At a given point in time, a station can be either in busy (B), idle (I) or sleep state (S). If the station is busy, then it has probabilities 0.5, 0.4, 0.1 to be in B, I or S states in the next step. If the station is currently idle, then it will transition to B, I or S state in the next step with probabilities 0.3, 0.4, 0.3. If the station is in sleep state, then it will be in B, I, or S states in the next step with probabilities 0.2, 0.3, 0.5. Let denote the station's state at time point n, then {, = 0,1, ... } is a three-state Markov chain (state 0 = B, state 1 = I, state 2 = S).

(a) Write down the transition probability matrix.

(b) If the station is currently busy, find the probability that it will be in sleep state after three steps.

(c) Is the Markov chain regular? State the reason for your answer.