Expected number of visits to state in Markov chain 
Consider a Markov chain on the states {$1, 2, 3$} with transition matrix $$P = \begin{pmatrix} 
.7 & .2 & .1\\
.4 & .4 & .2\\
.6 & .3 & .1\\
\end{pmatrix}$$
  This Markov chain is used to predict the magnitude of the next volcanic eruption, based on the magnitude of the last one. It is estimated that in an eruption of level 1 a volume of $79 m^3$ of lava is ejected, in an eruption of level 2 a volume of $316 m^3$ of lava is ejected, and in an eruption of level 3 a volume of $1580 m^3$ of lava is ejected.
(a) Estimate the average amount of lava thrown out during each eruption in a cycle of 300 consecutive eruptions.
(b) What is the expected number of eruptions between two eruptions of level 3?

I'm stuck on part (a). Surely we can find the expected number of steps to go from any one state to another. However, I'm unsure how to apply this information here since we're not given a starting state. Any push in the right direction is appreciated!
 A: (a) After a few iterations the occupation probabilities will quickly converge to the fixed point of the iteration $\pi_{n+1}=W\pi_n$, where $W=P^T$ is the transition matrix transpose.
Eventually, according to the Perron-Frobenius theorem, the fixed point will be reached, 
which obeys $W\pi_\infty =\pi_\infty$ ($1$).
The other two e-values have module less than 1, so whatever the initial state the final state will be the same.
Calling $\pi_\infty = (x\, y\, z)^T$, you can solve for the e-value equation ($1$) along with the normalization condition $x+y+z=1$. The results
 I found were
$\pi_\infty=(48/79\, 21/79\, 10/79)^T$.
Hence, the approximate volume of lava produced by eruption is simply
$\overline{V}= \pi_1(\infty)V_1+\pi_2(\infty)V_2+\pi_3(\infty)V_3=332 m^3$ (the lava volumes are multiples of 79)
(b) To calculate the expected number of eruptions between two eruptions of level $3$, we write down the iteration equations for the populations of levels $1$ and $2$. The level $3$ does not enter this Markov chain, since when the level $3$ is reached the iterations stop. The iteration equations are
\begin{align}
\pi_1(n+1) &= p_{11}\pi_1(n)+p_{21}\pi_2(n),\\
\pi_2(n+1) &= p_{12}\pi_1(n)+p_{22}\pi_2(n),
\end{align} 
where $p_{ij}$ are elements of the transition matrix $P$.
In matrix form we have
\begin{align}
\begin{pmatrix}
\pi_1(n+1)\\
\pi_2(n+1)
\end{pmatrix}
&=
M
\begin{pmatrix}
\pi_1(n)\\
\pi_2(n)
\end{pmatrix}=
M^n
\begin{pmatrix}
\pi_1(1)\\
\pi_2(1)
\end{pmatrix}
\qquad(1)
\end{align}
where
\[
M=\begin{pmatrix}
p_{11} &p_{21}\\
p_{12} & p_{22}
\end{pmatrix}.
\]
The initial occupation probabilities are
$$
\begin{pmatrix}
\pi_1(1)\\
\pi_2(1)
\end{pmatrix}
=
\begin{pmatrix}
p_{31}\\
p_{32}
\end{pmatrix}
$$
The average number of eruptions between two level $3$ eruptions is
$$\bar n=\sum_{n=1}^\infty n\mathcal{P}_n,$$
where the probability distribution is $\mathcal{P}_n=\pi_1(n)p_{13}+\pi_2(n)p_{23}$.
From Eq. (1) we obtain the occupation probabilities
$$
\pi(n)= \left(a\lambda_-^{n-1}+b\lambda_+^{n-1}\right)\pi(1),
$$
where $\lambda_\pm$ are the e-values of $M$, which are given by
\[
\lambda_\pm=\dfrac{-Tr M\pm\sqrt{Tr M^2-4\det M}}{2}.
\]
Hence, we find
\[
\mathcal{P}_n=\left(a\lambda_1^{n-1}+b\lambda_2^{n-1}\right)(p_{31}p_{13}+p_{32}p_{23})
\]
