Using survival models to estimate bus travel times and associated uncertainties Zhengyao Yu a , Jonathan S. Wood b , Vikash V. Gayah a,⇑ a Department of Civil and Environmental Engineering, The Pennsylvania State University, University Park, PA 16802, United States b Department of Civil and Environmental Engineering, South Dakota State University, Brookings, SD 57007, United States article info Article history: Received 28 February 2016 Received in revised form 19 October 2016 Accepted 12 November 2016 Keywords: Bus travel time prediction Accelerated failure time survival models Bus travel time reliability Real-time transit information systems abstract Transit agencies often provide travelers with point estimates of bus travel times to down- stream stops to improve the perceived reliability of bus transit systems. Prediction models that can estimate both point estimates and the level of uncertainty associated with these estimates (e.g., travel time variance) might help to further improve reliability by tempering user expectations. In this paper, accelerated failure time survival models are proposed to provide such simultaneous predictions. Data from a headway-based bus route serving the Pennsylvania State University-University Park campus were used to estimate bus travel times using the proposed survival model and traditional linear regression frameworks for comparison. Overall, the accuracy of point estimates from the two approaches, measured using the root-mean-squared errors (RMSEs) and mean absolute errors (MAEs), was similar. This suggests that both methods predict travel times equally well. However, the survival models were found to more accurately describe the uncertainty associated with the predictions. Furthermore, survival model estimates were found to have smaller uncer- tainties on average, especially when predicted travel times were small. Tests for transfer- ability over time suggested that the models did not over-fit the dataset and validated the predictive ability of models established with historical data. Overall, the survival model approach appears to be a promising method to predict both expected bus travel times and the uncertainty associated with these travel times. Ó 2016 Elsevier Ltd. All rights reserved. 1. Introduction and background Travel time reliability is a key indicator of transit service quality and has a strong impact on transit ridership (Paine et al., 1967; Golob et al., 1972; Prashker, 1979). Previous research suggests that travel time reliability may be valued more highly than travel time itself among transit users (Bates et al., 2001; Brownstone and Small, 2005). Furthermore, negative experi- ences related to unreliable transit service discourage users from continuing to use public transportation (Carrel et al., 2013). Thus, maintaining travel time reliability is extremely important for improving transit competitiveness. Unfortunately, transit agencies have a difficult time maintaining reliable travel times as bus transit systems are inher- ently unstable (Newell and Potts, 1964; Newell, 1974). The mechanism that causes this instability is the passenger arrival and service process: the time that a bus spends serving passengers at a stop generally increases with the time between the current and preceding bus arrivals to that stop. For this reason, a bus arriving late to a stop spends more time serving http://dx.doi.org/10.1016/j.trc.2016.11.013 0968-090X/Ó 2016 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: zuy107@psu.edu (Z. Yu), jonathan.wood@sdstate.edu (J.S. Wood), gayah@engr.psu.edu (V.V. Gayah). Transportation Research Part C 74 (2017) 366–382 Contents lists available at ScienceDirect Transportation Research Part C journal homepage: www.elsevier.com/locate/trc