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no longer collect data. The scientist comes every L units of time to check the
equipment. If the equipment has been disturbed then she can usually ﬁx it
quickly so we will assume the the repairs take 0 time. (a) What is the limiting
fraction of time the machine is working? (b) Suppose that the data that is being
collected is worth a dollars per unit time, while each inspection costs c < a.
Find the best value of the inspection time L.
Age and Residual Life
3.19. Consider the discrete renewal process with fj = P (t1 = j ) and Fi =
P (t1 > i). (a) Show that the age chain has transition probability
q (j, j + 1) = Fj +1
Fj q (j, 0) = 1 Fj +1
fj +1
=
Fj
Fj for j 0 (b) Show that if Et1 < 1, the stationary distribution ⇡ (i) = P (t1 > i)/Et1 .
(c) Let p(i, j ) be the transition probability for the renewal chain. Verify that It
should be clear by comparing the numerical examples above that there is a close
relationship between q is the dual chain of p, i.e., the chain p run backwards.
That is,...
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 Spring '10
 DURRETT
 The Land

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