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Consecutive-ksystems have been studied extensively in reliability engineering. Linear and circular consecutive-k-out-of-n: F systems with shared component(s) have been studied recently by Lin et al. and Yin and Cui . They considered two adjacent subsystems overlapping with one (multiple) shared component(s), respectively, and obtained system reliability formulas by summing the reliability values for all disjoint cases. As their method is computationally intensive, it would be of interest to develop a simpler and more efficient method for the computation of the reliability function of such systems instead of requiring to list all disjoint cases. In this work, by employing the finite Markov chain imbedding approach, we develop unified formulas as products of matrices for evaluating system reliabilities by redefining the state space of the Markov chain. The results developed here decrease the complexity in the computation of system reliability. Furthermore, the new method is also employed to obtain reliability formulas for Markov-dependent cases. A case study of communication systems is finally presented and some numerical examples are presented to illustrate the developed model and the corresponding results.

Critical infrastructures are vital in supporting the essential services of society. However, they are susceptible to damage when confronted with disruptive events like earthquakes. This fosters a pressing demand for research geared towards building a resilient critical infrastructure. To furnish crucial benchmarks for enhancing system resilience, we developed a framework for assessing the resilience of critical infrastructures, considering their intricate multi-component configurations. Our framework conceptualizes overall resilience as a synthesis of four principal attributes: resistant, absorption, adaptive, and recovery resilience. Aiming at quantifying these attributes, the temporal performance behavior of the system is modeled via Markov processes. On this basis, resilience measures are defined concerning attributes within the framework, each incorporating both inherent and acquired aspects. The calculation formulas for these measures are derived analytically based on aggregated stochastic processes. This investigation also proves that, within the proposed framework, series systems are no more resilient than parallel systems. A case study of a potable water system subject to earthquakes is showcased to exemplify the devised assessment framework and calculation method. Meanwhile, some managerial insights are provided to designers and managers responsible for critical infrastructures. Managerial Relevance Statement–This article introduces a framework for assessing the resilience of critical infrastructures (CIs) in response to disruptive events. The findings indicate a progressive decline in both the inherent resistant and absorption resilience of CIs with increased operational duration, leading inevitably to interruptions. This highlights the imperative for CI designers to focus on enhancing inherent adaptive and recovery capabilities of CIs to counter such interruptions. Meanwhile, the study finds that both the acquired adaptive and recovery resilience initially increase before diminishing over the lifespan of CIs. This insight offers guidance for managers aiming to prolong the operational lifespan of CIs, emphasizing initial resource allocation to prevent potential interruptions, and recommending subsequent resource reallocation to facilitate rapid recovery from such interruptions. In addition, the research proves that parallel systems exhibit superior resilience compared to series systems, given identical components. This suggests a significant variance in the improvement level of systems with different structures when equivalent recovery resources are applied, underscoring the strategic importance of structural considerations in resilience enhancement efforts.