addressing long retiming cycles and limited scalability of traditional systems. By introducing Newellian coordinates and a stochastic point-queue compatible with trajectory observations, the method reconstructs stationary traffic patterns and supports predictive evaluation under alternative signal parameters. Diagnosis via gradients and line searches quantifies optimality gaps (e.g., green split imbalance, cycle length, offsets) and guides practical adjustments. Field results confirm that offset optimization reduces delay and stops and can increase corridor speeds, validating the modeling and optimization pipeline. The approach can be deployed broadly since trajectory data are available across networks and resilient to single-sensor failures. While primarily designed for periodic retiming, the framework can inform real-time adjustments in high-risk scenarios (e.g., over-saturation, spillback) as connectivity improves. Overall, OSaaS demonstrates a scalable, sustainable pathway to modernize signal timing practices using existing connected vehicle data.

The paper introduces a trajectory-aligned stochastic traffic flow framework (Newellian coordinates, point-queue model, and PTS diagram) and an operational system (OSaaS) for detector-free, large-scale signal retiming. By aggregating multi-day trajectories at low penetration, the method estimates penetration and arrival profiles, reconstructs recurrent traffic states, diagnoses suboptimal parameters, and optimizes cycle, splits, and offsets. A citywide deployment in Birmingham, MI, achieved substantial reductions in delay and stops and improved speeds, particularly on targeted corridors. The system enables frequent, data-driven retiming cycles and is readily scalable to other fixed-time signal networks. Future work includes enhancing real-time responsiveness, extending to more congested/oversaturated conditions, and integrating broader objectives (e.g., multimodal priorities, fairness, energy/emissions).

• Assumes homogeneous deterministic driving behavior (Newell's model), focusing uncertainty on stochastic demand and sparse observation; heterogeneity is ignored.
• Designed for undersaturated, stationary conditions within time-of-day periods; residual queues may persist stochastically, but the framework targets stationary cycles and may be less accurate under persistent over-saturation.
• Low penetration limits real-time state estimation accuracy; method relies on aggregating sufficient historical data for recurrent patterns.
• Parameter estimation assumes random sampling of observed trajectories and accurate map-matching; errors in GNSS or matching could affect estimates.
• Field implementation primarily adjusted offsets; side-street impacts under other adjustments (e.g., split changes) not extensively evaluated in main text.