  Staircases play an essential role in crowd dynamics, allowing pedestrians to
flow across large multi-level public facilities such as transportation hubs,
and office buildings. Achieving a robust 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. In this work, we analyze data from
an unprecedentedly high statistics year-long pedestrian tracking campaign, in
which we anonymously collected millions of trajectories across a staircase
within Eindhoven train station (NL). Made possible thanks to a
state-of-the-art, faster than real-time, computer vision approach hinged on 3D
depth imaging, and YOLOv7-based depth localization. We consider both
free-stream conditions, i.e. pedestrians walking in undisturbed, and trafficked
conditions, uni/bidirectional flows. We report the position vs density,
considering the crowd as a 'compressible' physical medium. 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 = 0.6 m 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. Notably, averages and
modes of velocity distribution turn out to be substantially different. Our
results, including probabilistic parametrizations based on few variables, are
key towards improved facility design and realistic simulation of pedestrians on
staircases.
