Reliability of several component sets with inspections at random times

We consider a random process which represents a system of components with constant failure rates and subjected to inspections at times defining a renewal process. We give an analytic method for calculating the reliability function, its Laplace transform and the mean time to failure. These formulas are computable if the Laplace transform of the inter-arrival law of the renewal process is explicit. We also study the asymptotic behavior of the reliability and determine the asymptotic failure rate of the system.