# Ind eng 173 practice first midterm 3 a certain

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IND ENG 173, Practice First Midterm 3. A certain computer can crash on any day with probability p (0 , 1), independently of previous days. If it crashes, it gets sent at the end of the day to receive maintenance. If the computer goes m days without crashing it will be sent for maintenance at the end of that day regardless of whether it crashed during the day or not. Let X n denote the number of days that the computer has operated without maintenance at the beginning of its n th day of operation; X n ∈ { 0 , 1 , 2 , . . . , m - 1 } . The state 0 corresponds to the machine just having received maintenance. (a) [10 pts] Write down the one-step transition probabilities (in terms of p and m ). (b) [10 pts] Write down a system of linear equations (in terms of p and m ) that when solved will give the stationary probabilities. Do not solve it.
IND ENG 173, Practice First Midterm 5 (c) [15 pts] Solve the system of equations from part (a) and compute ( π 0 , π 1 , π 2 , . . . , π m - 1 ).