# Large deviations of the long term distribution of a non Markov process

## TL;DR

This paper establishes a large deviation principle for the long-term queue length distribution in ergodic generalized Jackson networks, linking it to the quasipotential and idempotent probability theory.

## Contribution

It introduces a novel connection between large deviations, quasipotential, and idempotent distributions in queueing networks.

## Key findings

- Long-term queue distribution obeys the Large Deviation Principle.
- The deviation function is given by the quasipotential.
- The quasipotential relates to the unique long-term idempotent distribution.

## Abstract

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 processes. The proof draws on developments in queueing network stability and idempotent probability.