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An important aspect of the quality of a public transport service is itsreliability, which is defined as the invariability of the service attributes.Preventive measures taken during planning can reduce risks of unreliabilitythroughout operations. In order to tackle reliability during the serviceplanning phase, a key piece of information is the long-term prediction of thedensity of the travel time, which conveys the uncertainty of travel times. Weintroduce a reliable approach to one of the problems of service planning inpublic transport, namely the Multiple Depot Vehicle Scheduling Problem (MDVSP),which takes as input a set of trips and the probability density function(p.d.f.) of the travel time of each trip in order to output delay-tolerantvehicle schedules. This work empirically compares probabilistic models for theprediction of the conditional p.d.f. of the travel time, as a first steptowards reliable MDVSP solutions. Two types of probabilistic models, namelysimilarity-based density estimation models and a smoothed Logistic Regressionfor probabilistic classification model, are compared on a dataset of more than41,000 trips and 50 bus routes of the city of Montr\'eal. The result of a vastmajority of probabilistic models outperforms that of a Random Forests model,which is not inherently probabilistic, thus highlighting the added value ofmodeling the conditional p.d.f. of the travel time with probabilistic models. Asimilarity-based density estimation model using a $k$ Nearest Neighbors methodand a Kernel Density Estimation predicted the best estimate of the trueconditional p.d.f. on this dataset.

Predicting the probability distribution of bus travel time to move towards reliable planning of public transport services