SO this question is probably really easy, I am just struggling in understanding how to do it
It goes like this: we have a system with 3 components, at time $t=0$, component 1 is active and the other two are not. When 1 fails then 2 becomes active, when 2 fails then 3 becomes active. When 3 fails than all the parts are replaced and 1 is active again. Let $T_k$ be the lifetime of component $k$, then $\quad T_k\sim Exp(\lambda_k), \quad$ and $\quad \forall j,i \quad$ $T_j$ and $T_i$ are independent. Let the number of the component that is active be the state number, denoted by $X_t$ where $t$ is time. Then write Kolmogorov backward equations.
Now I am not sure how to write $X_t$ in general as $T_k$ is exponential so naturally I don't know to get Kolmogorov's equation. (All I know is discrete time markov chains, I am tying to study ahead on continuous time markov chains)
All that I know is that (sorry I don't know how to properly draw circles and such in mathjax) $$(1) \rightarrow^{\lambda_1} (2)\rightarrow^{\lambda_2} (3)\rightarrow^{\lambda_3} (1)$$ and that $Pr(X_0=1)=1$. As well as that the generating matrix is $$ G= \begin{bmatrix} -\lambda_1 & \lambda_1 & 0 \\ 0 & -\lambda_2 & \lambda_2 \\ \lambda_3 & 0 & -\lambda_3 \\ \end{bmatrix} $$
So I was able to answer this myself, unless someone can explain better than me, I will have my answer being selected here.
Since I know the Generating matrix is: $$ G= \begin{bmatrix} -\lambda_1 & \lambda_1 & 0 \\ 0 & -\lambda_2 & \lambda_2 \\ \lambda_3 & 0 & -\lambda_3 \\ \end{bmatrix} $$ And Kolmogorov backward equations is given by $ \frac{dp_ij(t)}{dt} = \sum_{k=0}^\infty v_{ik}p_{kj} $.
Where $p_{ij}(t)$ is the probability of moving from state i to j at time t. And $v_{ij}$ is the infinitesimal parameter (which is given by the Generating matrix) of jump i to j.
You can show that: $$ \forall j, \quad \frac{dp_{1j}(t)}{dt} = -\lambda_1 p_{1j} + \lambda_1 p_{2j} \\ \ \quad \frac{dp_{2j}(t)}{dt} = -\lambda_2 p_{2j} + \lambda_2 p_{3j} \\ \ \quad \frac{dp_{3j}(t)}{dt} = \lambda_3 p_{1j} - \lambda_3 p_{3j} $$ Kolmogorov backward equations^ for all i,j .