On exponential ergodicity of multiclass queueing networks

One of the key performance measures in queueing systems is the decay rate of the steady-state tail probabilities of the queue lengths. It is known that if a corresponding fluid model is stable and the stochastic primitives have finite moments, then the queue lengths also have finite moments, so that the tail probability ℙ(⋅>s) decays faster than s ⁻ⁿ for any n. It is natural to conjecture that the decay rate is in fact exponential.