We view a $\bullet/M/K$ node having $K$ exponential servers of service rate $\mu$ as a map on the space of stationary ergodic arrival processes of rate $\lambda, \lambda < K\mu$. It is well known that the Poisson process of rate $\lambda$ is a fixed point of this map. We prove there is no other fixed point.
"Uniqueness of Stationary Ergodic Fixed Point for $A \cdot / M/ K Node$." Ann. Appl. Probab. 3 (1) 154 - 172, February, 1993. https://doi.org/10.1214/aoap/1177005512