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March 2014 Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps
Matthieu Jonckheere, Seva Shneer
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Adv. in Appl. Probab. 46(1): 59-75 (March 2014). DOI: 10.1239/aap/1396360103

Abstract

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 RN. 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.

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Matthieu Jonckheere. Seva Shneer. "Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps." Adv. in Appl. Probab. 46 (1) 59 - 75, March 2014. https://doi.org/10.1239/aap/1396360103

Information

Published: March 2014
First available in Project Euclid: 1 April 2014

zbMATH: 1286.93189
MathSciNet: MR3189048
Digital Object Identifier: 10.1239/aap/1396360103

Subjects:
Primary: 93E15
Secondary: 60K25

Keywords: Birth-and-death process , fluid limit , positive recurrence , transience

Rights: Copyright © 2014 Applied Probability Trust

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Vol.46 • No. 1 • March 2014
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