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Unformatted text preview: IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2011, Professor Whitt Reversibility 1. Four Problems (1) The Knight Errant (The Random Knight) A knight is placed alone on one of the corner squares of a chessboard (having 8 × 8 = 64 squares). What is the expected total number of moves required for the knight to first return to its initial position, if we assume that the knight moves randomly, taking each of its legal moves in each step with equal probability? (2) A Big Closed Maze for Markov Mouse Suppose that the closed maze for Markov mouse is enlarged to be 10 × 20 instead of 3 × 3; i.e., it now has 10 × 20 = 400 rooms instead of 3 × 3 = 9 rooms, but still arranged in a rectangular fashion, with doors connecting neighboring rooms. Now there are 10 rows of rooms, with 20 rooms in each row. There are doors connecting neighboring rooms on each row. And there are doors connecting neighboring rooms on each column. Suppose that, just as before, the mouse moves randomly according to a Markov chain, moving to one of the available neighboring rooms on each move, with each of the available alternatives chosen with equal probability. Suppose that the mouse starts in Room 1 (in the upper lefthand corner). What is the expected total number of moves required for the mouse to first return to this initial room?...
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- Spring '11
- Operations Research, Probability theory, Markov chain, Random walk, total number, Πi, irreducible ﬁnite-state Markov