Estimating left behind patterns in congested metro systems: a Bayesian model

Purpose During rush hours, many passengers find it difficult to board the first train due to the insufficient capacity of metro vehicles, namely, left behind phenomenon. In this paper, a data-driven approach is presented to estimate left-behind patterns using automatic fare collection (AFC) data and train timetable data. Design/methodology/approach First, a data preprocessing method is introduced to obtain the waiting time of passengers at the target station. Second, a hierarchical Bayesian (HB) model is proposed to describe the left behind phenomenon, in which the waiting time is expressed as a Gaussian mixture model. Then a sampling algorithm based on Markov Chain Monte Carlo (MCMC) is developed to estimate the parameters in the model. Third, a case of Beijing metro system is taken as an application of the proposed method. Findings The comparison result shows that the proposed method performs better in estimating left behind patterns than the existing Maximum Likelihood Estimation. Finally, three main reasons for left behind phenomenon are summarized to make relevant strategies for metro managers. Originality/value First, an HB model is constructed to describe the left behind phenomenon in a target station and in the target direction on the basis of AFC data and train timetable data. Second, a MCMC-based sampling method Metropolis–Hasting algorithm is proposed to estimate the model parameters and obtain the quantitative results of left behind patterns. Third, a case of Beijing metro is presented as an application to test the applicability and accuracy of the proposed method.