Proof in order to compute the limiting fraction of

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Unformatted text preview: if we earn a reward of ⇢ dollar every ⌧ units of time then in the long run we earn ⇢/⌧ dollars per unit time. To get from this to the answer given in 3.3, note that the answer there only depends on the means Eri and Eti , so the general answer must be ⇢/⌧ = Eri /Eti This device can be applied to remember many of the results in this chapter: when the answer only depends on the mean the limit must be the same as in the case when the times are not random. To illustrate the use of Theorem 3.3 we consider Example 3.4. Long run car costs. Suppose that the lifetime of a car is a random variable with density function h. Our methodical Mr. Brown buys a new car as soon as the old one breaks down or reaches T years. Suppose that a new car costs A dollars and that an additional cost of B dollars to repair the vehicle is incurred if it breaks down before time T . What is the long-run cost per unit time of Mr. Brown’s policy? Solution. The duration of the ith cycle, ti , has ZT Z1 Eti = th(t) dt + T h(t) dt 0 T since the length of the cycle will be ti if the car’s lif...
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