A characterization of exponential and geometrical distributions and its stability bound (Q914277)

Consider a renewal type system where components are replaced either on failure or preventively after random times, independent of the lifetimes. Let R(x) be the limit as \(t\to \infty\) of the probability that no failure occurs in the time interval \([t,t+x]\) and let F be the lifetime distribution function. Inequalities relating R(x) and 1-F(x) are derived for lifetime distributions with increasing and decreasing failure rates and an associated characterization of exponential and geometric distributions is given; for the latter the author obtains some stability results.