The invariant distribution for the large transition matrix of a Markov Chain?

Dear all,

I am kind of thinking how to calculate the invariant distribution for the
large transition matrix
of a Markov Chain.  The general Lumpability is not applicable here.
Actually, each column
in the transition matrix is just like a Binomial distribution with
probability P(i), where i is the
index of the column.   Can you give me some suggestion on how to calculate
the invariant
distribution efficiently?  I am more interested in the mean and the variance
of the states, any
simpler ways to calculate them?

Thanks a lot.

Kevin