Two-Time-Scale Hybrid Traffic Models for Pedestrian Crowds

This paper introduces new models to describe pedestrian crowd dynamics in a typical unidirectional environment, such as corridors, pathways, and railway platforms. Pedestrian movements are represented in a two-dimensional space that is further divided into narrow virtual lanes. Consequently, pedestrians either move in a lane following each other or change lanes, when it is desirable. Within this framework, the motions of pedestrians are modeled as a two-dimensional and two-time-scale hybrid system. A pedestrian’s movement along the crowd direction is labeled as the <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> direction and modeled by a real-valued process, a solution of a differential equation in continuous time, the lane change is labeled as the <inline-formula> <tex-math notation="LaTeX">$y$ </tex-math></inline-formula> direction. In contrast to the <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> direction dynamics, the movements in the <inline-formula> <tex-math notation="LaTeX">$y$ </tex-math></inline-formula> direction only happen at some time epoch. Although the movements are still on the same time horizon as the <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> direction movements, with a slight abuse of notation and for simplicity and convenience, we use discrete time as the time indicator, and model the movements by a recursive equation taking values in a finite set. Under common assumptions of crowd movements, we prove that the crowd movements in the <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> direction will converge to a uniform distance distribution and the convergence rate is exponential. Furthermore, by using a velocity-distance function to represent the common crowd and traffic congestion scenarios, we show that all pedestrians will asymptotically move with a uniform group speed. In the <inline-formula> <tex-math notation="LaTeX">$y$ </tex-math></inline-formula> direction, when pedestrians naturally wish to change to faster lanes, we show that the numbers in each virtual lanes converge to a balanced distribution and hence achieves asymptotic consensus as shown typically in a crowd behavior. Stability and convergence analysis is carried out rigorously by using properties of circular matrices, stability of networked systems, and stochastic approximations. Simulation studies are used to demonstrate the main properties of our modeling approach and establish its usefulness in representing pedestrian dynamics.