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 non-stationary Gaussian vector process crossing into a time-dependent convex polyhydral set is derived. The method is demonstrated to give good results by examples.