In Example 12.13, at the Writing Center of a college, pk > 0 is the probability that a new computer needs to be replaced after k semesters. For a computer in use at the end of the nth semester, let Xnbe the number of additional semesters it remains functional. Then {Xn: n = 0, 1,...} is a Markov chain with transition probability matrix
Show that {Xn: n = 0, 1,...} is irreducible, recurrent, and aperiodic. Find the long-run probability that a computer selected randomly at the end of a semester will last at least k additional semesters.
Example 12.13
The computers in the Writing Center of a college are inspected at the end of each semester. If a computer needs minor repairs, it will be fixed. If the computer has crashed, it will be replaced with a new one. For k ≥ 0, let pk > 0 be the probability that a new computer needs to be replaced after k semesters. For a computer in use at the end of the nth semester, let Xnbe the number of additional semesters it will remain functional. Let Y be the lifetime, in semesters, of a new computer installed in the lab. Then
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