Stochastic

# The bulb is then replaced and the cycle starts again

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online