Mathematics – Probability
Scientific paper
2009-10-30
Mathematics
Probability
18 pages, 4 figures
Scientific paper
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 $\mathbb R^N$, 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.
Jonckheere Matthieu
Shneer Seva
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