We prove that the long term distribution of the queue length process in an ergodic generalised Jackson network obeys the Large Deviation Principle with a deviation function given by the quasipotential. The latter is related to the unique long term idempotent distribution, which is also a stationary idempotent distribution, of the large deviation limit of the queue length process. The proof draws on developments in queueing network stability and idempotent probability.
"Large deviations of the long term distribution of a non Markov process." Electron. Commun. Probab. 24 1 - 11, 2019. https://doi.org/10.1214/19-ECP243