High-statistics pedestrian dynamics on stairways and their probabilistic fundamental diagrams

Staircases play an essential role in crowd dynamics, allowing pedestrians to flow across large multi-level public facilities such as transportation hubs, shopping malls, and office buildings. Achieving a robust quantitative understanding of pedestrian behavior in these facilities is a key societal necessity. What makes this an outstanding scientific challenge is the extreme randomness intrinsic to pedestrian behavior. Any quantitative understanding necessarily needs to be probabilistic, including average dynamics and fluctuations. To this purpose, large-scale, real-life trajectory datasets are paramount. In this work, we analyze the data from an unprecedentedly high statistics year-long pedestrian tracking campaign, in which we anonymously collected millions of trajectories of pedestrians ascending and descending stairs within Eindhoven Central train station (The Netherlands). This has been possible thanks to a state-of-the-art, faster than real-time, computer vision approach hinged on 3D depth imaging, sensor fusion, and YOLOv7-based depth localization. We consider both free-stream conditions, i.e. pedestrians walking in undisturbed, and trafficked conditions, unidirectional/bidirectional flows. We report on Eulerian fields (density, velocity and acceleration), showing how the walking dynamics changes when transitioning from stairs to landing. We then investigate the (mutual) positions of pedestrian as density changes, considering the crowd as a ''compressible'' physical medium. In particular, we show how pedestrians willingly opt to occupy fewer space than available, accepting a certain degree of compressibility. This is a non-trivial physical feature of pedestrian dynamics and we introduce a novel way to quantify this effect. As density increases, pedestrians strive to keep a minimum distance d approximate to 0.6 m (two treads of the staircase) from the person in front of them. Finally, we establish first-of-kind fully resolved probabilistic fundamental diagrams, where we model the pedestrian walking velocity as a mixture of a slow and fast-paced component (both in non-negligible percentages and with density-dependent characteristic fluctuations). Notably, averages and modes of velocity distribution turn out to be substantially different. Our results, of which we include probabilistic parametrizations based on few variables, are key towards improved facility design and realistic simulation of pedestrians on staircases.