  This paper presents two novel approaches for uncertainty estimation adapted
and extended for the multi-link bus travel time problem. The uncertainty is
modeled directly as part of recurrent artificial neural networks, but using two
fundamentally different approaches: one based on Deep Quantile Regression (DQR)
and the other on Bayesian Recurrent Neural Networks (BRNN). Both models predict
multiple time steps into the future, but handle the time-dependent uncertainty
estimation differently. We present a sampling technique in order to aggregate
quantile estimates for link level travel time to yield the multi-link travel
time distribution needed for a vehicle to travel from its current position to a
specific downstream stop point or transfer site.
  To motivate the relevance of uncertainty-aware models in the domain, we focus
on the connection assurance application as a case study: An expert system to
determine whether a bus driver should hold and wait for a connecting service,
or break the connection and reduce its own delay. Our results show that the
DQR-model performs overall best for the 80%, 90% and 95% prediction intervals,
both for a 15 minute time horizon into the future (t + 1), but also for the 30
and 45 minutes time horizon (t + 2 and t + 3), with a constant, but very small
underestimation of the uncertainty interval (1-4 pp.). However, we also show,
that the BRNN model still can outperform the DQR for specific cases. Lastly, we
demonstrate how a simple decision support system can take advantage of our
uncertainty-aware travel time models to prioritize the difference in travel
time uncertainty for bus holding at strategic points, thus reducing the
introduced delay for the connection assurance application.
