I have the following Markov chain and my goal is to find the expected convergence time from $A_1$ to any absorbing state $F_1,F_2,\dots$.
My attempt was to transform the chain into a lumped chain with the states $\{A,F,S_1,S_2,\dots\}$ with $A=\{A_1,\dots\}$, $F=\{F_1,\dots\}$,... . Since the initial state now is $A$, I thought I should only find the expected convergence time to $F$.
Now the lumped chain is a symmetric random walk ($p=q=0.5$) with a single absorbing boundary. I know that the chain will converge to the absorbing state, but I cannot find the expected convergence time since the state space is infinity. Is there a way that I can solve this?
Thanks!