  We consider a wide class of increasing L\'evy processes perturbed by an
independent Brownian motion as a degradation model. Such family contains almost
all classical degradation models considered in the literature. Classically
failure time associated to such model is defined as the hitting time or the
first-passage time of a fixed level. Since sample paths are not in general
increasing, we consider also the last-passage time as the failure time
following a recent work by Barker and Newby. We address here the problem of
determining the distribution of the first-passage time and of the last-passage
time. In the last section we consider a maintenance policy for such models.
