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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. © Applied Probability Trust 2014.

Registro:

Documento: Artículo
Título:Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps
Autor:Jonckheere, M.; Shneer, S.
Filiación:CONICET, Mathematics Department, Facultad de Ciencias Exactas y Naturales, Buenos Aires, Argentina
Department of AMS, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
Palabras clave:Birth-and-death process; Fluid limit; Positive recurrence; Transience; Dynamical systems; Birth and death process; Deterministic dynamical systems; Fluid limits; Geometric interpretation; Instability condition; Piece-wise constants; Positive recurrence; Transience; Lyapunov functions
Año:2014
Volumen:46
Número:1
Página de inicio:59
Página de fin:75
DOI: http://dx.doi.org/10.1239/aap/1396360103
Título revista:Advances in Applied Probability
Título revista abreviado:Adv Appl Probab
ISSN:00018678
CODEN:AAPBB
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018678_v46_n1_p59_Jonckheere

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Citas:

---------- APA ----------
Jonckheere, M. & Shneer, S. (2014) . Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps. Advances in Applied Probability, 46(1), 59-75.
http://dx.doi.org/10.1239/aap/1396360103
---------- CHICAGO ----------
Jonckheere, M., Shneer, S. "Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps" . Advances in Applied Probability 46, no. 1 (2014) : 59-75.
http://dx.doi.org/10.1239/aap/1396360103
---------- MLA ----------
Jonckheere, M., Shneer, S. "Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps" . Advances in Applied Probability, vol. 46, no. 1, 2014, pp. 59-75.
http://dx.doi.org/10.1239/aap/1396360103
---------- VANCOUVER ----------
Jonckheere, M., Shneer, S. Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps. Adv Appl Probab. 2014;46(1):59-75.
http://dx.doi.org/10.1239/aap/1396360103