The multi-depot scheduling problem (MDVSP) is one of the most studied problem in public transport service planning. It consists of assigning buses to each timetabled trip while respecting vehicle availability at each depot. Although service quality, and especially reliability, is the core of most transport agencies, the MDVSP is more often than not solved solely in a cost-efficient way. This work introduces a data-driven model to the reliable MDVSP with stochastic travel time (R-MDVSP-STT). The reliability of a schedule is assessed and accounted for by propagating delays using the probability mass function of the travel time of each timetabled trip. We propose a heuristic branch-and-price algorithm to solve this problem and a labeling algorithm with stochastic dominance criterion for the associated subproblems. The solutions obtained are compared based on three passenger-oriented metrics - under normal and extraordinary circumstances. Computational results on real-life instances show that our method can efficiently find good trade-offs between operational costs and reliability, improving the reliability of the solutions with little cost increase.