State Distributions of Multi-State Consecutive- $k$ Systems

As a flexible, accurate tool for representing practical industrial and military systems, multi-state systems have been paid great attention to in both academic and engineering fields. Multi-state consecutive-k systems are extensions of binary consecutive-k systems. In this paper, by using the finite Markov chain imbedding approach, we present a unified formula with the product of matrices for evaluating state distributions of two kinds of multi-state consecutive-k systems. Moreover, the analytic expressions are given for the expectation, the probability mass function, and the distribution of the waiting time for the appearance of a state of multi-state consecutive- k systems. The formulas obtained in this paper will be useful for start-up demonstration tests, quality control, and so on. Finally, numeral examples are given, which show the flexibility and effectiveness of our model.