(A) Explanation Definition used: “Standard Forms for Absorbing Markov Chains: A transition matrix for an absorbing Markov chain is in standard form if the rows and columns are labelled so that all the absorbing states precede all the nonadsorbing states. (There may be more than one standard form.) Any standard form can always be partitioned into four submatrices: P = [ I 0 R Q ] where I is an identity matrix and 0 is a zero matrix. Theorem used: Limiting Matrices for Absorbing Markov Chains: If a standard form P for an absorbing Markov chain is partitioned as then P k approaches a limiting matrix P ¯ as k increases, where P ¯ = [ I 0 F R 0 ] The matrix F is given by F = ( I − Q ) − 1 and is called the fundamental matrix for P. The identity matrix used to form the fundamental matrix F must be the same size as the matrix Q ” . Calculation: According to the given data, the transition matrix P is: P = [ 0 0.50 0.25 0.25 0.50 0 0.50 0 0 0 1 0 0 0 0 1 ] In transition matrix row first, second, third, fourth represents states F, B, L, R respectively. Similarly, in transition matrix row first, second, third, fourth represents states F, B, L, R respectively where, rat is placed in room F or room B , also room L and R contains food. If a standard form P for an absorbing Markov chain is partitioned as P = [ I 0 R Q ] , Then P k approaches a limiting matrix P ¯ as k increases, where P ¯ = [ I 0 F R 0 ] . The matrix F is given by F = ( I − Q ) − 1 and is called the fundamental matrix for P. The identity matrix used to form the fundamental matrix F must be the same size as the matrix Q. The standard transition matrix P is: P = [ 1 0 0 0 0 1 0 0 0.25 0.25 0 0.50 0.50 0 0.50 0 ] In standard transition matrix row first, second, third, fourth represents states L, R, F, B respectively. Similarly, in transition matrix row first, second, third, fourth represents states L, R, F, B respectively. I = [ 1 0 0 1 ] 0 = [ 0 0 0 0 ] R = [ 0 (B) To determine The average number of exits that a rat placed in room B will choose until it finds food when the exit from room B to room R is blocked.

Pearson eText for College Mathematics for Business, Economics, Life Sciences, and Social Sciences -- Instant Access (Pearson+)

14th Edition

ISBN: 9780137553341

Author: Raymond Barnett, Michael Ziegler

Publisher: PEARSON+

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Chapter 15.3, Problem 68E

(A)

To determine

The long-run probability that a rat placed in room B ends up in room R when the exit from room B to room R is blocked.

(B)

To determine

The average number of exits that a rat placed in room B will choose until it finds food when the exit from room B to room R is blocked.

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Chapter 15.3, Problem 67E

Chapter 15, Problem 1RE

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Chapter 15 Solutions

Pearson eText for College Mathematics for Business, Economics, Life Sciences, and Social Sciences -- Instant Access (Pearson+)

Ch. 15.1 - Prob. 1MP Ch. 15.1 - Prob. 2MP Ch. 15.1 - Prob. 3MP Ch. 15.1 - Refer to Example 4. At the end of each year the...Ch. 15.1 - Prob. 1ED Ch. 15.1 - Prob. 2ED Ch. 15.1 - Prob. 1E Ch. 15.1 - Prob. 2E Ch. 15.1 - Prob. 3E Ch. 15.1 - Prob. 4E

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