Finding absorption time of a markov chain

Question

I have the following problem.

Consider $N$ cells in a line. The first one is set to $1$ and the rest are set to $0$. At each step one of the elements that is $0$ is randomly picked. If it is adjacent to one that has a $1$, then it is set to $1$; otherwise it remains $0$. The process is repeated until all cells are filled. Draw the possible states and the transition rates. What is the expected absorption time?

I think the transition probabilities are pretty easy to figure out, but how do you find the expected absorption time? If it was a numeric problem I know I could do that using the fundamental matrix. But I am not sure how you can do that here. I also tried to derive a difference equation but didn't really work out

Any ideas will be appreciated

Thanks

Answer 1

Let $t_i$ denote the absorption time if we start from state $i$.

There is only one absorption state $M$.

Hence $$t_M=0.$$

For $i \neq M$, by law of total expectation,

$$t_i = \sum_{k=0}^Np_{ik}t_k+1$$

Now we just have to solve a linear system to compute the absorption time.