Derivation of a fundamental diagram for urban traffic flow

Helbing, Dirk · 2009 · OpenAlex-citations

DOI: 10.1140/epjb/e2009-00093-7

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Summary

This paper addresses the lack of theoretical frameworks for understanding fundamental relationships between macroscopic traffic variables—such as density, utilization, velocity, and travel time—in urban traffic flows. While empirical measurements have historically relied on phenomenological fit curves, such as the Bureau of Public Roads’ capacity constraint function, these approaches often suffer from theoretical inconsistencies, particularly regarding capacity limits. The author aims to derive expected fundamental relationships from a model of traffic flows at intersections, providing a systematic understanding of recently measured urban fundamental diagrams. This work contrasts with density-based approaches by pursuing a utilization-based method common in queueing theory. The methodology involves analyzing cyclically signalized intersections to derive relationships for undersaturated and congested traffic conditions. The model defines key variables including cycle time, lost service time (amber and red phases), and green time fractions. It establishes the utilization of service capacity as the ratio of arrival flow to outflow capacity. By modeling vehicle trajectories and queue dynamics, the author derives formulas for average delay time, average queue length, and average travel time as functions of utilization and the number of delayed vehicles. The analysis incorporates an efficiency coefficient to account for the synchronization between traffic light signals and vehicle platoons, allowing the model to adjust for stochastic arrivals and signal offsets. The study finds that average delay time is proportional to the maximum queue length and depends on the utilization and efficiency of the traffic operation. Specifically, the paper derives a fundamental relationship expressing average travel time as a function of utilization, free flow travel time, and lost service time. It demonstrates that under uniform arrivals, the average delay can be calculated based on the fraction of delayed vehicles and the signal cycle parameters. The results show that the average vehicle density relates to utilization and average travel time via Little’s Law. The derived formulas allow for the calculation of average speed and travel time based on the utilization of road sections and the number of delayed vehicles, providing a theoretical basis for the shape of urban fundamental diagrams. The significance of this work lies in providing a theoretically consistent derivation of fundamental diagrams for urban traffic, moving beyond empirical fitting. By linking macroscopic variables through utilization and queueing theory, the model offers a clearer understanding of traffic dynamics at intersections. This approach facilitates the prediction of travel times and speeds based on measurable parameters like utilization and signal timing. The findings suggest that urban traffic flow characteristics can be systematically understood and modeled, aiding in traffic planning and the optimization of signal control strategies. The paper also connects the utilization-based approach with density-based methods, offering a comprehensive framework for analyzing urban traffic congestion and capacity.

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• traffic density