The reliability of correlated two unit systems

The classic problem of the extremes of two unit systems is revisited by using an explicit correlation coefficient (rho) between the units. Two unit systems have long been studied in reliability literature as these systems serve as the foundation of the serial and parallel systems. Often, the units are assumed to be independent, but this is not always the case in reality. The effect and inclusion of correlation in reliability analysis is often an unclear area. The purpose of this paper is two-fold: i) to provide clear probability density, mean, and variance expressions for two unit systems where the bivariate exponential and bivariate normal distributions hold and ii) to show how the available simulation schemes compare to some known analytical solutions. Results are stated in terms of the correlation coefficient, and include probability density function (PDF) and parameter expressions needed for reliability analysis and comments on the performance of simulation methods. As simulation works well for known cases, it can be used with high confidence in cases where analytical solutions do not exist