Given a finite-state Markov chain $M$ and assume there is a terminate state $f$ which is reached with prob. 1, I am interested in the distribution of hitting time $T$ of $f$, namely, $T$ is defined as a RV which is the number of steps before reaching $f$, and $P(T\geq N)$ for natural number $N$ is the probability of reaching $f$ by at least $N$ steps. In particular, I want to obtain a nontrivial bound of $N$ such that $P(T\geq N)\leq \epsilon$, in terms of the transition probabilities of $M$ and $\epsilon$.

Is there any existing research on this problem, as it seems to be a very natural problem to consider.

Thanks.