Absorbing Markov Chain

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This abstract example of an absorbing Markov chain provides three basic measurements: The fundamental matrix is the mean number of times the process is in state given that it started in state . The absorption probability matrix shows the probability of each transient state being absorbed by the two absorption states, 1 and 7. The mean time for each transient state to be absorbed is shown in the absorption time matrix. The graph shows all the transient states, 2, 3, 4, 5, 6, and 8, and the two absorbing states, 1 and 7, together with all the probabilities allowed between the states.

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You can vary the probability of the transition from 2 to 1 with the slider. Because the sum of the probabilities is 1, for each state's total transitions, the transition probability for the transition from 2 to 3 is internally computed to sum to 1 when you drag the slider.

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Contributed by: Conrad A. Benulis (March 2011)
Open content licensed under CC BY-NC-SA

Details

The absorbing Markov chain shown above the original caption was designed with some simple symmetry as can be seen by observing the patterns in the original matrices. State 8 shows an isolated state that is immediately absorbed after 1 clock cycle. The extremes of the slider affect the fundamental matrix and the absorption of most of the other states. Although it can be discarded after extractions, the full probability state matrix in canonical form is provided in the program. The methods used to calculate all fundamental and absorption matrices are done using standard industry methods.