I have a model like:
A B C
A 0.80 0.10 0.10
B 0.20 0.75 0.05
C 0.10 0.10 0.80
How do I get the average time from B to A? I understand that there's a formula:
$$ m_{ij}=1+\sum_{k\neq j}p_{ik}m_{kj} $$
But I'm not sure what to do with the recursion. Can k=i?
I started with
$ m_{BA} = 1 + p_{BC} m_{CA} $
$ m_{CA} = 1 + p_{CB} m_{BA} $
First of all: Yes, the summation is over all $k \neq j$; so in particular, we have to consider $k=i$ (whenever $i \neq j$).
For simplicity of notation, let's identify
$$A \longleftrightarrow 1 \qquad B \longleftrightarrow 2 \qquad C \longleftrightarrow 3.$$
Then the mentioned formula gives
$$m_{21} = 1+ p_{22} m_{21} + p_{23} m_{31} = 1+ \frac{3}{4} m_{21} + \frac{1}{20} m_{31} \tag{1}$$
and
$$m_{31} = 1+ p_{32} m_{21} + p_{33} m_{31} = 1+ \frac{1}{10} m_{21} + \frac{4}{5} m_{31} \tag{2}$$
Equation $(1)$ is equivalent to
$$\frac{1}{4} m_{21} = 1+ \frac{1}{20} m_{31} \tag{1'}$$
and equation $(2)$ is equivalent to
$$\frac{1}{5} m_{31} = 1+ \frac{1}{10} m_{21} \tag{2'}$$
i.e. we have to solve a system of linear equations (2 equations, 2 unknowns). It is not difficult to see that
$$m_{21} = \frac{50}{9} \qquad \quad m_{31} = \frac{70}{9}.$$