Transformation to achieve unit transition rate in a continuous time Markov chain

Question

I have a continuous time Markov chain (CTMC) defined by a transition matrix $P$ and where all transition times go as a exponential random variables with transition rate $\gamma$. I would like to transform the matrix $P$ some how so that my process now has transition rate $1$ and the same dynamics. Does anyone know how to do this?

Answer 1

The transition matrices of a CTMC are $P(t)=\mathrm e^{tQ}$. To multiply the transition rates by a ratio $r$, define some new transition matrices by $\bar P(t)=\mathrm e^{rtQ}=P(rt)$. In your case, $\bar P(t)=P(t/\gamma)$.