A unified palm measure framework for failure-event-based component importance measures in nonrepairable, repairable, and phase-mission systems

This paper proposes a Palm-based framework for component importance measures conditioned on system failure events, uniformly applicable to nonrepairable, repairable, and phase-mission systems. The formulation does not require renewal, stationary, or Markovian assumptions. We classify measures into transition- and state-based importance. The former captures component failure transitions, recovering Birnbaum and Barlow–Proschan importance, while the latter provides an event-based interpretation of Fussell–Vesely importance. This framework yields well-defined finite-time and phase-conditioned measures for nonstationary systems, establishing a rigorous probabilistic foundation for failure-event-based analysis.