On Exponential Ergodicity of Multiclass Queueing Networks
  One of the key performance measures in queueing systems is the exponential 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 \pr(\cdot >s) decays faster than s^{-n} for any n. It is natural to conjecture that the decay rate is in fact exponential.   In this paper an example is constructed to demonstrate that this conjecture is false. For a specific stationary policy applied to a network with exponentially distributed interarrival and service times it is shown that the corresponding fluid limit model is stable, but the tail probability for the buffer length decays slower than s^{-\log s}. 
David Gamarnik and Sean Meyn