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The structure signature of a system made up of $n$ components havingcontinuous and i.i.d. lifetimes was defined in the eighties by Samaniego as the$n$-tuple whose $k$-th coordinate is the probability that the $k$-th componentfailure causes the system to fail. More recently, a bivariate version of thisconcept was considered as follows. The joint structure signature of a pair ofsystems built on a common set of components having continuous and i.i.d.lifetimes is a square matrix of order $n$ whose $(k,l)$-entry is theprobability that the $k$-th failure causes the first system to fail and the$l$-th failure causes the second system to fail. This concept was successfullyused to derive a signature-based decomposition of the joint reliability of thetwo systems. In the first part of this paper we provide an explicit formula tocompute the joint structure signature of two or more systems and extend thisformula to the general non-i.i.d. case, assuming only that the distribution ofthe component lifetimes has no ties. We also provide and discuss a necessaryand sufficient condition on this distribution for the joint reliability of thesystems to have a signature-based decomposition. In the second part of thispaper we show how our results can be efficiently applied to the investigationof the reliability and signature of multistate systems made up of two-statecomponents. The key observation is that the structure function of such amultistate system can always be additively decomposed into a sum of classicalstructure functions. Considering a multistate system then reduces toconsidering simultaneously several two-state systems.

Joint signature of two or more systems with applications to multistate systems made up of two-state components