I am given a transition matrix equal to
$$P = \begin{bmatrix} 0 & 0 & 0 & 3/4 & 0 & 1/4 \\ 1/2 & 0 & 1/2 & 0 & 0 & 0 \\ 0 & 2/3 & 0 & 0 & 0 & 1/3 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix}$$
I am asked what $r_{30}$, $r_{33}$, and $E[N_0|X_0=3]$.
In general, I am confused what $r_{ij}$ really means in the context of a Markov Chain. I understand it is the probability that, starting in state i, the chain will eventually hit j at some time. I also understand that the calculation is $r_{ij} = \sum_{m=1}^\infty P(T_j=m|X_0=i)$ where $T_j$ is the hitting time of state j; however, I am really unsure how to calculate this.
I believe $E[N_0|X_0=3]$ is equal to 0 since state 0 is recurrent (<- please correct me if I am wrong: state 0 is recurrent because, starting at state 0, you can go to state 3 with a probability of 3/4 and from state 3 go to state 5 with a probability of 1 and from state 5 back to state 0 with a probability of 1 OR go from state 0 to state 5 with a probability of 1/4 and back to state 0 with a probability of 1) and I am guessing $r_{30}$ is equal to 0, therefore, $E[N_0|X_0=3]$ is equal to 0.
We do not technically have an assigned book for class; however, we do have excerpts from Ross's "Probability Models" (specifically chapter 4) if that helps at all.
The substochastic matrix obtained by considering only transition between states in $S:=\{0,3,5\}$ is $$ Q=\begin{pmatrix} 0&\frac34&\frac14\\ 0&0&1\\ 1&0&0 \end{pmatrix}. $$ Since the rows of $Q$ each sum to $1$, we see that this is a closed class of recurrent states; that is, conditioned on $\{X_n\in S\}$ for some nonnegative integer $n$ we have $\mathbb P(X_k\in S)=1$ for all $k>n$. It is clear then that $r_{ij}=1$ for $i,j\in S$.
Let $N_0=\inf\{n>0:X_n=0\}$. Since $P_{35}=P_{50}=1$, it follows immediately that $\mathbb P(N_0=2\mid X_0=3)=1$ and hence $\mathbb E[N_0\mid X_0=3]=2$.