Question

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 $$ \left[\int_{0}^{T} x f(x) d x+T(1-F(T))\right]^{-1} $$ (b) the long-run rate at which machines in use fail equals $$ \frac{F(T)}{\int_{0}^{T} x f(x) d x+T[1-F(T)]} $$

Short answer

The long-run rates at which machines are replaced and machines in use fail are: a) Replacement Rate: $$ \left[\int_{0}^{T} x f(x) d x+T(1-F(T))\right]^{-1} $$ b) Failure Rate: $$ \frac{F(T)}{\int_{0}^{T} x f(x) d x+T[1-F(T)]} $$

Step-by-step solution

Calculate the expected time until a machine is replaced

To find the expected time until a machine is replaced, we have to consider two possibilities: 1. The machine fails before reaching the age of \(T\) years. 2. The machine reaches the age of \(T\) years. For the first possibility, we need to find the expected time it takes for a machine to fail. This can be obtained by integrating the lifespan \(x\) multiplied by the density function \(f(x)\) from 0 to \(T\): $$ \int_{0}^{T} x f(x) d x $$ For the second possibility, the machine reaches the age of \(T\) years and is replaced. The probability that a machine reaches the age of \(T\) years is given by \((1 - F(T))\). Taking the weighted sum of these two possibilities gives the expected time until a machine is replaced: $$ E[T] = \int_{0}^{T} x f(x) d x + T(1-F(T)) $$

Calculate the long-run rate at which machines are replaced

The long-run rate at which machines are replaced is just the reciprocal of the expected time until a machine is replaced. So, $$ \text{Replacement Rate} = \frac{1}{E[T]} = \left[\int_{0}^{T} x f(x) d x+T(1-F(T))\right]^{-1} $$

Calculate the long-run rate at which machines in use fail

The long-run rate at which machines in use fail can be obtained by taking the product of the probability that a machine fails before reaching the age of \(T\) years and the long-run rate at which machines are replaced: $$ \text{Failure Rate} = F(T) \times \text{Replacement Rate} = \frac{F(T)}{\int_{0}^{T} x f(x) d x+T[1-F(T)]} $$ Thus, we have derived the long-run rates at which machines are replaced and machines in use fail as: a) Replacement Rate: $$ \left[\int_{0}^{T} x f(x) d x+T(1-F(T))\right]^{-1} $$ b) Failure Rate: $$ \frac{F(T)}{\int_{0}^{T} x f(x) d x+T[1-F(T)]} $$