M-M-s-K queue ($M/M/s/K$)

An $M/M/s/K$ queue has arrivals forming a Poisson process of rate $\lambda$, $s$ servers with independent service times having an exponential distribution of rate $\mu$, and total system capacity $K\ge s$. Its occupancy is the finite birth-death process on $\{0,\dots ,K\}$ with

Put $\rho =\lambda /\mu$. The unique stationary distribution of an $M/M/s/K$ queue iswhere $Z$ is the sum of the displayed weights. Adjacent terms satisfy detailed balance for a birth-death process, so the queue is reversible in equilibrium.