2 consider two machines that are maintained by a single...

2. Consider two machines that are maintained by a single repairman. Machine i functions for an exponentially distributed amount of time with rate $\lambda_i$ before it fails. The repair times for each unit are exponential with rate $\mu_i$. If both machines have failed, they are repaired in the order in which they failed. Formulate a Markov chain model for this situation with state space {0,1,2, 12, 21}. (Here 0 means that both machine are working, 1 or 2 means that the appropriate machine is not working and 12 and 21 mean that both machines are not working, with the first number showing which failed first.) Give the arrow diagram with the jump rates.

Transcript

00:02 For part 8, we first draw the arrow diagram.

00:07 Here with the state, state has five states, which is 0, 1, 2, 1 2, 1 2, and 2, and 2 1.

00:28 So from 0, we can go to 1 and 2, which means that the machine 1 or machine 2.

00:38 Failed so this has rate lambda 1 and here it has rate lambda 1 and here it has rate lambda 2 now starting from 1 we can go back to 0 which means that the machine 1 is repaired we can go to 1 2 which means that machine 2 failed before machine 1 is repaired.

01:24 So going to 0 it has rate, mu 1, and go into 1 2 it has rate lambda 2.

01:37 So similarly from 2 we have rate mu 2, going to 0 and the rate lambda 1, going to 2 1, going to 2 1.

01:57 Starting from 1 -2 because 1 failed before 2 fell so the repairman is working on repairing machine 1.

02:15 The next step is machine 1 is repair but machine 2 is still down so it can go to 2.

02:26 And the rate here it is mu1.

02:38 Similarly, we have this arrow from 2 1 and the rate is mu2 here.

02:49 Now we have finished the arrow diagram.

02:54 Next is the generator matrix.

02:59 So we first list the states we have 0.

03:05 1, 2, 1 2, 1 2, 1.

03:10 2, 1.

03:12 And here we also have 0, 1, 2, 1 2, 2, 2, 1.

03:19 So we have a 5 by 5 matrix.

03:28 Each row means a starting point and a, each column means an endpoint.

03:35 So starting from 0, we can go to 1 or 2.

03:41 With rated lambda 1, lambda 2 respectively.

03:45 So we put the lambda 1, lambda 2 here, and here it is 0.

03:52 Each row sums to 0.

03:57 So here we put negative lambda 1 plus lambda 2.

04:12 Next, starting from 1, we can go to 0 or 1 2.

04:20 So go to 0, it has rate mu 1.

04:23 Go to 1 2 it has rate lambda 2 and this is 0 0 and here it is negative mu 1 plus lambda 2 now starting from 2 we can go to 0 or 2 1 go to 0 it has rate mu 2 going to 21 it has rate lambda 1 and so here is 0 here is 0 and here it is negative mu 2 plus lambda 1 starting from 1 2 we can only go to 2 with rate mu 1 so mu 1 here and negative mu 1 here and 0 0 0 here starting from 2 -1, we can only go to 1 with rate mu -2...

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