Consider there is an absorbing DTMC $X_n$ and it has $m$ absorbing states and $n$ nonabsorbing states.
My question is if the $X_n$ starts from a transient state $s$ and ends in a specific absorbing state $t$, what is the distribution of the time $X_n$ visit arc $i-j$, $N_{s,ij,t}$? If we dont demand this $X_n$ ends in $t$, what is the distribution of $N_{s,ij}$?
I want to know the $Prob\left\{ N_{s,ij,t}>l \right\} $ and the $Mean\left\{ N_{s,ij,t} \right\} $, I found the $Mean\left\{ N_{s,ij,t} \right\} $ can be derived by absorbing DTMC with reward, but it can not give us any information of its distribution.
I thought there will be a classic conclusion of this problem, but I was astonished I did not find any conclusion or result of arcs from Markov textbook. I will be very glad if there is anyone can give me any references or the result of this problem.