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A comparative study of Macroscopic Fundamental Diagrams of arterial road networks governed by adaptive traffic signal systems
Lele Zhang, Timothy M. Garoni, Jan de Gier
TL;DR
Urban MFDs may depend on spatial heterogeneity, signal control, and changing boundary demand rather than aggregated density alone. Using stochastic cellular-automaton simulations, the paper compares SCATS variants with self-organizing traffic lights across stationary and time-dependent scenarios. It finds that SOTL produces higher MFDs than SCATS, while time-dependent demand generates hysteresis correlated with density heterogeneity.
Problem
The paper examines whether urban-network MFDs depend only on aggregated density or also on spatial heterogeneity, demand conditions, and signal-system choice.
Method
The study simulates arterial networks with a stochastic cellular automaton under different boundary conditions and compares SCATS-F, SCATS-L, and SOTL signal systems.
Results
SOTL always produces a higher MFD than SCATS, increasing network capacity and producing higher congested-regime flows; time-dependent demand produces hysteresis correlated with density heterogeneity.
Takeaways & Limitations
Adaptively uniformizing network density is associated with better network performance and higher capacity within the simulated scenarios.
Takeaways & Limitations
Single-link flow-density relations can exhibit significant scatter even in the simplest case, and some simulated hysteresis loops are comparable to error bars and cannot be firmly established.
Abstract
from arXiv · showhide
Using a stochastic cellular automaton model for urban traffic flow, we study and compare Macroscopic Fundamental Diagrams (MFDs) of arterial road networks governed by different types of adaptive traffic signal systems, under various boundary conditions. In particular, we simulate realistic signal systems that include signal linking and adaptive cycle times, and compare their performance against a highly adaptive system of self-organizing traffic signals which is designed to uniformly distribute the network density. We find that for networks with time-independent boundary conditions, well-defined stationary MFDs are observed, whose shape depends on the particular signal system used, and also on the level of heterogeneity in the system. We find that the spatial heterogeneity of both density and flow provide important indicators of network performance. We also study networks with time-dependent boundary conditions, containing morning and afternoon peaks. In this case, intricate hysteresis loops are observed in the MFDs which are strongly correlated with the density heterogeneity. Our results show that the MFD of the self-organizing traffic signals lies above the MFD for the realistic systems, suggesting that by adaptively homogenizing the network density, overall better performance and higher capacity can be achieved.
1. Introduction
The paper asks when urban road networks exhibit Macroscopic Fundamental Diagrams and how spatial heterogeneity and signal control affect them. It motivates simulations comparing adaptive signal systems under stationary and time-varying conditions.
- MFDs relate network-aggregated demand and performance, extending the single-link relationship between flow and density to urban networks.Empirical work has reported such relationships in congested urban networks, including Yokohama.
- MFD existence has been linked to slowly varying, distributed demand and homogeneous infrastructure, while spatial density variation may also shape the diagram.
- Both clockwise and anticlockwise hysteresis loops can appear when boundary input/output rates are substantially higher than rates in the network bulk.This setting includes commuter corridors and may also arise with perimeter control.
- The simulations use a stochastic cellular automaton that models vehicles statistically while retaining realistic intersection signal phasing.The model was designed to study arbitrary traffic signal systems on arbitrary networks efficiently.
- The study compares SCATS variants with self-organizing traffic lights and examines both time-independent and time-dependent boundary conditions.The paper focuses on how signal adaptivity and changing demand affect MFDs and hysteresis.
2. Cellular Automata Model
The NetNaSch model represents an urban road network as stochastic cellular automata connected at signalized intersections, with open boundaries and specified turning, entry, and exit processes. The study uses an 8 × 8 grid and several stationary or time-varying demand scenarios.
- Cellular Automata Model: NetNaSch represents roads as directed-graph links whose lanes follow stochastic Nagel–Schreckenberg dynamics, with lane changes and network-dependent inputs and outputs.
- Cellular Automata Model: Vehicles randomly select desired outgoing links, but may revise those choices after excessive waiting caused by spillback.The simulations use nG = 6 green signals as the waiting threshold.
- Boundary Conditions: Open boundary conditions let vehicles enter and exit stochastically according to prescribed rates, while internal sources and sinks model processes such as parking garages.
