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Traffic flow densities in large transport networks

Published online by Cambridge University Press:  17 November 2017

Christian Hirsch*
Affiliation:
Ludwig-Maximilians-Universität München
Benedikt Jahnel*
Affiliation:
Weierstrass Institute
Paul Keeler*
Affiliation:
Weierstrass Institute
Robert I. A. Patterson*
Affiliation:
Weierstrass Institute
*
* Postal address: Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstraße 39, 80333 München, Germany. Email address: hirsch@math.lmu.de
** Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany.
** Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany.
** Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany.

Abstract

We consider transport networks with nodes scattered at random in a large domain. At certain local rates, the nodes generate traffic flows according to some navigation scheme in a given direction. In the thermodynamic limit of a growing domain, we present an asymptotic formula expressing the local traffic flow density at any given location in the domain in terms of three fundamental characteristics of the underlying network: the spatial intensity of the nodes together with their traffic generation rates, and of the links induced by the navigation. This formula holds for a general class of navigations satisfying a link-density and a sub-ballisticity condition. As a specific example, we verify these conditions for navigations arising from a directed spanning tree on a Poisson point process with inhomogeneous intensity function.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2017 

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