  We consider a simple but important class of metastable discrete time Markov
chains, which we call perturbed Markov chains. Basically, we assume that the
transition matrices depend on a parameter $\varepsilon$, and converge as
$\varepsilon$. We further assume that the chain is irreducible for
$\varepsilon$ but may have several essential communicating classes when
$\varepsilon$. This leads to metastable behavior, possibly on multiple time
scales. For each of the relevant time scales, we derive two effective chains.
The first one describes the (possibly irreversible) metastable dynamics, while
the second one is reversible and describes metastable escape probabilities.
Closed probabilistic expressions are given for the asymptotic transition
probabilities of these chains, but we also show how to compute them in a fast
and numerically stable way. As a consequence, we obtain efficient algorithms
for computing the committor function and the limiting stationary distribution.
