Mean time to failure of systems with dependent components

We consider first a system of two components that are possibly dependent. Two simple reliability structures can be formed: series system and parallel (redundant) system. The exact formulas and two-sided bounds for the MTTF of systems are provided. We extend some of the results of Kotz et al. (2003) [8] on the effect of redundancy or competing risk when component lifetimes are no longer independent. The main results are illustrated by some useful distributions. The new (r,s)(r,s)-out-of-n system is also discussed. In addition, a new concept of dependence ordering (more diagonal dependent) between bivariate distributions is introduced and used to compare the system lifetimes. Finally, the extension of the main results to the three-component systems and multi-component systems are also briefly investigated.