How many cars in the city are too many? Towards finding the optimal modal split for a multi-modal urban road network

Allister Loder, Lea Bressan, Maria J. Wierbos, Henrik Becker, Andy Emmonds, Martin Obee, Victor L. Knoop, Monica Menendez, Kay W. Axhausen

Research output: Contribution to journalArticleScientificpeer-review

8 Citations (Scopus)
27 Downloads (Pure)


Interactions among different modes or vehicle classes in urban road networks affect the network performance in different and complex ways. Thus, an answer to the question of “how many cars are too many for a city?” is not trivial. However, multi-modal macroscopic fundamental diagrams (MFD) offer a novel opportunity to answer this question. So far, no methodology exists to estimate multi-modal MFDs resulting from arbitrary multi-modal interactions. In this paper, we propose a methodology to capture additional delays in the shape of the MFD and derive an approach for estimating multi-modal MFDs thereof. The influence on the MFD shape is established using the two-fluid theory of urban traffic by defining pairwise copula functions between travel times of each mode. In contrast to many existing approaches, the presented approach retains individual mode's speed information. We show the applicability of the approach with a tri-modal case of bicycles, buses, and cars with empirical data from Amsterdam (The Netherlands) and London (United Kingdom). Although the approach is not limited to this specific tri-modal case, we use the example to discuss the initial policy question by deriving optimal modal splits for a given accumulation of travelers. Last, we compare the new approach to existing estimation methods for bi-modal MFDs describing car and bus traffic.

Original languageEnglish
Article number665006
Number of pages12
JournalFrontiers in Future Transportation
Publication statusPublished - 2021


  • macroscopic fundamental diagram
  • two-fluid theory
  • multi-modal
  • bicycle
  • bus


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