A system is preventively replaced by an identical new one at time points \(\tau, 2 \tau, \ldots\) If failures happen in between, then the failed system is replaced by an identical new one as well. The latter replacement actions are called emergency replacements. This replacement policy is called block replacement. The costs for preventive and emergency replacements are \(c_{p}\) and \(c_{e}, 0

\[F(t)=P(L \leq t)=\left(1-e^{-\lambda t}\right)^{2}, t \geq 0\]

(1) Determine the renewal function \(H(t)\) of the ordinary renewal process with cycle length distribution function \(F(t)\).
(2) Based on the renewal reward theorem (7.148), give a formula for the long-run maintenance cost rate \(K(\tau)\) under the block replacement policy.
(3) Determine an optimal \(\tau=\tau *\) with regard to \(K(\tau)\) for \(\lambda=0.1, c_{e}=180, c_{p}=100\).
(4) Under otherwise the same assumptions, determine the cost rate if the system is only replaced after failures and compare it with the one obtained under (3).