On Exponential Ergodicity of Multiclass Queueing Networks

David Gamarnik and Sean Meyn

**Abstract:**

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 P( ** ^{.}** >

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 ^{-}*

**Reference:**

@unpublished{gammey05a,

Title = {On Exponential Ergodicity in Multiclass Queueing Networks},

Author = {Gamarnik, D. and Meyn, S. P.},

Note = {{Asymptotic Analysis of Stochastic Systems}, invited session at the {INFORMS} Annual Meeting. And, submitted for publication, 2006.},

Year = {November 13-16, 2005}}