Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 2 / 26I / a / Solution

The arrows in the PDF give $C_i\to C_{i+1}$ at rate $\lambda$, $W_i\to W_{i-1}$ at rate $\mu$ for $i\ge 1$, and switches $C_i\to W_i$ and $W_i\to C_i$ at rates $\alpha$ and $\beta$. All rates are positive. One can travel upwards in level in the $C$ phase and downwards in the $W$ phase, switching at either end. Hence any state reaches any other through a finite positive-rate path: the continuous-time Markov chain is irreducible. Its total jump rates are uniformly bounded, so it is also nonexplosive.