A system is subjected to an age renewal policy with renewal interval. τ Contrary to example 3.21, it is assumed that renewals occur in negligible time and that preventive and emergency renewals give...

A system is subjected to an age renewal policy with renewal interval. τ Contrary to example 3.21, it is assumed that renewals occur in negligible time and that preventive and emergency renewals give rise to the respective constant costs c_p
and c_e
with 0
_p
e. Further, let F(t) be the distribution function of the system lifetime T and λ(t) be the corresponding failure rate.

(1) Determine the maintenance cost rate (total maintenance cost per unit time) K(τ) for an unbounded running time of the system. (Note Total maintenance cost' includes replacement and repair costs.)

(2) Give a necessary and sufficient condition for the existence of an optimal renewal interval τ∗.

(3) Determine τ∗ if T has a uniform distribution over the interval  [0,z].