I have a (time-homogeneous, discrete-time) Markov chain with $K+1$ states $\{0,1,\ldots,K\}$. The last state $K$ is an absorbing state, all other states are transient states. Furthermore, from each state $i$ only transitions into state $i$ itself or "higher" states $i+j$ are possible. All entries in the upper triangle (including the diagonal) are strictly positive, and I have an exponential generating function for them, but the resulting expressions for the transition probabilities are clumsy, involving sums of binomials and the like, and so I won't give them here. The system will usually start in state 0. When $\mathbf{P}$ is the transition matrix of the full system and $\mathbf{P'}$ is the sub-matrix of $\mathbf{P}$ which one gets after removing the absorbing state $K$, then it is well-known that the vector $\mathbf{k}$ of the average absorption times is a solution of $ \mathbf{k} = (\mathbf{I}-\mathbf{P'})^{-1}\cdot\mathbf{1} = \left(\sum_{n=0}^\infty\mathbf{P'}\right)\cdot\mathbf{1} $ where $\mathbf{1}$ is a vector of all ones.
I would like to get some good estimates of the average absorption time when starting in state 0, in particular for growing $K$. In my particular system the row-sum norm (which is the matrix infinity-norm, i believe) comes extremely close to one for growing $K$, and estimates based on that (calculating geometric series based on the row-sum norm) differ wildly from numerical estimates. Similarly, some of the diagonal elements of $\mathbf{P'}$ get very close to one as well, so any norm-based approach will suffer from similar problems.
My question is: has this particular type (upper-triangular, one absorbing state) of Markov chain been analysed before and are there any methods giving good bounds on the average absorption time?