Characteristic function of time spent of moving between two states in a CTMC

Question

I have a CTMC with S={0,1,2} and that has generator $$ G= \begin{bmatrix} -1 & 0 & 1 \\ 1 & -1 & 0 \\ 1 & 1 & -2 \\ \end{bmatrix} $$

I want to find the Characteristic function of the time spent moving between state 2 and 0.

I realise that the holding time will be exp(-$g_{ii}$)-distributed, but that is about it.

Maybe should I use the fact that CF of a rv X is an expectance, and try to use total law of expectance, but no luck.

Try to find the transistion probabilities and go from there?

Answer 1

Thought for a bit, and I have an answer! (Don't know if it's against the rules, but it might be helpful for someone else)

Looking at the jump chain one sees that $p_{21}$=$p_{20}$=1/2 and that $p_{10}$=1. Therefore, it is possible to see that CF(T)=E[$e^{jx*exp(2)}$] (1/2+(1/2)E[$e^{jx*exp(1)}$] ) ! (Since the we have to go from 1 to 0)

Happy to hear if someone has a better way to do this.