Approximating the randomized hitting time distribution of a non-stationary gamma process

Abstract

The non-stationary gamma process is a non-decreasing stochasticprocess with independent increments. By this monotonic behavior thisstochastic process serves as a natural candidate for modellingtime-dependent phenomena such as degradation. In condition-basedmaintenance the first time such a process exceeds a random thresholdis used as a model for the lifetime of a device or for the randomtime between two successive imperfect maintenance actions. Thereforethere is a need to investigate in detail the cumulative distributionfunction (cdf) of this so-called randomized hitting time. We firstrelate the cdf of the (randomized) hitting time of a non-stationarygamma process to the cdf of a related hitting time of a stationarygamma process. Even for a stationary gamma process this cdf has ingeneral no elementary formula and its evaluation is time-consuming.Hence two approximations are proposed in this paper and both have aclear probabilistic interpretation. Numerical experiments show thatthese approximations are easy to evaluate and their accuracy dependson the scale parameter of the non-stationary gamma process. Finally,we also consider some special cases of randomized hitting times forwhich it is possible to give an elementary formula for its cdf.approximation;condition based maintencance;first hitting time;non-stationary gamma process;random threshold