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It is common practice to have standby redundant units in mechanical and electrical systems so as to attain a high degree of reliability. Suppose two machines are available, one in use and one on a standby basis. The probability that a machine that is in use fails during a unit period is p(q=1-q). It takes three units of time to repair a failed machine. Define a process with the states identified by different combinations of two elements, number of machines in working condition; expended repair time. Thus, the states are 20, 10, 11, 12, 01, and 02. Show that this process is a Markov chain, and determine its transition probability matrix.

I am having difficulty in deciding how to define the states 0,1,2 because of the two elements requirements. Please help me! Thank you! :)

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$ij$ means that there are $i$ machines currently working and $j$ units of time have elapsed since the repair wait started.

  • 21 and 22 are impossible since if two machines are working, there is no waiting for repairs.

  • 00 is also impossible since if no machines are working, then first one failed and then the other (the second machine can't fail while it's not being used), so we are already waiting at least 1 unit of time for repair.

For instance, in state 11 we can go to 12 if all goes well, or 02 if the second machine also fails. In 02 we must go to 10 (assuming only one machine can be repaired at a time).

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