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# Markov Chain and Steady State Probabilities

1. A production process contains a machine that deteriorates rapidly in both quantity and output under heavy usage, so it is inspected at the end of each day. Immediately after inspection the condition of the machine is noted and classified into one of four possible states:

State
0: good as new
1: Operable - minimum deterioration
2: Operable - major deterioration
3: Inoperable and replaced by a good new machine

The process can be modeled as a Markov Chain with the following transition matrix:

State...............0................1................2...............3
0.....................0................7/8............1/16..........1/16
1.....................0................3/4.............1/8...........1/8
2.....................0.................0...............1/2............1/2
3.....................1.................0................0...............0

a) find the steady state probabilities of being in each state
b) if the daily costs of being in states 0, 1, 2, 3 are \$0, \$1000, \$3000, \$6000 respectively, what is the expected long run daily cost?
c) reclassify state 3 as an absorbing state and calculate the expected number of days for a 'good as new' machine to become inoperable.

#### Solution Summary

The steady state probability is calculated from the transition matrix.

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