We consider one-dimensional directed trap models and suppose that the
trapping times are heavy-tailed. We obtain the inverse of a stable subordinator
as scaling limit and prove an aging phenomenon expressed in terms of the
generalized arcsine law. These results confirm the status of universality
described by Ben Arous and \v{C}ern\'y for a large class of graphs.Comment: 16 pages, accepted for publication in "Markov processes and Related
Fields