The first exit time stochastic theory applied to estimate the life-time of a complicated systemThe first exit time stochastic theory applied to estimate the life-time of a complicated system Peer-Reviewed Journal Publication Δημοσίευση σε Περιοδικό με Κριτές 2021-04-152020enWe develop a first exit time methodology to model the life time process of a complicated system. We assume that the functionality level of a complicated system follows a stochastic process during time and the end of the functionality of the system comes when the functionality function reaches a zero level. After solving several technical details including the Fokker-Planck equation for the appropriate boundary conditions we estimate the transition probability density function and then the first exit time probability density of the functionality of the system reaching a barrier during time. The formula we arrive is essential for complicated system forms. A simpler case has the form called as Inverse Gaussian and was first proposed independently by Schrödinger and Smoluchowsky in the same journal issue (1915) to express the probability density of a simple first exit time process hitting a linear barrier. Applications to the health state of biological systems (the human population and the Mediterranean flies) and to the functionality life time of cars are done.http://creativecommons.org/licenses/by/4.0/Methodology and Computing in Applied Probability2241601–1611 Skiadas Christos Σκιαδας Χρηστος Skiadas, Charilaos Springer Hitting time First exit time Inverse Gaussian Extended Inverse Gaussian First order approximation Second order approximation Fractional derivatives Complicated systems Human populations Health state Health state model Death probability density Car functional state