Parallel System Approach for Vector Out-Crossing

The mean crossing rate of a stochastic process out-crossing a safe domain is calculated using methods from time-independent reliability theory. The method is developed from Madsen’s formula which expresses the mean up-crossing rate of a scalar process through a specified level as a parallel system sensitivity measure. The method is applicable to stationary as well as nonstationary stochastic vector processes provided the random variables describing the process and the time-derivative process at time t can be mapped jointly into a set of independent standardized random normal variables. This is identical to the restriction imposed on the random variables using the first and second-order reliability methods (FORM, SORM), and is not very restrictive. Thus, quite general stochastic models can be treated. Also, closed-form results have been developed. In this paper the mean crossing rate for the stationary Gaussian process crossing into a polyhedral convex set is given. The method is demonstrated to give good results by examples.