Reliability Analysis of $k$ -out-of- $n$ Systems With Single Cold Standby Using Pearson Distributions

k-out-of-n systems with cold standby units are typically studied for unit lifetime distributions that allow analytical tractability. Often, however, these distributions differ significantly from reality. In this paper, we present an analytical approach to compute the mean failure time for k -out-of-n systems with a single cold standby unit for the wide class of lifetime distributions that can be captured by the Pearson distribution. The method requires the first four statistical moments of the unit´s lifetime distribution to be given, and computes the mean failure time using the Pearson distribution as an intermediate vehicle during the numerical integration. Experimental results for various instances of the Weibull distribution show that the numerical accuracy of the approach is high, with less than 0.5 percent error across a large range of k -out-of-n systems.