Question 7.EX.22: The lifetime of a car has a distribution H and probability d......
The lifetime of a car has a distribution H and probability density h. Ms. Jones buys a new car as soon as her old car either breaks down or reaches the age of T years. A new car costs C_{1} dollars and an additional cost of C_{2} dollars is incurred whenever a car breaks down. Assuming that a T -year-old car in working order has an expected resale value R(T ), what is Ms. Jones’ long-run average cost?
Cost of a cycle =C_1+C_2 I-R(T)(1-I)
I=\left\{\begin{array}{ll}1, & \text { if } X<T \\0, & \text { if } X \geqslant T\end{array} \text { where } X=\right.\text { life of car }
Hence,
E[cost of a cycle] = C_{1} + C_{2}H(T ) − R(T )[1 − H(T )]
Also,
\begin{aligned}E[\text { time of cycle }] & =\int E[\text { time } \mid X=x] h(x) d x \\& =\int_0^T x h(x) d x+T[1-H(T)]\end{aligned}
Thus the average cost per unit time is given by
\begin{gathered}\frac{C_1+C_2 H(T)-R(T)[1-H(T)]}{\int_0^T x h(x) d x+T[1-H(T)]} \end{gathered}