The number of failed components upon system failure when the lifetimes are discretely distributed

Abstract

The number of failed components at the time when the system fails is an important quantity which can be effectively used in the determination of the optimal number of spares. This paper is concerned with the distribution and expected value of this quantity when the lifetimes of a given coherent system are discretely distributed. In particular, the distribution of the corresponding random quantity is derived for all coherent systems of order three and four. The mean number of the failed components upon system failure is exactly derived for a linear consecutive-2-out-of-n:F structure. The mean of the quantity under concern is also computed for series and parallel systems consisting of disjoint modules. The latter computation provides an efficient way to obtain the corresponding mean for a larger system via the modules which have smaller number of components.