Robots have been programmed to traverse the maze shown in the figure and at each

Question

Robots have been programmed to traverse the maze shown in the figure and at each junction randomly choose which way to go. Construct the transition matrix for the Markov chain that models this situation. This is a 4 times 4 matrix whose entry in the (i, j) position is the probability a robot moves from position v_j to position v_i (each column is labeled by one vertex position). For example, the entry in the (2, 3) position should be 1/4, since there is a 1 in 4 chances that a robot located at v_3 moves towards v_2. Suppose we start with 15 robots at each junction. Find the steady state distribution of robots. Assume it takes each robot the same amount of time to travel between two adjacent junctions.

Explanation / Answer

transition matrix

1 1/3 1/4 0

1/2 1 1/4 1/3

1/2 1/3 1 1/3

0 1/3 1/4 1

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