Alternating arrival and service queue ($\theta =\lambda (\mu +\beta )/[\mu (\lambda +\alpha )]$)

A continuous-time Markov chain may alternate between an arrival phase $C$, with upward transitions at rate $\lambda$, and a service phase $W$, with downward transitions at rate $\mu$ when the level is positive. Switching rates $C\to W$ and $W\to C$ are $\alpha ,\beta$. Its stationary balance equations give a geometric invariant tail with ratio $\theta =\lambda (\mu +\beta )/[\mu (\lambda +\alpha )]$. A stationary probability exists exactly when $\mu \alpha >\lambda \beta$, consistent with the stationary average service rate $\mu \alpha /(\alpha +\beta )$ exceeding the arrival rate $\lambda \beta /(\alpha +\beta )$.