Multi-state linear three-dimensional consecutive <i>k</i> -type systems

Consecutive $k$ -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.