  The structure signature of a system made up of $n$ components having
continuous 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 component
failure causes the system to fail. More recently, a bivariate version of this
concept was considered as follows. The joint structure signature of a pair of
systems 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 the
probability 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 successfully
used to derive a signature-based decomposition of the joint reliability of the
two systems. In the first part of this paper we provide an explicit formula to
compute the joint structure signature of two or more systems and extend this
formula to the general non-i.i.d. case, assuming only that the distribution of
the component lifetimes has no ties. We also provide and discuss a necessary
and sufficient condition on this distribution for the joint reliability of the
systems to have a signature-based decomposition. In the second part of this
paper we show how our results can be efficiently applied to the investigation
of the reliability and signature of multistate systems made up of two-state
components. The key observation is that the structure function of such a
multistate system can always be additively decomposed into a sum of classical
structure functions. Considering a multistate system then reduces to
considering simultaneously several two-state systems.
