The non-stationary gamma process is a non-decreasing stochastic
process with independent increments. By this monotonic behavior this
stochastic process serves as a natural candidate for modelling
time-dependent phenomena such as degradation. In condition-based
maintenance the first time such a process exceeds a random threshold
is used as a model for the lifetime of a device or for the random
time between two successive imperfect maintenance actions. Therefore
there is a need to investigate in detail the cumulative distribution
function (cdf) of this so-called randomized hitting time. We first
relate the cdf of the (randomized) hitting time of a non-stationary
gamma process to the cdf of a related hitting time of a stationary
gamma process. Even for a stationary gamma process this cdf has in
general no elementary formula and its evaluation is time-consuming.
Hence two approximations are proposed in this paper and both have a
clear probabilistic interpretation. Numerical experiments show that
these approximations are easy to evaluate and their accuracy depends
on the scale parameter of the non-stationary gamma process. Finally,
we also consider some special cases of randomized hitting times for
which it is possible to give an elementary formula for its cdf
