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

Autores
Jonckheere, Matthieu Thimothy Samson; Shneer, Seva
Año de publicación
2014
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
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.
Fil: Jonckheere, Matthieu Thimothy Samson. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentina
Fil: Shneer, Seva. Heriot-Watt University; Reino Unido
Materia
Stability analysis
Birth and death processes
Lyapunov functions
Stochastic networks
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/18829
Acceder
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network_name_str CONICET Digital (CONICET)
spelling Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumpsJonckheere, Matthieu Thimothy SamsonShneer, SevaStability analysisBirth and death processesLyapunov functionsStochastic networkshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We 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.Fil: Jonckheere, Matthieu Thimothy Samson. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; ArgentinaFil: Shneer, Seva. Heriot-Watt University; Reino UnidoCambridge University Press2014-03info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/18829Jonckheere, Matthieu Thimothy Samson; Shneer, Seva; Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps; Cambridge University Press; Advances In Applied Probability; 46; 1; 3-2014; 59-750001-86781475-6064CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1017/S0001867800006935info:eu-repo/semantics/altIdentifier/url/https://www.cambridge.org/core/journals/advances-in-applied-probability/article/stability-of-multi-dimensional-birth-and-death-processes-with-state-dependent-0-homogeneous-jumps/CA2AC6770A5C9D69BA0DC593156DAAC6info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/0910.5851info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T09:48:10Zoai:ri.conicet.gov.ar:11336/18829instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-03 09:48:10.394CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps
title Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps
spellingShingle Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps
Jonckheere, Matthieu Thimothy Samson
Stability analysis
Birth and death processes
Lyapunov functions
Stochastic networks
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
dc.creator.none.fl_str_mv Jonckheere, Matthieu Thimothy Samson
Shneer, Seva
author Jonckheere, Matthieu Thimothy Samson
author_facet Jonckheere, Matthieu Thimothy Samson
Shneer, Seva
author_role author
author2 Shneer, Seva
author2_role author
dc.subject.none.fl_str_mv Stability analysis
Birth and death processes
Lyapunov functions
Stochastic networks
topic Stability analysis
Birth and death processes
Lyapunov functions
Stochastic networks
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv 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.
Fil: Jonckheere, Matthieu Thimothy Samson. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentina
Fil: Shneer, Seva. Heriot-Watt University; Reino Unido
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.
publishDate 2014
dc.date.none.fl_str_mv 2014-03
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/18829
Jonckheere, Matthieu Thimothy Samson; Shneer, Seva; Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps; Cambridge University Press; Advances In Applied Probability; 46; 1; 3-2014; 59-75
0001-8678
1475-6064
CONICET Digital
CONICET
url http://hdl.handle.net/11336/18829
identifier_str_mv Jonckheere, Matthieu Thimothy Samson; Shneer, Seva; Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps; Cambridge University Press; Advances In Applied Probability; 46; 1; 3-2014; 59-75
0001-8678
1475-6064
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1017/S0001867800006935
info:eu-repo/semantics/altIdentifier/url/https://www.cambridge.org/core/journals/advances-in-applied-probability/article/stability-of-multi-dimensional-birth-and-death-processes-with-state-dependent-0-homogeneous-jumps/CA2AC6770A5C9D69BA0DC593156DAAC6
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/0910.5851
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Cambridge University Press
publisher.none.fl_str_mv Cambridge University Press
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 13.13397

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