Quantized Consensus of Unidirectional Pedestrian Flow based on Two-Time-Scale Hybrid Models

Abstract

The convergence of pedestrian flow in a unidirectional corridor is studied based on the two-time-scale hybrid traffic models for pedestrian crowds. In the two-time-scale framework, the corridor is divided into a set of virtual lanes. The width of each lane is equal to the diameter of a pedestrian which is modeled as a disk. Being situated in a corridor, the pedestrians can either walk in a lane following each other or change to the adequate lanes. The in-lane movements of pedestrians are viewed as a one-line multi-agent dynamic system in continuous time. The lane changing behavior of pedestrians is modeled as a Markov decision process. Pedestrians will change lanes according to the transition probabilities which are functions depending on pedestrian velocities in the related lanes. Furthermore, the quantized consensus state of pedestrian flow is defined. We show that when pedestrians wish to change to faster lanes, the numbers of pedestrians in each lane converge to a balanced distribution (quantized consensus state) as shown typically in pedestrian flows. Numerical simulations and theoretical analysis are used to demonstrate the convergence properties of this method. Simulation results show that the model can capture the main features of unidirectional pedestrian flows.