We study the positive recurrence of multi-dimensional birth-and-death
processes describing the evolution of a large class of stochastic systems, a
typical example being the randomly varying number of flow-level transfers in a
telecommunication wire-line or wireless network.
We first provide a generic method to construct a Lyapunov function when the
drift can be extended to a smooth function on RN, using an
associated deterministic dynamical system. This approach gives an elementary
proof of ergodicity without needing to establish the convergence of the scaled
version of the process towards a fluid limit and then proving that the
stability of the fluid limit implies the stability of the process. We also
provide a counterpart result proving instability conditions.
We then show how discontinuous drifts change the nature of the stability
conditions and we provide generic sufficient stability conditions having a
simple geometric interpretation. These conditions turn out to be necessary
(outside a negligible set of the parameter space) for piece-wise constant
drifts in dimension 2.Comment: 18 pages, 4 figure