Reliability Expression of Three-Dimensional Consecutive-k Systems

In the realm of medical diagnostics, as well as in various other fields, the accurate calculation of detection probabilities within three-dimensional spaces can be essential. This practical requirement has led to the development of three-dimensional consecutive-k systems. The assessment of the reliability of these systems, specifically the three-dimensional consecutive-(r1, r2, r3)-out-of-(n1, n2, n3):F systems, has been a subject of extensive study. There are two commonly employed and efficient approaches that are commonly used for computing their reliability: the Recursive Equation Approach (REA) and the Finite Markov Chain Imbedding Approach (FMCIA). To date, REA and FMCIA have been independently developed with limited discussion on their interrelationship. Therefore, this study has rederived the REA in accordance with existing studies, providing a clearer comprehension of both the similarities and differences between REA and FMCIA. In conclusion, although REA and FMCIA share a fundamental similarity, discernible computational differences highlight the practical distinction between these two approaches. Finally, this paper enumerates necessary improvements required for the efficient computation of system reliabilities, with the ultimate aim of enhancing the practical applicability of these approaches.