The concept of joint bivariate signature, introduced by Navarro et al. , is a useful tool for quantifying the reliability of two systems with shared components. As with the univariate system signature, introduced by Samaniego , its applications are limited to systems with only one type of components, which restricts its practical use. Coolen and Coolen-Maturi  introduced the survival signature, which generalizes Samaniego’s signature and can be used for systems with multiple types of components. This paper introduces a joint survival signature for multiple systems with multiple types of components and with some components shared between systems. A particularly important feature is that the functioning of these systems can be considered at different times, enabling computation of relevant conditional probabilities with regard to a system’s functioning conditional on the status of another system with which it shares components. Several opportunities for practical application and related challenges for further development of the presented concept are briefly discussed, setting out an important direction for future research.
Coolen-Maturi, T., Coolen, F. P., & Balakrishnan, N. (2021). The joint survival signature of coherent systems with shared components. Reliability Engineering & System Safety, 207, Article 107350. https://doi.org/10.1016/j.ress.2020.107350