scholarly journals Uniqueness and Decay Properties of Markov Branching Processes with Disasters

2014 ◽  
Vol 51 (3) ◽  
pp. 613-624 ◽  
Author(s):  
Anyue Chen ◽  
Kai Wang Ng ◽  
Hanjun Zhang
Keyword(s):  
Branching Process ◽  
Invariant Measures ◽  
Branching Processes ◽  
Decay Parameter ◽  
Markov Branching Process ◽  
Decay Properties ◽  
Invariant Vectors ◽  
Quasistationary Distributions

In this paper we discuss the decay properties of Markov branching processes with disasters, including the decay parameter, invariant measures, and quasistationary distributions. After showing that the corresponding q-matrix Q is always regular and, thus, that the Feller minimal Q-process is honest, we obtain the exact value of the decay parameter λC. We show that the decay parameter can be easily expressed explicitly. We further show that the Markov branching process with disaster is always λC-positive. The invariant vectors, the invariant measures, and the quasidistributions are given explicitly.

Uniqueness and Decay Properties of Markov Branching Processes with Disasters

2014 ◽  
Vol 51 (03) ◽  
pp. 613-624 ◽  
Author(s):  
Anyue Chen ◽  
Kai Wang Ng ◽  
Hanjun Zhang
Keyword(s):  
Branching Process ◽  
Invariant Measures ◽  
Branching Processes ◽  
Decay Parameter ◽  
Markov Branching Process ◽  
Decay Properties ◽  
Invariant Vectors ◽  
Quasistationary Distributions

In this paper we discuss the decay properties of Markov branching processes with disasters, including the decay parameter, invariant measures, and quasistationary distributions. After showing that the corresponding q-matrix Q is always regular and, thus, that the Feller minimal Q-process is honest, we obtain the exact value of the decay parameter λ C . We show that the decay parameter can be easily expressed explicitly. We further show that the Markov branching process with disaster is always λ C -positive. The invariant vectors, the invariant measures, and the quasidistributions are given explicitly.

scholarly journals Decay property of stopped Markovian bulk-arriving queues

2008 ◽  
Vol 40 (1) ◽  
pp. 95-121 ◽  
Author(s):  
Junping Li ◽  
Anyue Chen
Keyword(s):  
Generating Functions ◽  
Busy Period ◽  
Invariant Measures ◽  
Geometric Interpretation ◽  
Decay Property ◽  
Decay Parameter ◽  
Decay Properties ◽  
Invariant Vectors ◽  
Quasistationary Distributions

We consider decay properties including the decay parameter, invariant measures, invariant vectors, and quasistationary distributions of a Markovian bulk-arriving queue that stops immediately after hitting the zero state. Investigating such behavior is crucial in realizing the busy period and some other related properties of Markovian bulk-arriving queues. The exact value of the decay parameter λC is obtained and expressed explicitly. The invariant measures, invariant vectors, and quasistationary distributions are then presented. We show that there exists a family of invariant measures indexed by λ ∈ [0, λC]. We then show that, under some conditions, there exists a family of quasistationary distributions, also indexed by λ ∈ [0, λC]. The generating functions of these invariant measures and quasistationary distributions are presented. We further show that a stopped Markovian bulk-arriving queue is always λC-transient and some deep properties are revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper.

scholarly journals Decay property of stopped Markovian bulk-arriving queues

2008 ◽  
Vol 40 (01) ◽  
pp. 95-121 ◽  
Author(s):  
Junping Li ◽  
Anyue Chen
Keyword(s):  
Generating Functions ◽  
Busy Period ◽  
Invariant Measures ◽  
Geometric Interpretation ◽  
Decay Property ◽  
Decay Parameter ◽  
Decay Properties ◽  
Invariant Vectors ◽  
Quasistationary Distributions

We consider decay properties including the decay parameter, invariant measures, invariant vectors, and quasistationary distributions of a Markovian bulk-arriving queue that stops immediately after hitting the zero state. Investigating such behavior is crucial in realizing the busy period and some other related properties of Markovian bulk-arriving queues. The exact value of the decay parameter λCis obtained and expressed explicitly. The invariant measures, invariant vectors, and quasistationary distributions are then presented. We show that there exists a family of invariant measures indexed by λ ∈ [0, λC]. We then show that, under some conditions, there exists a family of quasistationary distributions, also indexed by λ ∈ [0, λC]. The generating functions of these invariant measures and quasistationary distributions are presented. We further show that a stopped Markovian bulk-arriving queue is always λC-transient and some deep properties are revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper.

scholarly journals Skeletal stochastic differential equations for superprocesses

2020 ◽  
Vol 57 (4) ◽  
pp. 1111-1134
Author(s):  
Dorottya Fekete ◽  
Joaquin Fontbona ◽  
Andreas E. Kyprianou
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Subcritical Case ◽  
Markov Branching Process ◽  
Continuous State ◽  
Common Framework ◽  
Infinite Line ◽  
Current Article ◽  
Lines Of Descent ◽  
Skeletal Representation

