Fundamental Diagram of Rail Transit and Its Application to Dynamic Assignment

Urban rail transit often operates with high service frequencies to serve heavy passenger demand during rush hours. Such operations can be delayed by train congestion, passenger congestion, and the interaction of the two. Delays are problematic for many transit systems, as they become amplified by this interactive feedback. However, there are no tractable models to describe transit systems with dynamical delays, making it difficult to analyze the management strategies of congested transit systems in general, solvable ways. To fill this gap, this article proposes simple yet physical and dynamic models of urban rail transit. First, a fundamental diagram of a transit system (3-dimensional relation among train-flow, train-density, and passenger-flow) is analytically derived by considering the physical interactions in delays and congestion based on microscopic operation principles. Then, a macroscopic model of a transit system with time-varying demand and supply is developed as a continuous approximation based on the fundamental diagram. Finally, the accuracy of the macroscopic model is investigated using a microscopic simulation, and applicable range of the model is confirmed.