# Generate PDF from CTMC

I have an irreducible continuous-time Markov chain (CTMC) with a finite state space. The CTMC also does not have any one-step transitions from any state to itself. I have the transition rate matrix $$Q$$, so I can solve for the limiting probabilities (null space of $$Q^T$$ - computed via the SVD) and stationary transition probabilities (matrix exponential $$e^{Qt}$$).

Given some time interval $$\Delta t$$, how do I determine the probability density function (PDF) for the number of times state $$i$$ has a one-step transition to state $$j$$ in such an interval? I'm looking for something like "state $$i$$ goes to state $$j$$ 3 times with probability 0.6 and 4 times with probability 0.4 in time interval $$\Delta t$$." Do I need any other information?

note: My matrix $$Q$$ is very large (over three million states), so I would like to avoid computing a matrix exponential. The matrix is also very sparse - each row only have five non-zeros elements, but I don't think that would help me too much in computing a matrix exponential.