The lifetime of a car is a continuous random variable having a distributionH and probability density h. Mr. Brown has a policy that he buys a new car as soon as his old one either breaks down or reaches the age of T years. Suppose that a new car costs C1 dollars and also that an additional cost of C2 dollars is incurred whenever Mr. Brown’s car breaks down. Under the assumption that a used car has no resale value, what is Mr. Brown’s long-run average cost?

If we say that a cycle is complete every time Mr. Brown gets a new car, then it follows from Proposition 7.3 (with costs replacing rewards) that his long-run average cost equals

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E[cost incurred during a cycle] E[length of a cycle]