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Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps. (English) Zbl 1286.93189

Summary: We study the conditions for positive recurrence and transience 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. First, using an associated deterministic dynamical system, we provide a generic method to construct a Lyapunov function when the drift is a smooth function on \(\mathbb R^{N}\). This approach gives an elementary and direct proof of ergodicity. We also provide instability conditions. Our main contribution consists of showing how discontinuous drifts change the nature of the stability conditions and of providing 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 piecewise constant drifts in dimension two.

MSC:

93E15 Stochastic stability in control theory
60K25 Queueing theory (aspects of probability theory)

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