Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps

TL;DR

This paper investigates the stability and instability conditions of multi-dimensional birth-and-death processes with state-dependent jumps, offering a new Lyapunov function construction and geometric stability criteria applicable to complex stochastic systems.

Contribution

It introduces a generic Lyapunov function construction method based on associated dynamical systems, simplifying stability analysis without fluid limit convergence, and extends results to discontinuous drifts.

Findings

Elementary proof of ergodicity for smooth drifts

Necessary and sufficient stability conditions for certain discontinuous drifts

Geometric interpretation of stability criteria in multi-dimensional systems

Abstract

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.