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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 longrun 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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 Spring '10
 DURRETT
 The Land

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