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**Traffic flow on networks.**
*(English)*
Zbl 1136.90012

AIMS Series on Applied Mathematics 1. Springfield, MO: American Institute of Mathematical Sciences (AIMS) (ISBN 978-1-60133-000-0/hbk). xvi, 243 p. (2006).

Transportation networks are ubiquitous. Due to the networked structure of these systems, it makes sense to define a mathematical description that is set on a graph. The flow on a single edge of this graph can be modelled by a conservation law. This approach has been introduced by Lighthill, Whitham and Richards in the 1950s. The book contains a description of conservation law based models of network flow. One crucial point is the model for the flow through junctions, where interactions between the flows on the adjacent edges occur. For the traffic flow, the model is based upon the following two ingredients (A) and (B): (A) A traffic distribution matrix containing the percentages of cars flowing from the \(i\)-th incoming road to the \(j\)-th outgoing one. (B) The rule that the number of cars passing the junction is the maximum possible respecting rule (A). Based upon the rules (A) and (B), unique solutions of the network dynamics are well-defined. The book develops a complete theory for car traffic. It also contains some results for telecommunication networks.

The titles of the ten chapters are: Introduction; conservation laws; macroscopic traffic models; networks; Lighthill-Whitham-Richards model on networks; Aw-Rascle models on networks; source destination models; an example of traffic regulation: circles vs lights; telecommunication networks; numerics on networks.

The titles of the ten chapters are: Introduction; conservation laws; macroscopic traffic models; networks; Lighthill-Whitham-Richards model on networks; Aw-Rascle models on networks; source destination models; an example of traffic regulation: circles vs lights; telecommunication networks; numerics on networks.

Reviewer: Martin Gugat (Erlangen)