AbstractIt is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritical superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is, individuals who produce eternal genealogical lines of descent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess. Such skeletal decompositions are equally well understood for continuous-state branching processes (CSBP).In a previous article [16] we developed an SDE approach to study the skeletal representation of CSBPs, which provided a common framework for the skeletal decompositions of supercritical and (sub)critical CSBPs. It also helped us to understand how the skeleton thins down onto one infinite line of descent when conditioning on survival until larger and larger times, and eventually forever.Here our main motivation is to show the robustness of the SDE approach by expanding it to the spatial setting of superprocesses. The current article only considers supercritical superprocesses, leaving the subcritical case open.

scholarly journals A note on Markov branching processes

1985 ◽  
Vol 17 (02) ◽  
pp. 463-464
Author(s):  
Fred M. Hoppe
Keyword(s):  
Laplace Transform ◽  
Simple Proof ◽  
Branching Process ◽  
Branching Processes ◽  
Markov Branching Process ◽  
The Laplace Transform

We present a simple proof of Zolotarev’s representation for the Laplace transform of the normalized limit of a Markov branching process and relate it to the Harris representation.

scholarly journals Extinction Times in Multitype Markov Branching Processes

2009 ◽  
Vol 46 (01) ◽  
pp. 296-307 ◽  
Author(s):  
Dominik Heinzmann
Keyword(s):  
Limit Theorem ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Markov Branching Process ◽  
Time To Extinction

In this paper, a distributional approximation to the time to extinction in a subcritical continuous-time Markov branching process is derived. A limit theorem for this distribution is established and the error in the approximation is quantified. The accuracy of the approximation is illustrated in an epidemiological example. Since Markov branching processes serve as approximations to nonlinear epidemic processes in the initial and final stages, our results can also be used to describe the time to extinction for such processes.

scholarly journals The Limit Behavior of Dual Markov Branching Processes

2008 ◽  
Vol 45 (1) ◽  
pp. 176-189 ◽  
Author(s):  
Yangrong Li ◽  
Anthony G. Pakes ◽  
Jia Li ◽  
Anhui Gu
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Limit Behavior ◽  
Positive Recurrence ◽  
Strong Ergodicity ◽  
Feller Property ◽  
Markov Branching Process ◽  
Invariant Distributions ◽  
Geometric Limit ◽  
Branching Property

A dual Markov branching process (DMBP) is by definition a Siegmund's predual of some Markov branching process (MBP). Such a process does exist and is uniquely determined by the so-called dual-branching property. Its q-matrix Q is derived and proved to be regular and monotone. Several equivalent definitions for a DMBP are given. The criteria for transience, positive recurrence, strong ergodicity, and the Feller property are established. The invariant distributions are given by a clear formulation with a geometric limit law.

scholarly journals Extinction Times in Multitype Markov Branching Processes

2009 ◽  
Vol 46 (1) ◽  
pp. 296-307 ◽  
Author(s):  
Dominik Heinzmann
Keyword(s):  
Limit Theorem ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Markov Branching Process ◽  
Time To Extinction

In this paper, a distributional approximation to the time to extinction in a subcritical continuous-time Markov branching process is derived. A limit theorem for this distribution is established and the error in the approximation is quantified. The accuracy of the approximation is illustrated in an epidemiological example. Since Markov branching processes serve as approximations to nonlinear epidemic processes in the initial and final stages, our results can also be used to describe the time to extinction for such processes.

scholarly journals A note on Markov branching processes

1985 ◽  
Vol 17 (2) ◽  
pp. 463-464 ◽  
Author(s):  
Fred M. Hoppe
Keyword(s):  
Laplace Transform ◽  
Simple Proof ◽  
Branching Process ◽  
Branching Processes ◽  
Markov Branching Process ◽  
The Laplace Transform

We present a simple proof of Zolotarev’s representation for the Laplace transform of the normalized limit of a Markov branching process and relate it to the Harris representation.

scholarly journals The Limit Behavior of Dual Markov Branching Processes

2008 ◽  
Vol 45 (01) ◽  
pp. 176-189
Author(s):  
Yangrong Li ◽  
Anthony G. Pakes ◽  
Jia Li ◽  
Anhui Gu
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Limit Behavior ◽  
Positive Recurrence ◽  
Strong Ergodicity ◽  
Feller Property ◽  
Markov Branching Process ◽  
Invariant Distributions ◽  
Geometric Limit ◽  
Branching Property

A dual Markov branching process (DMBP) is by definition a Siegmund's predual of some Markov branching process (MBP). Such a process does exist and is uniquely determined by the so-called dual-branching property. Its q-matrix Q is derived and proved to be regular and monotone. Several equivalent definitions for a DMBP are given. The criteria for transience, positive recurrence, strong ergodicity, and the Feller property are established. The invariant distributions are given by a clear formulation with a geometric limit law.

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