Unformatted text preview: answer, note that the weather cycles between sunny, smoggy,
and rainy spending independent exponentially distributed amounts of time with
means 3, 4, and 1, so the limiting fraction of time spent in each state is just
the mean time spent in that state over the mean cycle time, 8.
Example 4.11. Duke basketball. To “simulate” a basketball game we use
a four state Markov chain with four states
0 = Duke on o↵ense
1 = Duke scores 2 = UNC on o↵ense
3 = UNC scores and transition rate matrix
0
1
2
3 0
3
0
1
6 1
2
5
0
0 2
1
5
2.5
0 3
0
0
1.5
6 130 CHAPTER 4. CONTINUOUS TIME MARKOV CHAINS where the rates are per minute. To explain the rates:
The Duke o↵ense keeps the ball for an exponential amount of time with mean
1/3 minute, ending with a score with probability 2/3, and turning the ball
over to UNC with probability 1/3. After a score UNC needs an average of 1/5
minute to take the ball down the court and then they are on o↵ense.
The UNC o↵ense keeps the ball for an exponential amount of time with mean
2/5 minute, ending with a score with probability 0.6, and turning the ball over
to Duke with probability 0.4. After a score Duke needs an average of 1/6...
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 Spring '10
 DURRETT
 The Land

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