Stationary distribution of an M-M-s-K queue
= Stationary distribution of an M-M-s-K queue
Put $\rho=\lambda/\mu$. The unique stationary distribution of an $M/M/s/K$ queue is
$$
\pi_n=\frac1Z
\begin{cases}
\rho^n/n!,&0\leq n\leq s,\\
\rho^n/(s!s^{n-s}),&s<n\leq K,
\end{cases}
$$
where $Z$ is the sum of the displayed weights. Adjacent terms satisfy <detailed balance for a birth-death process>, so the queue is reversible in equilibrium.