Figure 1

(a) Part of an $l$-state semi-Markov chain with nearest neighbor transitions and trapping. The dynamics are governed by the directional waiting time PDFs, ${\psi }_{n\pm \ln }(t)={\omega }_{n\pm \ln }{\phi }_{n\pm \ln }(t)$ and ${\psi }_{Nn}(t)={\omega }_{Nn}{\phi }_{Nn}(t)$. A way to simulate such a system is to first draw a random number out of a uniform distribution, which determines the direction according to the transition probabilities. Once the direction is set, say, from state $n$ to state $j$, a time is drawn out of ${\phi }_{jn}(t)$, randomly and independently. This Gillespie kind of algorithm produces an uncorrelated waiting time $l$-state trajectory. (b) Part of a stochastic trajectory, generated from the algorithm above, with $l=3$, ${\omega }_{21}={\omega }_{23}=1$, ${\omega }_{12}={\omega }_{32}=1/2$, and ${\phi }_{jn}={\lambda }^{2}t{e}^{-\lambda t}$. $t^{\prime }{}_{-}$ and $t^{\prime }{}_{+}$ are, respectively, the backward and forward recurrence times when occupying state 2 at time $t^{\prime }$ and the next transition is to state 1. We call the corresponding waiting time a “junction waiting time.”Reuse & Permissions