A functional form with a physical meaning for the macroscopic fundamental diagram

Lukas Ambühl, Allister Loder, Michiel C.J. Bliemer, Monica Menendez, Kay W. Axhausen

Research output: Contribution to journalArticlepeer-review


The macroscopic fundamental diagram (MFD) relates vehicle accumulation and production of travel in an urban network with a well-defined and reproducible curve. Thanks to this relationship, the MFD offers a wide range of applications, most notably for traffic control. Recently, more and more empirical MFDs have been documented, providing further insights and facilitating their application in real urban networks. So far, however, no generally accepted functional form has been identified. This paper proposes a functional form for the MFD that is based on the smooth approximation of an upper bound of technologically feasible traffic states (uMFD). In this functional form, the uMFD can be either estimated from MFD measurements or defined a-priori, either analytically or with additional measurements in the network, while the smoothing to the uMFD is quantified with a single parameter λ. The uMFD can in principle be any multi-regime function, but we find that a trapezoidal shape with only four parameters, all physically meaningful, models the familiar shape of the MFD very well as shown with empirical data from Marseille, London, Lucerne, Yokohama, and Zurich. Further, we point to novel applications and analyses based on the interpretation of λ that would otherwise not be possible without this new functional form.

Original languageEnglish (US)
Pages (from-to)119-132
Number of pages14
JournalTransportation Research Part B: Methodological
StatePublished - Jul 2020


  • Functional form
  • Macroscopic fundamental diagram (MFD)
  • Smooth approximation

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Transportation


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