= Solution
In a time-homogeneous <continuous-time Markov chain>, a state's <holding time> is exponential with rate equal to its total outgoing <transition intensity>. Converting the specified times to months gives
$$
a+b=\frac1{96},\qquad \frac a{a+b}=\frac12,\qquad c=\frac1{36}.
$$
The exit-type <probability> follows by dividing its <transition intensity> by the total exit rate. Therefore
$$
\boxed{a=b=\frac1{192},\quad c=\frac1{36},\quad Q=\begin{pmatrix}-1/96&1/192&1/192\\0&-1/36&1/36\\0&0&0\end{pmatrix}\ \text{month}^{-1}.}
$$
These are starting values for numerical estimation, rather than further observations or constraints on the final fitted rates.
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