We aim to approximate the distribution of the first-passage time of a particle moving according to a Gaussian process with increasing trend, i. e., the distribution of the first time a particle described, e. g., by a state-space model such as a constant-velocity or constant-acceleration model, arrives at a fixed location. Since the known approaches from the literature either consider processes from different families or lead to highly
complex approximations, we seek a fast-to-compute method for the problem. Motivated by an engineering particle transport task for which we can assume that once a particle has arrived at this location it cannot move back, we derive an analytic approximation for the first-passage time probabilities and calculate its inverse cumulative distribution function analytically and the moments numerically. Furthermore, we propose a Gaussian approximation based on a linearization approach. The strengths and limitations of our methods are discussed and by comparison with Monte Carlo simulations, we show that in particular, the first one satisfies the requirements of engineering problems in terms of accuracy and computation time.