Suppose a cell can be in two types: type I or type II. A cell of type I can become a cell of type II after a time that is distributed exponentially with parameter $a$ and a cell of type II can become two cells of type I after a time that is distributed exponentially with parameter $b$.
How can I describe the population of the cells as a continuous Markov chain and what are the residence times and the transition probabilities?
What I know:
I find it very difficult to describe the population as you have to take account of both the number of I-cells and the number of II-cells.
Further I know that a transition rate matrix G is given by
$ g_{ij} = \left\{
\begin{array}{lr}
v_ip_{ij} & : i \neq j\\
-v_i & : i=j
\end{array}
\right.$
where the residence time in state $i$ is distributed exponentially with parameter $v_i$ and $p_{ij}$ are the transition probabilities.