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Article Linear Two-Dimensional Consecutive K-Type Systems in Multi-State Case(Elsevier Sci Ltd, 2026) Yi, He; Balakrishnan, Narayanaswamy; Li, XiangIn the context of consecutive k-type systems, multi-state system models are only considered in the onedimensional case and not in the two-dimensional case due to the complexity involved. In this paper, we consider several linear two-dimensional consecutive k-type systems in the multi-state case for the first time, as generalization of consecutive k-out-of-n systems and l-consecutive-k-out-of-n systems without/with overlapping. These systems include multi-state linear connected-(k, r)-out-of-(m, n): G systems, multi-state linear connected-(k, r)-or-(r, k)-out-of-(m, n): G systems, multi-state linear 1-connected-(k, r)-out-of-(m, n): G systems without/with overlapping, and multi-state linear 1-connected-(k, r)-or-(r, k)-out-of-(m, n): G systems without/with overlapping. We then derive their reliability functions by using the finite Markov chain imbedding approach (FMCIA) in a new way. We also present several examples to illustrate all the results developed here.Article Circular One/Two/Three-Dimensional Consecutive k-Type Systems(Springer, 2026) Yi, He; Balakrishnan, Narayanaswamy; Li, XiangIn this paper, several circular one/two/three-dimensional consecutive k-type systems are studied, including circular consecutive-k-out-of-n: F systems, circular l-consecutivek-out-of-n: F systems without/with overlapping, circular connected-(k(1), k(2))-outof-(n(1), n(2)): F systems, circular l-connected-(k(1), k(2))-out-of-(n(1), n(2)): F systems without/ with overlapping, circular connected-(k(1), k(2), k(3))-out-of-(n(1), n(2), n(3)): F systems, and circular l-connected-(k(1), k(2), k(3))-out-of-(n(1), n(2), n(3)): F systems without/with overlapping. Reliability functions of these systems are studied using finite Markov chain imbedding approach (FMCIA). Some illustrative examples are provided, and possible applications and generalizations of the established results are also mentioned.Article On the Notion of Discrete-Time Signature and Some Associated Properties and Results(Cambridge University Press, 2026) Balakrishnan, Narayanaswamy; Yi, He; Goroncy, AgnieszkaIn this work, by considering coherent systems comprising independent components with discrete lifetimes, we introduce the notion of discrete-time signature and then discuss some of its properties. With the use of the introduced signature, a stochastic ordering result is also established. We then introduce transformation formulas for the discrete-time signature to facilitate the comparison of systems of different sizes. Some examples are also presented to illustrate all the results developed here.Article A General Type of Linear Consecutive-K Systems(Springer, 2026) Yi, He; Balakrishnan, Narayanaswamy; Li, XiangIn this paper, some well-known consecutive k-type systems, including linear consecutive-k-out-of-n: F systems and linear l-consecutive-k-out-of-n: F systems without/with overlapping, are generalized by using more general failure patterns. Finite Markov chain imbedding approach (FMCIA) is applied in a new way for evaluating reliabilities of these generalized new systems. Some illustrative examples are provided for demonstrating the theoretical results established here and also for showing the efficiency of the computational process. Finally, some possible applications and generalizations are mentioned.Article Multi-State Linear Three-Dimensional Consecutive k-Type Systems(Cambridge Univ Press, 2026) Yi, He; Balakrishnan, Narayanaswamy; Li, XiangConsecutive $k$-type systems have become important in both reliability theory and applications; in spite of a large literature existing on them, three-dimensional consecutive $k$-type systems have not yet been studied for multi-state case. In this paper, we introduce several different types of multi-state linear three-dimensional consecutive $k$-type systems for the first time, with due consideration to possible overlapping of failure blocks. The finite Markov chain imbedding approach is then used for the derivation of their reliability functions with state spaces and transition matrices provided in a novel way, and the involved computational process is illustrated through several numerical examples. Finally, some possible applications of the work and potential extensions are pointed out.
