Analytical derivation of the outcrossing rate in time-variant reliability problems

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A Sampling Approach to Extreme Value Distribution for Time-Dependent Reliability Analysis (2013, methodology): “They include the Monte Carlo simulation (MCS) [4], the Gamma process method [5], the Markov chain method [6], the extreme value distribution method [7,8], the upcrossing rate method [9,10], and the composite limit state method [1].”
Time-Dependent Mechanism Reliability Analysis With Envelope Functions and First-Order Approximation (2014, methodology): “There are two basic types of methods for time-dependent reliability: extreme value methods [31-35] and first-passage methods [30, 33, 36-42].”
Time-Dependent Reliability of Dynamic Systems Using Subset Simulation With Splitting Over a Series of Correlated Time Intervals (2013, result): “The out-crossing rate approach was first introduced by Rice [8] followed by extensive studies [6, 9-13] under the assumption of the out-crossings being statistically independent and Poisson distributed.”