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Unformatted text preview: What is the longrun proﬁt per day of this inventory policy? How do we
choose s and S to maximize proﬁt?
Example 1.7. Repair chain. A machine has three critical parts that are
subject to failure, but can function as long as two of these parts are working.
When two are broken, they are replaced and the machine is back to working
order the next day. To formulate a Markov chain model we declare its state
space to be the parts that are broken {0, 1, 2, 3, 12, 13, 23}. If we assume that 5 1.1. DEFINITIONS AND EXAMPLES parts 1, 2, and 3 fail with probabilities .01, .02, and .04, but no two parts fail
on the same day, then we arrive at the following transition matrix:
0
1
2
3
12
13
23 0
.93
0
0
0
1
1
1 1
.01
.94
0
0
0
0
0 2
.02
0
.95
0
0
0
0 3
.04
0
0
.97
0
0
0 12 13
0
0
.02 .04
.01 0
0 .01
0
0
0
0
0
0 23
0
0
.04
.02
0
0
0 If we own a machine like this, then it is natural to ask about the longrun cost
per day to operate it. For example, we might ask:
Q. If we are going to operate the machine for 1800 days (...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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