Vector Process Out‐Crossing as Parallel System Sensitivity Measure

Abstract

The mean rate of vector processes out‐crossing safe domains is calculated using methods from time‐independent reliability theory. The method is founded on a result for scalar up‐crossing derived by Madsen. The out‐crossing is formulated as a zero down‐crossing of a continuously differentiable scalar process, and the mean crossing rate is obtained as a sensitivity measure of the probability for an associated parallel system domain. The vector process may be Gaussian, non‐Gaussian, stationary or nonstationary, and the failure function defining the boundary of the safe domain may be time‐dependent. A method for calculation of the expected number of crossings in a time interval through the introduction of an auxiliary uniformly distributed variable is presented. For stochastic failure surfaces the ensemble averaged rate is determined. A closed‐form expression for the mean crossing rate of a nonstationary Gaussian vector process crossing into a time‐dependent convex polyhedral set is derived. The method is demonstrated to give good results by examples.