 Hitting probabilities for a velocity-switching Markov ladder

On nonnegative levels, a chain moves right at rate $\lambda$ in state $C$, left at rate $\mu$ in state $W$, and switches $C\to W$ at rate $\alpha$ and $W\to C$ at rate $\beta$. The minimal nonnegative solution for hitting $W_0$ is identically one when $\mu \alpha \ge \lambda \beta$. When $\mu \alpha <\lambda \beta$, it is $h_{W_i}={\vartheta }^i$ and $h_{C_i}=b{\vartheta }^i$, where $b=\alpha (\mu +\beta )/[\beta (\lambda +\alpha )]<1$. These cases give recurrence and transience respectively.