A pure number to assess “congestion” in pedestrian crowds

The development of technologies for reliable tracking of pedestrian trajectories in public spaces has recently enabled collecting large data sets and real-time information about the usage of urban space and indoor facilities by human crowds. Such an information, nevertheless, may be properly used only with the aid of theoretical and computational tools to assess the state of the crowd. As shown in this work, traditional assessment metrics such as density and flow may provide only a partial information, since it is also important to understand how “regular” these flows are, as spatially uniform flows are arguably less problematic than strongly fluctuating ones. Recently, the Congestion Level (CL), based on the computation of spatial variation of the rotor of the crowd velocity field, has been proposed as an assessment metric to evaluate the state of the crowd. Nevertheless, the CL definition was lacking sound theoretical foundations and, more importantly, was of very difficult interpretation (it was difficult to understand “what” CL was measuring). As we believe that such theoretical shortcomings were limiting also its relevance to applied studies, in this work we clarify some aspects concerning the CL definition, and we show that such an assessment metric may be improved by defining a dimensionless Congestion Number (CN). As a first application of the newly defined CN indicator we first focus on the cross-flow scenario and, by using discrete and continuous toy models, idealised “limit scenarios”, more realistic simulations and finally data from experiments with human participants, we show that CN≪1 corresponds to a crowd with a regular and safe motion (even in high density and high flow settings), while CN≈1 indicates the emergence of a congested and possibly dangerous condition. We finally use the CN indicator to analyse and discuss different settings such as bottlenecks, uni-, bi- and multi-directional flows, and real-world data concerning the movement of pedestrians in the world's busiest railway station.