= Solution
For this <alternating arrival and service queue>, set $\theta=\lambda(\mu+\beta)/[\mu(\lambda+\alpha)]$. Direct multiplication gives $(\theta,\lambda/\mu)B=\theta(\theta,\lambda/\mu)$, so the initial row is a left <eigenvector> and $(\pi_{iC},\pi_{iW})=\theta^{i-1}(\pi_{1C},\pi_{1W})$. Consequently
$$
\pi_{iC}=\frac{\beta z}{\lambda+\alpha}\theta^i\quad(i\geq0),\qquad
\pi_{iW}=\frac{\beta\lambda z}{\mu(\lambda+\alpha)}\theta^{i-1}\quad(i\geq1),\qquad\pi_{0W}=z.
$$
These nonnegative probabilities can be normalized exactly when $\theta<1$, equivalently $\mu\alpha>\lambda\beta$. Summing the <geometric series> gives
$$
\boxed{z=\frac{\mu\alpha-\lambda\beta}{\mu(\alpha+\beta)},\qquad\text{positive recurrence}\iff\mu\alpha>\lambda\beta.}
$$
When $\theta\geq1$ no positive multiple of the invariant sequence is summable, so no <stationary distribution> exists. This rules out positive recurrence; absence of a stationary probability alone should not be used to distinguish null recurrence from transience.
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