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.

宏观基本图（MFD）将车辆在城市网络中的累积和行驶产生与定义明确且可再现的曲线联系起来。由于这种关系，MFD提供了广泛的应用程序，尤其是流量控制。最近，越来越多的经验MFD被记录下来，提供了进一步的见解并促进了它们在实际城市网络中的应用。但是，到目前为止，尚未确定普遍接受的功能形式。

本文提出了一种MFD的功能形式，该形式基于技术可行交通状态（uMFD）上限的平滑逼近。在这种功能形式中，可以通过分析或网络中的其他测量，通过MFD测量或先验定义uMFD，同时使用单个参数λ量化uMFD的平滑度。uMFD原则上可以是任何多区域函数，但我们发现只有四个参数的梯形形状在物理上都有意义，因此很好地模拟了MFD熟悉的形状，并显示了马赛，伦敦，卢塞恩，横滨的经验数据和苏黎世。此外，我们指出了基于λ的解释的新颖应用和分析 如果没有这种新的功能形式，那将是不可能的。