The bound of reliability for the system subject to shock breakdowns

For the shock model (t) = E{ _N(t)}, where {N(t), t 0} is a renewal process having the inter-arrival distribution F(t) with a finite mean, we discuss its bounds under the cases as below: (i) F(t) is IFR (DFR). (ii) F(t) is NBUE(NWUE) and obtain more concise results for some specific cases. These results are useful in life testing and quality control management.