The joint signature of parallel systems for different permutations of failure times

Abstract

The joint signature of m coherent systems is an extension of the joint signature of two systems introduced in 2010. It is interesting because the joint signature depends only on the mechanism of the systems’ composition rather than the underlying distribution function of the component lifetimes. It also depends on the permutation of the failure times. As there are m! permutations we need to compute m! joint signatures, many of which are the same. Even if the joint signatures for two different orderings of the system lifetimes are different, it is sometimes possible to obtain one of them via the other one by a transformation. In this article, we consider several parallel systems with independent and identically distributed component lifetimes and investigate how to classify the m! permutations of the failure times according to the same joint signature. Further, we partition the result into sets based on transformed joint signatures. Moreover, we establish a relationship between the different sets in the last partition so that the related joint signatures can be converted into each other by small modifications. By this procedure, we are able to compute the m! joint signatures together efficiently. Using several examples, we illustrate that our classifications save us a large number of calculations. We introduce an algorithm to determine the partitions and present a computer program for the algorithm. The case of series-parallel systems is quite complicated and is discussed briefly.