Some probability functions in reliability and their applications

There has been much research on the general failure model recently. In the general failure model, when the unit fails at its age t, Type I failure (minor failure) occurs with probability 1 - p(t) and Type II failure (catastrophic failure) occurs with probability p(t). In the previous research, some specific shapes (constant, non-decreasing, or bathtub-shape) on the probability function p(t) are assumed. In this article, general results on some probability functions are obtained and applied to study the shapes of p(t). The results are also applied to determining the optimal inspection and allocation policies in maintenance problems.