A min-plus derivation of the fundamental car-traffic law

We give deterministic and stochastic models of the traffic on a circular road without overtaking. From this model the mean speed is derived as an eigenvalue of the min-plus matrix describing the dynamics of the system in the deterministic case and as the Lyapunov exponent of a min-plus stochastic ma...

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Bibliographic Details
Published in:IEEE transactions on automatic control 2005-05, Vol.50 (5), p.699-705
Main Authors: Lotito, P.A., Mancinelli, E.M., Quadrat, J.-P.
Format: Article
Language:English
Subjects:
Algorithm design and analysis
Applied sciences
Automatic control
Cellular automata
Computer science
control theory
systems
Control theory. Systems
Discrete event systems
Dynamic programming
Exact sciences and technology
fundamental diagram
Lyapunov exponent
max-plus algebra
Optimal control
Parameter estimation
Sensitivity analysis
Stochastic processes
Terminology
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Summary:We give deterministic and stochastic models of the traffic on a circular road without overtaking. From this model the mean speed is derived as an eigenvalue of the min-plus matrix describing the dynamics of the system in the deterministic case and as the Lyapunov exponent of a min-plus stochastic matrix in the stochastic case. The eigenvalue and the Lyapunov exponent are computed explicitly. From these formulas, we derive the fundamental law that links the flow to the density of vehicles on the road. Numerical experiments using the MAXPLUS toolbox of SCILAB confirm the theoretical results obtained.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2005.848336