  Urban rail transit often operates with high service frequencies to serve
heavy passenger demand during rush hours. Such operations can be delayed by two
types of congestion: train congestion and passenger congestion, both of which
interact with each other. This delay is problematic for many transit systems,
since it can be amplified due to the interaction. However, there are no
tractable models describing them; and it makes difficult to analyze management
strategies of congested transit systems in general and tractable ways. To fill
this gap, this article proposes simple yet physical and dynamic model of urban
rail transit. First, a fundamental diagram of transit system (i.e., theoretical
relation among train-flow, train-density, and passenger-flow) is analytically
derived considering the aforementioned physical interaction. Then, a
macroscopic model of transit system for dynamic transit assignment is developed
based on the fundamental diagram. Finally, accuracy of the macroscopic model is
investigated by comparing to microscopic simulation. The proposed models would
be useful for mathematical analysis on management strategies of urban rail
transit systems, in a similar way that the macroscopic fundamental diagram of
urban traffic did.
