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Unformatted text preview: 3 + 2/9 + 4/27 = 19/27 in
agreement with the previous calculation.
Example 4.22. Return to o ce hours. With the machinery just developed
we can give a simple solution to one of the exercises in Chapter 2. Ron, Sue,
and Ted arrive at the beginning of a professor’s o ce hours. The amount of
time they will stay is exponentially distributed with means of 1, 1/2, and 1/3
hour, i.e., rates 1, 2, and 3. What is the expected time until all three students
are gone?
If we describe the state of the Markov chain by the rates of the students that
are left, with ; to denote an empty o ce, then the Qmatrix is
123
12
13
23
1
2
3 123
6
0
0
0
0
0
0 12
3
3
0
0
0
0
0 13
2
0
4
0
0
0
0 23
1
0
0
5
0
0
0 1
0
2
3
0
1
0
0 2
0
1
0
3
0
2
0 3;
00
00
10
20
01
02
33 Letting R be the previous matrix with the last column deleted, the ﬁrst row of
R 1 is
1/6 1/6 1/12 1/30 7/12 2/15 1/20
The sum is 63/60, or 1 hour and 3 minutes. The ﬁrst term is the 1/6 hour until
the ﬁrst student leaves. The next three are
11
·
23 11
·
34 11
·
65 which are the probability we visit...
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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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