Phase transitions and departure statistics of critically loaded queues: oscillating cumulants and generalized BRAVO

TL;DR

This paper investigates non-equilibrium phase transitions in queues, revealing oscillating departure cumulants near critical loads and generalizing the BRAVO effect, with implications for various boundary-driven systems.

Contribution

It introduces a novel analysis of departure statistics during phase transitions, showing oscillations in cumulants and extending the BRAVO effect to non-equilibrium queues.

Findings

Departure cumulants deviate from Poissonian values near phase transitions.

Cumulants oscillate with load, indicating non-trivial departure flow behavior.

Generalization of the BRAVO effect to non-equilibrium queue systems.

Abstract

Queueing theory is used for modeling biological processes, traffic flows and many more real-life situations. Beyond that, queues describe systems out of equilibrium and can thus be considered as minimal models of non-equilibrium statistical mechanics. We demonstrate that non-equilibrium phase transitions of queues in the steady state are accompanied by a nontrivial flow of departing customers. Our analytical results show that the cumulants of the departure statistics deviate strongly from Poissonian values and oscillate in the vicinity of phase transitions, i.e., if a critical load is approached. The load-dependent oscillations of the cumulants generalize the BRAVO effect (Balancing Reduces Asymptotic Variance of Outputs) in queues and may occur in other boundary-driven non-equilibrium systems.