The bulb is then replaced and the cycle starts again

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Unformatted text preview: e is ti < T , but T if the car’s life ti T . The reward (or cost) of the ith cycle has ZT Eri = A + B h(t) dt 0 104 CHAPTER 3. RENEWAL PROCESSES since Mr. Brown always has to pay A dollars for a new car but only owes the additional B dollars if the car breaks down before time T . Using Theorem 3.3 we see that the long run cost per unit time is RT A + B 0 h(t) dt Eri = RT R1 Eti th(t) dt + T h(t) dt 0 T Concrete example. Suppose that the lifetime of Mr. Brown’s car is uniformlu distributed on [0, 10]. This is probably not a reasonable assumption, since when cars get older they have a greater tendency to break. However, having confessed to this weakness, we will proceed with this assumption since it makes calculations easier. Suppose that the cost of a new car is A = 10 (thousand dollars), while the breakdown cost is B = 3 (thousand dollars). If Mr. Brown replaces his car after T years then the expected values of interest are T Eri = 10 + 3 = 10 + 0.3T 10 ✓ ◆ ZT t T T2 Eti = dt + T 1 = +T 10 20 0 10 T2 =T 10 0.05T 2 Combining the expressions for the Eri and Eti we see that the lo...
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