I am interested in the properties of a particle moving in a grid of cells.
I have a grid of cells modeled as a weighted undirected graph, such that every cell has eight neighbors. Some of the cells in the grid are labeled as termination cells such that if a particle is in that cell it is stuck/terminated.
I am then creating an Absorbing Markov Chain from this graph with the non-termination cells being transient states (t) and the termination cells absorption states (r).
What I am initially interested in is the expected time to termination for starting that random process from each cell.
According to this I make the matrix
$$P=\begin{pmatrix}Q & R\\ 0 & I_r\end{pmatrix}$$
where $Q$ is a t-by-t matrix, $R$ is a nonzero t-by-r matrix, $0$ is an r-by-t zero matrix, and $I_r$ is the r-by-r identity matrix. Thus, $Q$ describes the probability of transitioning from some transient state to another while $R$ describes the probability of transitioning from some transient state to some absorbing state.
I create the transition probabilities based on the weights in the graph. Since this is a large grid (at least 200x200) I have a P matrix around 40000x40000, thus to calculate the expected time to termination I solve $$(I - Q)x = 1$$ for $x$ where $1$ is a vector of all ones. Which is computationally faster than $$x = (I-Q)^{-1}1$$
What I would like to know is if I were to make on of the transient states an absorption state, which one should I pick to reduce the mean of the expected time to termination vector $x$? $$\frac{1}{t}\sum x$$
I am assuming this is possible given the graph and $x$ but maybe it is not?