I'm trying to model an irreversible, discrete Markov process. I have a set of states $S$ arranged in a tree-like structure (it is only possible to move from parent vertex to child vertex). I compute the transition rates of parent-to-child arcs (the transition rates are assumed to parametrize exponential random variables characterizing the time to transition), and I aggregate the transition rates in an upper-triangular, weighted Laplacian matrix $L$.
From what I understand, the weighted Laplacian $L$ is the transition rate matrix (infinitesimal generator) describing the rate a continuous Markov chain will transition between states. Based on this question, to simulate from the transition rate matrix, it seems that I need the transition probability matrix (a stochastic matrix).
How might I simulate trajectories from the transition rate matrix given in $L$? Since my process is irreversible, must I take into account the exponential distribution of transition times, or is there an alternative way to "convert" $L$ to a transition probability matrix?