Population Size Dependent Generalized Multitype Branching Processes

Abstract

In this paper, we introduce the population size dependent generalized multitype branching process. This is a Markovian model that allows us to study homogeneous multitype branching processes in a unified way. The basic properties for this model, transitions between its states, as well as the existence of a stationary limiting distribution, are investigated. Finally, we apply the obtained results to a new controlled multitype branching process.     

Branching/queueing networks: Their introduction and near-decomposability asymptotics

A new class of models, which combines closed queueing networks with branching processes, is introduced. The motivation comes from MIMD computers and other service systems in which the arrival of new work is always triggered by the completion of former work, and the amount of arriving work is variable. In the variant of branching/queueing networks studied here, a customer branches into a random and state-independent number of offspring upon completing its service. The process regenerates whenever the population becomes extinct. Implications for less rudimentary variants are discussed. The ergodicity of the network and several other aspects are related to the expected total number of progeny of an associated multitype Galton-Watson process. We give a formula for that expected number of progeny. The objects of main interest are the stationary state distribution and the throughputs. Closed-form solutions are available for the multi-server single-node model, and for homogeneous networks of infinite-servers. Generally, branching/queueing networks do not seem to have a product-form state distribution. We propose a conditional product-form approximation, and show that it is approached as a limit by branching/queueing networks with a slowly varying population size. The proof demonstrates an application of the nearly complete decomposability paradigm to an infinite state space. This revised version was published online in June 2006 with corrections to the Cover Date.     

Bisexual Galton-Watson Branching Processes in Random Environments

In this paper, we consider a bisexual Galton-Watson branching process whose offspring probability distribution is controlled by a random environment proccss. Some results for the probability generating functions associated with the process are obtained and sufficient conditions for certain extinction and for non-certain extinction are established.     

人口数相依的两性分支过程的极限性质

本文研究了后代分布依赖于人口数的两性Galton-Watson分支过程, 在对后代分布的适当假设下,对于上临界的情况, 我们研究了有关过程的几乎处处收敛的极限性质.     

On the reverse process of a critical multitype Galton-Watson process without variances

Limit theorems are established for the reverse process of a critical multitype Galton-Watson process without variances. These results are an extension of those of Nakagawa |5| in the case with finite variances.     

THE ASYMPTOTIC PROPERTIES OF SUPERCRITICAL BISEXUAL GALTON-WATSON BRANCHING PROCESSES WITH IMMIGRATION OF MATING UNITS

In this article the supercritical bisexual Galton-Watson branching processes with the immigration of mating units is considered. A necessary condition for the almost sure convergence, and a sufficient condition for the L1 convergence are given for the process with the suitably normed condition.     

Modelling a Multitype Branching Brownian Motion: Filtering of a Measure-Valued Process

In the framework of marked trees, a multitype branching brownian motion, described by measure-valued processes, is studied. By applying the strong branching property, the Markov property and the expression of the generator are derived for the process whose components are the measure-valued processes associated to each type particles. The conditional law of the measure-valued process describing the whole population observing the cardinality of the subpopulation of a given type particles is characterized as the unique weak solution of the Kushner‐Stratonovich equation. An explicit representation of the filter is obtained by Feyman–Kac formula using the linearized filtering equation.     

Convergence of critical multitype Galton-Watson branching processes

Allowing an offspring probability distribution that has infinite variances, we establish the convergence in finite-dimensional distributions of normalized critical multitype Galton-Watson branching processes with increasing initial population size in the two cases of not conditioning and of conditioning on non-extinction of the processes in the nth generation. Furthermore, if the offspring probability distribution has only finite variances, we show that some linear functions of the above processes weakly converge to the diffusions given by Feller, and by Lamperti and Ney.     

Galton-Watson分支过程的谱半径的概率刻画

谱半径是不可约马尔可夫链的一个很重要的特征数字.Galton-Watson分支过程是一类特殊的马尔可夫链,我们已经证明了在不可约的条件下,Galton-Watson分支过程的谱半径等于其对应的概率母函数f(s)在灭绝概率q点的导数值.本文主要从理论上刻画从过程的任何状态逃离速度的Galton-Watson分支过程的谱半径的概率意义.     

