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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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 Spring '10
 DURRETT
 The Land

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