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

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 netw...

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Autores principales: Jonckheere, M., Shneer, S.
Formato: JOUR
Materias:
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
Lyapunov functions
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00018678_v46_n1_p59_Jonckheere
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling todo:paper_00018678_v46_n1_p59_Jonckheere2023-10-03T13:52:12Z Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps Jonckheere, M. Shneer, S. 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 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. Fil:Jonckheere, M. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00018678_v46_n1_p59_Jonckheere
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic 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
spellingShingle 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
Jonckheere, M.
Shneer, S.
Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps
topic_facet 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
description 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.
format JOUR
author Jonckheere, M.
Shneer, S.
author_facet Jonckheere, M.
Shneer, S.
author_sort Jonckheere, M.
title Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps
title_short Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps
title_full Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps
title_fullStr Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps
title_full_unstemmed Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps
title_sort stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps
url http://hdl.handle.net/20.500.12110/paper_00018678_v46_n1_p59_Jonckheere
work_keys_str_mv AT jonckheerem stabilityofmultidimensionalbirthanddeathprocesseswithstatedependent0homogeneousjumps
AT shneers stabilityofmultidimensionalbirthanddeathprocesseswithstatedependent0homogeneousjumps
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