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 process. The proof draws on developments in queueing network stability and idempotent probability.
Highlights
Introduction and summaryIn a seminal contribution, Freidlin and Wentzell [5] obtained the Large Deviation Principle (LDP) for the stationary distribution of a diffusion process and showed that the deviation function, which is often referred to as the action functional or the rate function, is given by the quasipotential
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 a stationary idempotent distribution, of the large deviation limit of the queue length process
Summary
Freidlin and Wentzell [5] obtained the Large Deviation Principle (LDP) for the stationary distribution of a diffusion process and showed that the deviation function, which is often referred to as the action functional or the (tight) rate function, is given by the quasipotential. Their ingenious analysis relied heavily on the strong Markov property and involved an intricate study of attainment times. Geometric ergodicity of the queue length process enables us to conclude that the long term idempotent distribution is the large deviation limit of the long term queue length distributions
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