Consecutive 𠑘 -within- 𠑚 -out-of- 𠑛 :F System with Nonidentical Components

As a generalisation of consecutive 𠑘 -out-of- 𠑛 :F and 𠑘 -out-of- 𠑛 :F system models, a consecutive 𠑘 -within- 𠑚 -out-of- 𠑛 :F system consists of 𠑛 linearly ordered components and fails if and only if there are 𠑚 consecutive components which include among them at least 𠑘 failed components. In this paper, we study the survival function of a consecutive 𠑘 -within- 𠑚 -out-of- 𠑛 :F system consisting of independent but nonidentical components. We obtain exact expressions for the survival function when 2 𠑚 ≥ 𠑛 . A detailed analysis for consecutive 2-within- 𠑚 -out-of- 𠑛 :F systems is presented and the asymptotic behaviour of hazard rate of these systems is investigated using mixture representations.