Backbone decomposition for continuous-state branching processes with immigration

In the spirit of Duquesne and Winkel (2007) and Berestycki et al. (2011), we show that supercritical continuous-state branching process with a general branching mechanism and general immigration mechanism is equivalent in law to a continuous-time Galton-Watson process with immigration (with Poissonian dressing). The result also helps to characterise the limiting backbone decomposition which is predictable from the work on consistent growth of Galton-Watson trees with immigration in Cao and Winkel (2010).     

On a multitype Galton-Watson process with state-dependent immigration

We study the behavior of a multitype Galton-Watson process with state-dependent immigration. A limit theorem is obtained in the near-critical case. Supported by the Russian Foundation for Fundamental Research (grant Nos. 96-01-00338 and 96-15-96092) and INTAS-RFBR (grant No. 95-0099). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.     

SOME PROPERTIES OF GALTON-WATSON BRANCHING PROCESSES IN VARYING ENVIRONMENTS

This article deals with some properties of Galton-Watson branching processes in varying environments. A necessary and suffcient condition for relative recurrent state is presented, and a series of ratio limit properties of the transition probabilities are showed.     

Law of large numbers for supercritical superprocesses with non-local branching

In this paper we establish a weak and a strong law of large numbers for supercritical superprocesses with general non-local branching mechanisms. Our results complement earlier results obtained for superprocesses with only local branching. Several interesting examples are developed, including multitype continuous-state branching processes, multitype superdiffusions and superprocesses with discontinuous spatial motions and non-decomposable branching mechanisms.     

On the transience and recurrence of Lamperti's random walk on Galton-Watson trees

Science China Mathematics - In a Galton-Watson tree generated by a supercritical branching process with offspring N and EN:= m > 1, the conductance assigned to the edge between the vertex x...     

Some results on first order stochastic models and estimation for diffusion approximation of the multitype galton–watson process

Several discrete and continuous–time first order autoregressive models and first order autoregressive type models are studied and the solution of the stochastic differential equations are considered. A diffusion approximation of the multitype branching process is given. Furthermore, maximum likelihood estimates of the parameters in the emigration–immigrarion model and of the drift parameters in the diffusion approximation of the multitype branching model are obtained.     

Asymptotic expansions in the central limit theorem for a branching Wiener process

Science China Mathematics - We consider a branching Wiener process in ?d, in which particles reproduce as a super-critical Galton-Watson process and disperse according to a Wiener process....     

Polling systems and multitype branching processes

The joint queue length process in polling systems with and without switchover times is studied. If the service discipline in each queue satisfies a certain property it is shown that the joint queue length process at polling instants of a fixed queue is a multitype branching process (MTBP) with immigration. In the case of polling models with switchover times, it turns out that we are dealing with an MTBP with immigration in each state, whereas in the case of polling models without switchover times we are dealing with an MTBP with immigration in state zero. The theory of MTBPs leads to expressions for the generating function of the joint queue length process at polling instants. Sufficient conditions for ergodicity and moment calculations are also given.This work was done while the author was at the Centre for Mathematics and Computer Science (CWI) in Amsterdam, The Netherlands.     

The exact hausdorff dimension of a branching set

Summary We obtain a critical function for which the Hausdorff measure of a branching set generated by a simple Galton-Watson process is positive and finite.     

独立同分布随机环境中两性Galton-Watson分支过程灭绝概率的渐近行为

本文研究了独立同分布随机环境中的两性Galton-Watson分支过程,在上临界情形下,当k充分大时,qk≤ck<'-α>.     

ON THE EXTINCTION OF POPULATION-SIZE-DEPENDENT BISEXUAL GALTON-WATSON PROCESSES

In this article, the population-size-dependent bisexual Galton-Watson processes are considered. Under some suitable conditions on the mating functions and the offspring distribution, existence of the limit of mean growth rate per mating unit is proved. And based on the limit, a criterion to identify whether the process admits ultimate extinct with probability one is obtained.