Abstract
Let $T$ denote the life length of a series system of $n$ components having respective life lengths $T_1,\cdots, T_n$, not necessarily independent. We give necessary and sufficient conditions for the existence of a set of independent random variables $\{H_I\}, I$ a subset of $\{1,\cdots, n\}$, such that the life length of the original series system and the occurrence of its failure pattern (set of components whose simultaneous failure coincides with that of the system) have the same joint distribution as the life length of a derived series system of components having life lengths $\{H_I\}$ and the occurrence of the corresponding failure pattern of the derived system. We also exhibit explicitly the distributions of these independent random variables $\{H_I\}$. This extends the results of Miller while using more elementary methods.
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