Time-variant reliability problems exist widely in practical engineering and it is usually solved based on the outcrossing rate. The common method to compute the outcrossing rate is by dealing with a two-component parallel system reliability problem, and sequentially the time-variant reliability can be computed by Rice's formula which is based on the assumptions of the independence and Poisson distribution for the out-crossings. However, the Rice's formula is imperfect and can create a big error because of the imperfect assumptions. Therefore in this paper, the formula for computing time-variant reliability is rederived and modified which avoids the assumptions. An approximate relationship between the outcrossing rate and failure rate is first derived and then the formula for computing time-variant reliability based on the failure rate is also provided. Consequently, the formula for time-variant reliability based on the outcrossing rate can be rederived and modified which is from the perspective of failure rate. Lastly, several examples are provided to validate the accuracy and efficiency of the proposed method.