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The Spatial Variability of Vehicle Densities as Determinant of Urban Network Capacity
Amin Mazloumian, Nikolas Geroliminis, Dirk Helbing
TL;DR
Urban traffic dynamics are difficult to characterize because similar demand and average density can produce widely varying network flows and congestion. The paper develops a macroscopic simulation approach with flow quantization and memoryless routing, finding that spatial density variability is central to explaining these outcomes and obtaining robust network-level relationships.
Problem
Urban traffic measurements show scattered flow–density relationships and variable congestion, while the existence and invariance of a reliable macroscopic fundamental diagram remain unresolved.
Method
The paper combines macroscopic flow quantization with memoryless traffic-flow routing to simulate heterogeneous urban demand without detailed origin-destination tables or individual route assignments.
Results
The standard deviation of vehicle density is a key variable explaining average-flow variation, invariant macroscopic relationships, and outcomes ranging from high flow to gridlock at the same average density and demand.
Takeaways & Limitations
Accounting for spatial density variability can replace scattered congested-flow measurements with clearer functional relationships and provides an urban traffic-performance variable linked to full links.
Takeaways & Limitations
Further work is needed to test whether the reported functional relationships hold for more complex networks, evolving congestion, and real cities.
Abstract
from arXiv · showhide
Due to the complexity of the traffic flow dynamics in urban road networks, most quantitative descriptions of city traffic so far are based on computer simulations. This contribution pursues a macroscopic (fluid-dynamic) simulation approach, which facilitates a simple simulation of congestion spreading in cities. First, we show that a quantization of the macroscopic turning flows into units of single vehicles is necessary to obtain realistic fluctuations in the traffic variables, and how this can be implemented in a fluid-dynamic model. Then, we propose a new method to simulate destination flows without the requirement of individual route assignments. Combining both methods allows us to study a variety of different simulation scenarios. These reveal fundamental relationships between the average flow, the average density, and the variability of the vehicle densities. Considering the inhomogeneity of traffic as an independent variable can eliminate the scattering of congested flow measurements. The variability also turns out to be a key variable of urban traffic performance. Our results can be explained through the number of full links of the road network, and approximated by a simple analytical formula.
1 Introduction
Urban traffic studies seek a reliable macroscopic relationship between average flow and density, but similar demand can produce sharply different congestion outcomes. This paper uses simulations to examine whether spatial density inhomogeneity explains that variability and affects the existence of an invariant urban fundamental diagram.
- The fundamental diagram describes flow increasing with vehicle density until road capacity, then decreasing toward zero under congestion.
- Macroscopic fundamental diagrams aggregate individual detector measurements into a space-mean flow–density relationship, yet their universality and network specificity remain unresolved.
- Empirical and simulated traffic can vary substantially across days despite similar travel activities, traffic volumes, and origin-destination flows.
- Existing simulations address urban traffic, but the frequency and evolution of flow breakdowns and congestion spreading remain poorly understood.
- The paper investigates how spatial car-density inhomogeneity affects the shape, scatter, and existence of macroscopic flow–density relations in urban networks.
- Flow quantization, memoryless routing, and variability analysis enable simulations without detailed route assignments while revealing functional relationships in congested traffic.
- Spatial aggregation alone may not yield a well-defined average flow–density relation, because identical average density and demand can produce outcomes from free flow to gridlock.
2 Model
The model represents urban traffic on a periodic lattice network using section-based fluid dynamics, stochastic signal offsets, and flow restrictions that capture queues and spillovers. It adds single-vehicle flow quantization and memoryless destination routing to reproduce realistic fluctuations and simulate inhomogeneous demand.
- Network model: The simulated city center is a 30 × 30 lattice of one-way links with periodic boundaries and fixed-cycle traffic signals.Each link is 200 metres long; signals use 30 seconds of green and 36 seconds of amber plus red time.
- Underlying dynamics: The section-based traffic model conserves vehicles and represents jam formation and congestion spillovers through arrival, departure, and jam-front dynamics.A triangular fundamental diagram uses V 0 = 503km/h, c0 = −14.28 km/h, and maximum density κ of 140 vehicles/km.
- Underlying dynamics: Traffic signals enter the dynamics through permeability, while departure flows are bounded by downstream capacity and depend on free travel time or delayed vehicles.Permeability is 1 during green, 0 during red, and 0 for both flows during amber intervals.
