Informational Complexity of Inactivity Times of the Failed Components through Cumulative Residual Extropy and Jensen Divergence

This paper investigates the complexity of inactivity times of the failed components in coherent systems through the lens of cumulative residual extropy and its divergence-based extension, Jensen-cumulative residual extropy. Unlike classical reliability metrics that focus on system inactivity or mean residual life, our framework quantifies the hidden informational structure of inactivity times of the failed components at the coherent system still functioning though some of its components have failed. We derive closed-form expressions for the cumulative residual extropy of inactivity times of the failed components using system signatures and establish stochastic bounds and comparisons that highlight the impact of structural configuration. A novel divergence measure, the Jensen-cumulative residual extropy, is introduced to capture discrepancies between coherent systems and benchmark k-out-of-n structures. Numerical illustrations with Weibull-distributed lifetimes demonstrate the sensitivity of cumulative residual extropy and Jensen-cumulative residual extropy to redundancy patterns and dependence structures. Furthermore, by integrating cost considerations into the divergence framework, we provide a rigorous optimization scheme for selecting system signatures that jointly minimize informational complexity and economic expenditure. The proposed approach enriches the theoretical foundation of reliability analysis and offers practical guidelines for designing resilient, cost-effective, and information-efficient engineering systems.