## Abstract

Consider a terminal in which users arrive continuously over a finite period of time at a variable rate known in advance. A fleet of shuttles has to carry them over a fixed trip. What is the shuttle schedule that minimizes their waiting time? This is the question addressed in the present paper. We consider several versions that differ according to whether the shuttles come back to the terminal after their trip or not, and according to the objective function (maximum or average of the waiting times). We propose efficient algorithms with proven performance guarantees for almost all versions, and we completely solve the case where all users are present in the terminal from the beginning, a result which is already of some interest. The techniques used are of various types (convex optimization, shortest paths, ...). The paper ends with numerical experiments showing that most of our algorithms behave also well in practice.

## Keywords

Convex optimization Shortest paths Timetabling Transportation Waiting time## Mathematics Subject Classification

MSC 90B35## Notes

### Acknowledgements

The authors thank the reviewers for their comments, which helped improve the paper. They are also grateful to Eurotunnel for its explanations about the operation of the tunnel and the terminals, and for the data it provided.

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