Joint signatures of two or more semi-coherent systems with shared components

In this paper, for two or more semi-coherent systems with shared components, three signature measures–joint signature, joint survival signature and joint cumulative signature–are presented and their relationship to joint reliability and joint unreliability are also discussed. This relationship provides a joint (survival/cumulative) signature computational method which transforms related computation to the computation of joint reliability/unreliability of the systems. Based on that, further discussions on computation of the joint reliability are presented for two different types of consecutive-k systems, namely, linear (n, f, k): F systems and linear <n, f, k>: F systems, by using finite Markov chain imbedding approach (FMCIA) in a new and novel way. Besides, for modular systems that can be regarded as a series/parallel connection of two groups of subsystems with shared components, their joint (survival/cumulative) signature is derived based on the joint survival/cumulative signatures of the subsystems, and this provides another computational method for joint signatures. Related results are applied to calculate joint signatures of a series/parallel connection of (n, f, k): F subsystems and <n, f, k>: F subsystems by applying the two computational methods together. Finally, some potential applications and generalizations are mentioned.