The question is given as: Alice has 3 books, numbered 1 to 3, and picks a book according to a probability distribution q, where q(k) > 0 for any k ∈ {1, 2, 3}. Analyse the long-term behaviour of this chain; if at the end of the day she places it to the right-most position.
I'm not 100% Im doing right because it seems like its not going to any neat solution. But my thought is maybe solving:
$Π123 = a3(Π123 + Π132 + Π312)$
$Π213 = a3(Π321 + Π231 + Π213)$
$Π132 = a2(Π123 + Π132 + Π213)$
$Π312 = a2(Π312 + Π213 + Π321)$
$Π231 = a1(Π123 + Π213 + Π231)$
$Π321 = a1(Π132 + Π312 + Π321)$
$1 = Π123 + Π132 + Π312 + Π213 + Π231 + Π321$
Ie. that with a probability $a3$ she picks book 3 (we don't know the actual probabilities) and if the books are in the order $123$ or $132$ or $312$ originally then when she places the 3rd book in the right-most position she will end up with the order $123$. I did that for the rest of the options, but when I try to solve the simultaneous equations I get negative steady states, and I cant work out where my equations are incorrect.