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Stability of Multi-Dimensional Birth-and-Death Processes with State-Dependent 0-Homogeneous Jumps

Part of: Stochastic systems and control Special processes

Published online by Cambridge University Press:  22 February 2016

Matthieu Jonckheere  and
Seva Shneer
Matthieu Jonckheere*
Affiliation:
University of Buenos Aires
Seva Shneer*
Affiliation:
Heriot-Watt University
*
∗ Postal address: CONICET, Mathematics Department, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires (1428) Pabellón I, Ciudad Universitaria, Buenos Aires, Argentina.
∗∗ Postal address: Department of AMS, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: [email protected]
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Abstract

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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.

Keywords

Birth-and-death processpositive recurrencetransiencefluid limit

MSC classification

Primary: 93E15: Stochastic stability
Secondary: 60K25: Queueing theory
Type
Research Article
Information
Advances in Applied Probability , Volume 46 , Issue 1 , March 2014 , pp. 59 - 75
Copyright
© Applied Probability Trust 

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