## Abstract

One of the first mathematical models of a physical network interacting with human behavior was the model of road traffic equilibria with variable flow (demand) formulated by Martin Beckmann and colleagues in 1954. Beckmann applied the recently-proved theorem of Kuhn and Tucker to incorporate an assumption and two hypotheses concerning road traffic into a single mathematical formulation. The model considers a road network consisting of nodes and links. Associated with each directional link is an increasing function relating its travel time, or generalized travel cost, to its flow. The behavioral hypotheses represented by the model are as follows: 1. All used routes from node p to node q have equal travel times, and no unused route has a lower travel time; 2. The total flow over all routes from node p to node q is determined by a decreasing function of this minimum and equal, or equilibrium, travel time. In large-scale implementations of the model, nodes p and q represent small areas called zones, at which flows originate and terminate; other nodes represent intersections on the road network. The formulation minimizes an artificial function, subject to definitional constraints. The optimality conditions of this model correspond to the above two hypotheses. Subsequently, more general formulations were investigated based on variational inequality, nonlinear complementarity and fixed point theory. Beckmann’s formulation and its descendents considered traffic flows over a relatively long period of time, during which network conditions may be regarded as constant. The peak commuting period in the morning or evening is a typical example. Such models are static, and the flows departing from and arriving at nodes are constant over the time period. Models that consider shorter periods of time, and for which the departure and arrival rates are variables, are dynamic. These models seek to represent the effect of changing network conditions during a longer time period, including accidents and other incidents disrupting flow. Although Beckmann did not propose an algorithm for solving his formulation, in the 1970s researchers began to solve large-scale traffic equilibria. Until recently, these solutions were rather approximate, and did not reveal the structure of the solution, especially with regard to the number and pattern of equilibrium routes. In 2003, Bar-Gera and Boyce proposed an algorithm that reveals this structure for the first time. Subsequently, they began to explore the properties of this solution for large-scale implementations, such as for the Chicago region. The initial results of these explorations for the Chicago region were unexpected and regarded as “astonishing” by one informed observer. One result examined is the relation between the number of routes between a pair of zones and the frequency with which this number occurs in the network. The authors observed that the number of routes increases greatly as the level of congestion increases. This chapter seeks to introduce traffic network equilibrium models to scholars from a broad range of backgrounds, mainly focusing on static models of urban road traffic. Findings on the solution properties of static models for a large network for three congestion levels are presented. A discussion of the applicability of the findings to other types of networks, such as electrical power and supply chain networks, concludes the paper.

Original language | English |
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Title of host publication | International Series in Operations Research and Management Science |

Publisher | Springer New York LLC |

Pages | 305-335 |

Number of pages | 31 |

State | Published - 1 Jan 2007 |

### Publication series

Name | International Series in Operations Research and Management Science |
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Volume | 102 |

ISSN (Print) | 0884-8289 |

## Keywords

- Congestion
- Traffic equilibria

## ASJC Scopus subject areas

- Software
- Computer Science Applications
- Strategy and Management
- Management Science and Operations Research
- Applied Mathematics