  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.
