Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 26J / d / Solution

Let $S_{+}=\{j:r_j>0\}$, nonempty since $R>0$. Every pair of distinct states in $S_{+}$ communicates directly, and no transition leaves $S_{+}$ for a zero-rate state. Hence $S_{+}$ is the unique closed communicating class. Each zero-rate state forms an open singleton: it has total departure rate $R>0$, goes into $S_{+}$, and can never be revisited. This includes the case that $S_{+}$ itself is a single absorbing state.