- Network Geometry: The simulated network is a regular 8 × 8 square grid with two lanes and an additional right-turning lane on each link.Bulk and boundary links are generally 750 m long, while turning lanes are 120 m.
3. Observables
The study constructs network-level density, flow, and spatial-heterogeneity observables by aggregating stochastic link measurements over five-minute bins. These measures test whether flow depends only on density or also on spatial variation.
- Link Observables: Link density is the occupied-cell fraction, while link flow counts vehicle crossings during each simulation time step.Network boundary links are excluded from the aggregated network observables.
- Temporal Aggregation: Instantaneous link density and flow are binned over five-minute intervals to study dynamics on traffic-cycle timescales.The physical time variable is measured in intervals of b = 5 minutes.
- Network Observables: Network density and flow are spatial averages, while hρ and hJ measure deviations of link-level density and flow from their network averages.Each heterogeneity measure is zero only when all corresponding link observables are equal.
- Performance Indicators: Both density heterogeneity and flow heterogeneity provide valuable indicators of network performance.
- Estimation: The expected observables and standard errors are estimated from 10 to 30 independent simulations for each signal-system and boundary-condition choice.
4. Traffic Signal Systems
The paper compares two adaptive SCATS variants with self-organizing traffic lights. SCATS adapts cycle, split, and linking parameters, whereas SOTL selects phases according to current demand and idle time.
- Systems Compared: The study compares SCATS-L, SCATS-F, and SOTL as three distinct traffic signal systems.SCATS-L includes linking and adaptive cycle lengths; SCATS-F removes linking; SOTL is self-organizing.
- SCATS: SCATS adaptively controls linking offset, cycle length, and split time using recent traffic conditions.Its cycle length is selected from a volume ratio based on measured traffic volume and phase split time.
- SCATS Linking: SCATS-L coordinates consecutive nodes through subsystems with shared cycle lengths, master nodes, slave nodes, and linking offsets.
- SCATS Variants: SCATS-F lets each node choose its cycle length and split plan independently, whereas SCATS-L imposes subsystems and linking.
- SOTL: SOTL is acyclic and selects the phase with the highest current demand while accounting for how long phases have remained idle.With demand defined by vehicles on phase inlinks, SOTL is intended to minimize network density heterogeneity.
5. Simulations: Time-independent boundary conditions
Under time-independent boundary conditions, traffic signal systems approach stationary MFDs whose capacity and transient behavior depend on signal adaptivity, density heterogeneity, and network design parameters.
- Stationary behavior: By hours 5 or 6, flow and density reach approximately stationary values, with SOTL relaxing faster than SCATS near capacity.The longest relaxation times occur near capacity, while SOTL achieves stationarity faster than SCATS.
- Signal-system comparison: 0.43 maximum flow for SOTL versus approximately 0.39 for SCATS-F and SCATS-L represents a 10% increase in network capacity.SOTL peaks near density 0.34, whereas SCATS peaks near density 0.19; low-density performance is similar.
- Heterogeneity: Flow and density heterogeneity are strongly anticorrelated, with higher density heterogeneity associated with lower flow at a given network density.This relationship persists during transients and helps explain MFD behavior.
- Heterogeneity: SOTL produces lower density heterogeneity than SCATS-L, consistent with its design objective of adaptively homogenizing network density.Its MFD lies above the SCATS-L MFD, supporting adaptive density homogenization as an effective control approach.
- Internal sources and sinks: Increasing γ and δ translates the SCATS-L and SOTL MFDs toward higher densities, with the shift more pronounced for SCATS-L.For SOTL, capacity is marginally higher with non-zero γ and δ; SCATS-L capacity changes remain unclear because of statistical noise.
- Turning probabilities: Increasing turning probability pT leaves low-density branches invariant but makes high-density branches decay more rapidly.For pT = 0.15 and 0.2, SCATS capacities remain close and below the corresponding SOTL capacity.
6. Simulations: Time-dependent boundary conditions
Time-dependent demand produces transient MFD behavior: hysteresis emerges around demand peaks, with loop orientation and magnitude linked to spatial density heterogeneity. Across loading scenarios, SOTL maintains lower heterogeneity and higher capacity than SCATS systems, although some weak loop differences are not firmly established.
