Residual Lifetime of Consecutive $k$ -out-of- $n$ Systems Under Double Monitoring

The consecutive k -out-of-n systems are important structures in reliability engineering due to their applications in various real life situations. In this paper, we study the residual lifetime of these systems under the condition that the total number of failed components at time t₁ is r , and at time t₂ (t₂ >; t₁) the system is still working. We obtain explicit expressions for the survival function of the corresponding residual life when the components are exchangeably dependent. We also obtain a signature-based mixture representation for a coherent system satisfying a certain property.