2. Two machines are running continuously, with their operating states described by two Markov chains. For each machine, the possible states are {1, 2, 3, 4, 5, 6}. The state of the first machine on day t is Xt. The state of the second machine on day t is Yt. The transition matrices for (Xo, X₁,...) and (Yo, Y₁,...) are P and Q respectively where: /1/3 2/3 0 0 0 10 0 1/3 4/9 2/9 0 0 0 0 1/3 0 1/3 1/3 P = ; 0 0 0 1/3 4/9 2/9 Q = 0 0 1 0 0 0 0 0 2/3 1/3 0 0 00 00 0 00 0 1/2 1/2 0 2/3 1/3 2/3 0 2/3 0 0 1/3 0 0 0 0 0 1/3/ 100 0 0 0 100 0 0 0 Supose that you (correctly) calculate that the vector w = (1/4 1/4 1/6 1/12 5/36 1/9) is both: ⚫ a limiting distribution for the Markov chain (Xo, X1, ...) ⚫ the unique equilibrium distribution but not a limiting distribution for the Markov chain (Yo, Y₁,...) (a) Using non-mathematical language, describe two things about the behaviour involving state 3 which apply to both machines. (b) Using non-mathematical language, describe one aspect of the behaviour of the first machine involving state 3 which does not apply to the second machine and explain the reason for this difference between the two machines.

2. Two machines are running continuously, with their operating states described by two Markov chains. For each machine, the possible states are {1, 2, 3, 4, 5, 6}. The state of the first machine on day t is Xt. The state of the second machine on day t is Yt. The transition matrices for (Xo, X₁,...) and (Yo, Y₁,...) are P and Q respectively where: /1/3 2/3 0 0 0 10 0 1/3 4/9 2/9 0 0 0 0 1/3 0 1/3 1/3 P = ; 0 0 0 1/3 4/9 2/9 Q = 0 0 1 0 0 0 0 0 2/3 1/3 0 0 00 00 0 00 0 1/2 1/2 0 2/3 1/3 2/3 0 2/3 0 0 1/3 0 0 0 0 0 1/3/ 100 0 0 0 100 0 0 0 Supose that you (correctly) calculate that the vector w = (1/4 1/4 1/6 1/12 5/36 1/9) is both: ⚫ a limiting distribution for the Markov chain (Xo, X1, ...) ⚫ the unique equilibrium distribution but not a limiting distribution for the Markov chain (Yo, Y₁,...) (a) Using non-mathematical language, describe two things about the behaviour involving state 3 which apply to both machines. (b) Using non-mathematical language, describe one aspect of the behaviour of the first machine involving state 3 which does not apply to the second machine and explain the reason for this difference between the two machines.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter2: Matrices

Section2.5: Markov Chain

Problem 47E: Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.

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