On large queue lengths in generalised Jackson networks
Anatolii A. Puhalskii
TL;DR
The work addresses the problem of characterizing rare events for queue lengths in subcritical generalised Jackson networks by establishing a rigorous large deviations principle (LDP) for the stationary distribution. It develops a trajectorial LDP framework for delayed renewal processes, proves exponential tightness, and identifies the LD limit as the time limit of an idempotent distribution, culminating in a quasipotential-based rate function V(x). The main result states that the normalized stationary queue-length vector hat Q / n satisfies an LDP with rate n and deviation function V, which is defined via an optimization over admissible trajectories. This provides a rigorous, principled description of tail behavior in complex, multi-station networks and supports performance assessments under rare-event scenarios.
Abstract
This paper proves a large deviation principle (LDP) for the stationary distribution of queue lengths in a subcritical generalised Jackson network assuming a Cramer condition on the interarrival and service times. The deviation function is given by the quasipotential.