Quantifying out-of-station waiting time in oversaturated urban metro systems

Metro systems in megacities such as Beijing, Shenzhen and Guangzhou are under great passenger demand pressure. During peak hours, it is common to see oversaturated conditions (i.e., passenger demand exceeds network capacity), which bring significant operational risks and safety issues. A popular control intervention is to restrict the entering rate during peak hours by setting up out-of-station queueing with crowd control barriers. The \textit{out-of-station waiting} can make up a substantial proportion of total travel time but is not well-studied in the literature. Accurate quantification of out-of-station waiting time is important to evaluating the social benefit and cost of service scheduling/optimization plans; however, out-of-station waiting time is difficult to estimate because it is not a part of smart card transactions. In this study, we propose an innovative method to estimate the out-of-station waiting time by leveraging the information from a small group of transfer passengers -- those who transfer from nearby bus routes to the metro station. Based on the estimated transfer time for this small group, we first infer the out-of-station waiting time for all passengers by developing a Gaussian Process regression with a Student-$t$ likelihood and then use the estimated out-of-station waiting time to build queueing diagrams. We apply our method to the Tiantongyuan North station of Beijing metro as a case study; our results show that the maximum out-of-station waiting time can reach 15 minutes, and the maximum queue length can be over 3000 passengers. Our results suggest that out-of-station waiting can cause significant travel costs and thus should be considered in analyzing transit performance, mode choice, and social benefits. To the best of our knowledge, this paper is the first quantitative study for out-of-station waiting time.