Let the initial state of a MC, $X_{0}=i$, be fixed.
Definition: $T=\left\{n \in \mathbb{Z^{+}}:X_{n}=i \right\}$ is the time of the first visit from i to i.
Definition: $\pi\left(i\right)$ is the expected amount of time a MC remains in a state i.
If a MC makes k=K number of visits to a state i, starting at state i, the expected time for one visit to a state i, starting at state i, is $\frac{1}{K} \sum_{\mathscr{k=1}}^{K}$= $\frac{T^{1}+ \cdot \cdot \cdot T^{K}}{K}=E[T]$
For K number of visits to a state i starting from state i, the amount of time spent by the MC is $K E[T]$.
Claim: In time n, the number of visits made by a MC from a state i to a state i is $n \pi\left(i\right)$ such that $n=K E[T]$.
How do I understand that $n \pi\left(i\right)$ is the number of visits made by a MC?
Any help is appreciated.
Edit @Michael:
There are about K visits to the state i in $kE[T]$ steps/ time of the MC. But, in n steps we expect about $n\pi\left(i\right)$ visits to the state i. Hence, setting $n=KE[T]$, we get $E[T]=\pi^{-1}\left(i\right)$