Consecutive-k-out-of-n: F systems with imperfect protection measures

Consecutive-k-out-of-n: F systems are widely used in reliability modeling to describe systems whose failure is triggered by the occurrence of k consecutive component failures. In this paper, we introduce a consecutive-k-out-of-n: F structure equipped with imperfect protection blocks. The system is partitioned into consecutive blocks of size k, each of which is operative with a predetermined probability. Under the proposed design, whenever a protection block is operative, the failure rate of the components it contains is reduced; otherwise, the components retain a higher baseline failure rate. This modeling framework captures both local dependence and stochastic protection effects. We first develop a recursive scheme for computing the system's reliability, while a Markov embedding, which leads to an efficient matrix-based representation, is also established. The asymptotic behavior of the system's reliability is also under investigation. In addition, we consider a cost-constrained optimization problem, where the protection parameters are selected to maximize reliability under limited resources. Several numerical results and graphical representations illustrate the impact of protection effectiveness and system size, while the performance improvement of the optimized protected structure compared to its unprotected counterpart is also confirmed under different design scenarios.