Reliability analysis for a generalized sparse connection multi-state consecutive-k-out-of-n linear system

The utilization of binary states is a common practice in reliability analysis. However, in complex systems such as aviation, spaceflight, and watercraft, the prevalence of multi-state systems and units is more pronounced. Consequently, extensive research has been undertaken to explore reliability analysis with a focus on multi-state consecutive-k models across various disciplines. Notably, the concept of sparse connection has received considerable attention due to its relevance in practical applications like wireless communication, cloud computing networks, and oil pipeline systems. This paper aims to propose a generalized multi-state consecutivek: G system model that incorporates sparse connection. Both the system and the units are allowed to be in one of the (M + 1) possible states, wherein the order numbers 0, 1, 2, ..., M indicate the performance measurement of the system or the units, i.e., 0 denotes the worst performance state while M denotes the best performance state. We classify the model into increasing, decreasing and non-monotonic consecutive-k systems corresponding to different mission requirements in practical applications. This paper utilizes the finite Markov chain imbedding (FMCI) method to derive explicit system state distributions for three types of multi-state consecutive-k-out-of-n: G systems. Subsequently, numerical examples are provided to demonstrate the proposed models and the corresponding results.