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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paper:paper_00018678_v46_n1_p59_Jonckheere2023-06-08T14:21:41Z Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps Jonckheere, Matthieu Thimothy Samson 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. 2014 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018678_v46_n1_p59_Jonckheere 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, Matthieu Thimothy Samson 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. |
| author |
Jonckheere, Matthieu Thimothy Samson |
| author_facet |
Jonckheere, Matthieu Thimothy Samson |
| author_sort |
Jonckheere, Matthieu Thimothy Samson |
| 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 |
| publishDate |
2014 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018678_v46_n1_p59_Jonckheere http://hdl.handle.net/20.500.12110/paper_00018678_v46_n1_p59_Jonckheere |
| work_keys_str_mv |
AT jonckheerematthieuthimothysamson stabilityofmultidimensionalbirthanddeathprocesseswithstatedependent0homogeneousjumps |
| _version_ |
1768546374303875072 |