Multi-state Signatures for Multi-state Systems with Binary/Multi-state Components

Signature theory, as an important part of reliability theory, provides an efficient tool for modeling and analyzing various properties of reliability systems. By now, signature theory has become exhaustive for binary-state systems, but for multi-state systems which are commonly encountered in practice, there are still a lot of issues to examine. In this work, we review important research works that have been carried out on signatures of systems with a special focus on signature concepts, their properties, computational methods, and some multi-state signatures for multi-state systems. We also summarize work that we have done recently on multi-state signatures, including their definitions, properties, transformation formulas and module structures. Finally, we present a number of examples to illustrate various notations and associated results described here.