= Alternating arrival and service queue
{title2=$\theta=\lambda(\mu+\beta)/[\mu(\lambda+\alpha)]$}
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 $\lambda\beta/(\alpha+\beta)$.
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