A Bayesian Framework for Post-disruption Travel Time Prediction in Metro Networks

TL;DR

A Bayesian spatiotemporal modeling framework for post-disruption train travel times that explicitly captures train interactions, headway imbalance, and non-Gaussian distributional characteristics observed during recovery periods is developed.

Abstract

Disruptions are an inherent feature of transportation systems, occurring unpredictably and with varying durations. Even after an incident is reported as resolved, disruptions can induce irregular train operations that generate substantial uncertainty in passenger waiting and travel times. Accurately forecasting post-disruption travel times therefore remains a critical challenge for transit operators and passenger information systems. This paper develops a Bayesian spatiotemporal modeling framework for post-disruption train travel times that explicitly captures train interactions, headway imbalance, and non-Gaussian distributional characteristics observed during recovery periods. The proposed model decomposes travel times into delay and journey components and incorporates a moving-average error structure to represent dependence between consecutive trains. Skew-normal and skew-$t$ distributions are employed to flexibly accommodate heteroskedasticity, skewness, and heavy-tailed behavior in post-disruption travel times. The framework is evaluated using high-resolution track-occupancy and disruption log data from the Montréal metro system, covering two lines in both travel directions. Empirical results indicate that post-disruption travel times exhibit pronounced distributional asymmetries that vary with traveled distance, as well as significant error dependence across trains. The proposed models consistently outperform baseline specifications in both point prediction accuracy and uncertainty quantification, with the skew-$t$ model demonstrating the most robust performance for longer journeys. These findings underscore the importance of incorporating both distributional flexibility and error dependence when forecasting post-disruption travel times in urban rail systems.

Figures (42)

• Figure 1: Map of the Montréal metro system. The network comprises four metro lines, of which only the Green and Orange lines are analyzed in this study. The operating direction of each line is indicated. Stations excluded from the Orange line are shown in gray. The blue-shaded area represents the downtown region, and stations located within this area are referred to as downtown stations.
• Figure 2: Space--time diagram illustrating a representative post-disruption travel time for train $i$ from station $j$ to station $k$. The travel time is decomposed into a delay component, $D_{i,j}$, defined as the interval between the time the disruption is reported as resolved and the departure of train $i$ from station $j$, and a journey component, $J_{i,j,k}$, corresponding to the time required for train $i$ to travel from station $j$ to its arrival at station $k$.
• Figure 3: Process times of trains at Guy-Concordia station on the Green metro line based on the hour of the day. Dwell times (left) exhibit substantial temporal variability, with noticeably higher variance during peak periods, while their mean remains relatively stable aside from slight increases at rush hours. The running times of tunnel segment after the station platform (right) remain highly stable throughout the day, showing no dependence on time of day and maintaining nearly constant variability. The x-axis indicates the hour of day, with post-midnight operations displayed by adding 24 to the corresponding hour.
• Figure 4: Headways at Guy-Concordia station (left) and full Green line journey times (right) as a function of the hour of day. Headways show a pronounced temporal pattern corresponding to operational adjustments between peak and off-peak periods. Line journey times exhibit higher mean and variance during peak hours, yet remain relatively stable over the rest of the day. The x-axis denotes the hour of day, with post-midnight operations represented by adding 24 to the corresponding hour.
• Figure 5: Empirical variance of journey times for all train trajectories in 2018 on the Green line of the Montréal metro, shown for direction 1. The horizontal axis represents the traveled distance from the origin station, measured by the number of stations traveled, while the vertical axis indicates the variance of the corresponding journey times.