## Abstract

Literature shows that–under specific conditions–the Macroscopic Fundamental Diagram (MFD) describes a crisp relationship between the average flow (production) and the average density in an entire network. The limiting condition is that traffic conditions must be homogeneous over the whole network. Recent works describe hysteresis effects: systematic deviations from the MFD as a result of loading and unloading.

This article proposes a two dimensional generalization of the MFD, the so-called Generalized Macroscopic Fundamental Diagram (GMFD), which relates the average flow to both the average density and the (spatial) inhomogeneity of density. The most important contribution is that we show this is a continuous function, of which the MFD is a projection. Using the GMFD, we can describe the mentioned hysteresis patterns in the MFD. The underlying traffic phenomenon explaining the two dimensional surface described by the GMFD is that congestion concentrates (and subsequently spreads out) around the bottlenecks that oversaturate first. We call this the nucleation effect. Due to this effect, the network flow is not constant for a fixed number of vehicles as predicted by the MFD, but decreases due to local queueing and spill back processes around the congestion “nuclei”. During this build up of congestion, the production hence decreases, which gives the hysteresis effects.

This article proposes a two dimensional generalization of the MFD, the so-called Generalized Macroscopic Fundamental Diagram (GMFD), which relates the average flow to both the average density and the (spatial) inhomogeneity of density. The most important contribution is that we show this is a continuous function, of which the MFD is a projection. Using the GMFD, we can describe the mentioned hysteresis patterns in the MFD. The underlying traffic phenomenon explaining the two dimensional surface described by the GMFD is that congestion concentrates (and subsequently spreads out) around the bottlenecks that oversaturate first. We call this the nucleation effect. Due to this effect, the network flow is not constant for a fixed number of vehicles as predicted by the MFD, but decreases due to local queueing and spill back processes around the congestion “nuclei”. During this build up of congestion, the production hence decreases, which gives the hysteresis effects.

Original language | English |
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Pages (from-to) | 236-250 |

Number of pages | 15 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 438 |

Issue number | November |

DOIs | |

Publication status | Published - 2015 |

## Keywords

- Macroscopic Fundamental Diagram
- Network fundamental diagram
- Traffic inhomogeneity
- Traffic dynamics