Question 7.EX.8: A machine in use is replaced by a new machine either when it......
A machine in use is replaced by a new machine either when it fails or when it reaches the age of T years. If the lifetimes of successive machines are independent with a common distribution F having density f , show that
(a) the long-run rate at which machines are replaced equals
(b) the long-run rate at which machines in use fail equals
\frac{F(T)}{\int_{0}^{T}{}xf (x) dx +T [1 −F(T )]}(a) The number of replaced machines by time t constitutes a renewal process. The time between replacements equals T, if lifetime of new machine is \geqslant T ; x, if lifetime of new machine is x, x<T. Hence,
E \text { [time between replacements }]=\int_0^T x f(x) d x+T[1-F(T)]
and the result follows by Proposition 3.1.
(b) The number of machines that have failed in use by time t constitutes a renewal process. The mean time between in-use failures, E[F], can be calculated by conditioning on the lifetime of the initial machine as E[F]= E[E[F \mid lifetime of initial machine ]]. Now
E[F \mid \text { lifetime of machine is } x]= \begin{cases}x, & \text { if } x \leqslant T \\ T+E[F], & \text { if } x>T\end{cases}
Hence,
E[F]=\int\limits_0^T x f(x) d x+(T+E[F])[1-F(T)]
or
E[F]=\frac{\int_0^T x f(x) d x+T[1-F(T)]}{F(T)}
and the result follows from Proposition 3.1.