- Simulation setup: Time-dependent simulations model a 20-hour weekday with morning and afternoon demand peaks under either boundary loading or uniform loading.The two scenarios differ in whether internal sources and sinks are absent or match boundary input and output rates.
- Transient MFDs: The initial four hours follow the stationary MFD, but hysteresis emerges as the morning demand peak approaches around hour 6.The network begins empty, remains uncongested, and reaches maximum flow before transient density-flow dependence develops.
- Uniform loading: Uniform loading produces clockwise MFD hysteresis accompanied by oppositely oriented anticlockwise hysteresis in density heterogeneity.This correspondence is reported for all three signal systems studied.
- Boundary loading: Boundary loading produces multiple loops, including anticlockwise high-density loops and clockwise low-density loops, especially for SOTL.The high-density behavior reflects differing spatial heterogeneity during loading and recovery, while low-density behavior resembles uniform loading.
- Signal-system comparison: SOTL has lower density heterogeneity and higher capacity than both SCATS systems under both boundary and uniform loading.Its stronger density homogenization is associated with pronounced anticlockwise loops under boundary loading.
- Signal-system comparison: Some apparent anticlockwise loops in SCATS boundary-loading MFDs are comparable to simulation error bars, so their existence cannot be firmly established.The associated comparison remains weaker than the clearly observed SOTL behavior.
7. Discussion
The study finds that MFDs depend on demand heterogeneity, anisotropy, turning rates, and signal-system choice, with SOTL consistently outperforming SCATS. Time-dependent demand produces hysteresis correlated with spatial density heterogeneity.
- The study examines MFDs across anisotropy, demand uniformity, turning probabilities, and three signal systems: SCATS-F, SCATS-L, and SOTL.
- Time-independent MFDs exist under non-uniform demand, but their shapes depend on the nature of that non-uniformity.
- More uniformly distributed demand produces similar capacities, but those capacities occur at higher densities.
- Anisotropic exogenous demand causes a steep flow drop just beyond the MFD maximum.
- As turning rates increase, capacity and jamming occur at lower densities, capacity decreases, and the congested MFD branch declines.
- Time-dependent demand produces clear hysteresis strongly correlated with spatial density heterogeneity, with behavior depending on loading and unloading uniformity.
- SOTL produces a higher MFD than SCATS, increasing network capacity and congested-regime flows through density uniformization.
- Because signal-system choice strongly affects MFD shape, MFDs can serve as a metric for comparing traffic signal-system performance.
Appendix A.1. SCATS
SCATS adapts cycle lengths using a master-node volume ratio and allocates phase splits according to demand, while linked slave nodes follow their masters. The strategy constrains volume ratios through threshold-based cycle adjustments and fixed timing safeguards.
- SCATS adapts cycle length using a master node’s volume ratio R.
- Algorithm 1 increases or decreases cycle length when R crosses 0.95 or 0.85, with special transitions between minimum and stopper cycles.
- The SCATS cycle parameters are MIN = 44 seconds, STOPPER = 64 seconds, MAX = 130 seconds, and STEP = 6 seconds.
- The strategy attempts to keep the volume ratio within [0.85, 0.95] by responding to high demand or wasted green time.
- For non-subsystem nodes, the volume ratio is maximized across inlinks and phases, while slave nodes use the cycle length of their master.
- Phase split demand is d(P) = max_l V(l, P), and phase changes impose a 2-second delay; simulations use a minimum split time of 5 seconds.
- Initial cycle lengths are set to the minimum, while adaptive split times make the initial split condition unimportant.
Appendix A.2. SOTL
SOTL is an acyclic, demand-responsive signal system that selects phases using threshold exceedance, demand strength, and idle time. Its operation allows phase ordering to adapt at each node and time step.
- SOTL imposes no fixed phase ordering and uses node idle time as a clock for signal decisions.
- Among eligible phases, SOTL selects the phase with maximal κ, then resolves ties using the longest idle time and random choice if necessary.
- When idle time exceeds the minimum split time, SOTL considers phases whose threshold functions exceed θ.