simulating a discrete markov process from a reducible transition rate matrix

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?

The simplest way is this (it does not require the structure of a tree). Assume: \begin{align} S &= \mbox{State space}\\ (q_{ij}) &= \mbox{transition rates} \\ v_i &= \sum_{j \in S: j \neq i} q_{ij} \end{align}
• When you enter a new state $$i$$, independently generate an exponentially distributed random variable $$T_i$$ with parameter $$v_i$$. The time $$T_i$$ is the time you stay in state $$i$$.
• After time $$T_i$$, independently transition to a new state $$j \in S$$ according to transition probabilities: $$P_{ij} = \left\{\begin{array}{ll} \frac{q_{ij}}{v_i} \quad & \mbox{ if j \neq i}\\ 0 & \mbox{ if i=j}\end{array}\right.$$ Notice that these probabilities indeed sum to 1: $$\sum_{j\in S}P_{ij} = 1$$.
• A related method of "uniformization" allows for self-transitions with rates $q_{ii}\geq 0$ and treats $v_i=v$ for all $i \in S$ (constant) where $q_{ii}$ is simply chosen to impose this constant condition. The advantage of such uniformization is that if you skip the generation of the exponential variables $T_i$ and just stay in each state for one unit of time, then the steady state in this "embedded discrete time chain" is the same as the true steady state. – Michael May 20 at 21:06