- Flow quantization: Deterministic and continuously varying turning factors produce an unrealistic sharp transition between free flow and gridlock across simulated density levels.The study compares deterministic factors, low variability in [0.25, 0.75], and high variability in [0, 1], using 500 realizations for each case.
- Flow quantization: Flow quantization assigns each vehicle a binary turn-or-straight decision, producing a realistic macroscopic fundamental diagram under uniform demand.The method discretizes outflows into units equivalent to single vehicles; with density above 30 vehicles/km, average network flow drops over time.
- Route-Choice: A memoryless routing protocol directs flows to destination areas by approximating shortest paths with a minimum number of turns, enabling inhomogeneous-demand scenarios without individual route assignments.Vehicles generally drive straight unless doing so increases their Manhattan distance to the destination area.
3 Macroscopic Fundamental Diagram
Simulations show that urban network flow is not determined by average density alone: at intermediate densities, flow varies substantially over time and across runs. Accounting for spatial density variability and full links substantially reduces this scattering and clarifies network performance.
- Flow–density relationship: 500-run simulations across densities of 20–120 vehicles/km show that intermediate-density network flow is highly variable and generally decreases over time.For low or very high densities, flow variability is negligible; intermediate densities exhibit pronounced temporal variation.
- Flow variability: At 35 vehicles/km, flow variability is negligible, whereas at higher densities it becomes significant and can span from zero to maximum flow.The histograms compare 500 runs at 35, 40, 45, and 50 vehicles/km after 1 and 3 hours.
- Spatial variability: Conditioning flow–density data on spatial variability S reduces the remaining flow scatter to less than 50 vehicles/h.The analysis selects signal cycles within ±0.1 vehicles/link of chosen S values, including 8, 10, and 12 vehicles/link.
- Spatial variability: For each fixed average density K, average flow Q follows a unique monotonically decreasing relationship with variability S, with gridlock possible at higher K.At small densities, flow does not break down as S increases; at higher densities, increasing S can drive the network to gridlock.
- Full links and capacity: The number of full links provides an almost linear explanation of network flow because each full link blocks upstream departures.A link is classified as full above 98% of its maximum occupancy, κL; simulated F_max values agree with the analytical estimates.
4 Simulations with Trip Generation and Termination
Destination-focused demand creates predictable spatial concentration, yet average network flow remains determined by average density together with congestion variability or full links.
- Figure 10 varies the fraction of vehicles routed to the destination area from 5% to 50% while maintaining constant network density.
- Destination-area routing concentrates vehicles spatially, providing a controlled test of predictable congestion distributions.Vehicles are directed toward a destination area so they concentrate around it.
- A fixed removal probability models trip termination inside the destination area while preventing removals from blocked road sections.
- Random-walker path lengths are used to calculate the termination probability and test the destination-flow relationship.
- The average network flow is determined by average vehicle density and either density variability or the number of full links, even under nonuniform demand.
5 Time-Dependent Scenarios
Time-dependent scenarios test whether the density–flow relationships persist when network vehicle volumes vary. The simulations examine growing demand and compare flow against density variability or full links.
- Figure 11 expresses average network flow as a function of average vehicle density and either density variability or the number of full links.
- Open-system simulations represent time-varying traffic volumes by making trip generation independent of trip termination.
- At K = 60 vehicles/km, network flows are always low.
6 Summary, Conclusions, and Outlook
The paper identifies spatial density variability as central to urban traffic performance and develops macroscopic simulation techniques to study it. The authors emphasize spillover mechanisms while marking complex networks and real-city application as open questions.
- 6 Summary, Conclusions, and Outlook: Spatial density variability is required to explain flow variation, support invariant macroscopic relationships, and characterize performance under differing origin-destination flows.
- 6 Summary, Conclusions, and Outlook: Density inhomogeneity increases spillover probability, substantially reducing network flow.
- 6 Summary, Conclusions, and Outlook: Flow quantization and memoryless routing reproduce realistic network-flow variability without detailed origin-destination tables or complicated route assignments.
- 6 Summary, Conclusions, and Outlook: Further work must test whether the functional relationships hold in more complex networks and determine how congestion spreads over time.
- 6 Summary, Conclusions, and Outlook: Extensive real-life experiments are needed to test simulation assumptions and assess whether flow depends on density and full links across demand profiles.