A Fluid Model of a Traffic Network with Information Feedback and Onramp Controls

J. W. Helton
F. P. Kelly
R. J. Williams
I. Ziedins


Unlimited access to a motorway network can, in overloaded conditions, cause a loss of throughput. Ramp metering, by controlling access to the motorway at onramps, can help avoid this loss of throughput. The queues that form at onramps are dependent on the metering rates chosen at the onramps, and these choices affect how the capacities of different motorway sections are shared amongst competing flows. In this paper we perform an analytical study of a fluid, or differential equation, model of a linear network topology with onramp queues. The model allows for adaptive arrivals, in the sense that the rate at which external traffic enters the queue at an onramp can depend on the current perceived delay in that queue. The model also includes a ramp metering policy which uses global onramp queue length information to determine the rate at which traffic enters the motorway from each onramp. This ramp metering policy minimizes the maximum delay over all onramps and produces equal delay times over many onramps. The paper characterizes both the dynamics and the equilibrium behavior of the system under this policy. While we consider an idealized model that leaves out many practical details, an aim of the paper is to develop analytical methods that yield interesting qualitative insights and might be adapted to more general contexts. The paper can be considered as a step in developing an analytical approach towards studying more complex network topologies and incorporating other model features.

This article is published in the Applied Mathematics and Optimization Journal, 84 (2021), 175-214, DOI: 10.1007/s00245-021-09766-8, under the CC BY Creative Commons licence. For a copy of this article, click here. Note that for use under the CC BY licence, the original source including authors, title, journal, year and DOI: 10.1007/s00245-021-09766-8 should be cited and a link provided to the license. Copyright is held by J. W. Helton, F. P. Kelly, R. J. Williams and I. Ziedins.