Research On Several Classes Of The Generalized Markov Branching Models

In this Ph.D. thesis, we investigate several classes of generalized Markov branching models which include collision branching models with immigration, collision branching models with immigration and resurrection, n-type branching models, n-type branching models with immigration. We further research the uniqueness criteria, the absorbing protperty, the explosion property and the decay property of these branching models. This Ph.D. thesis is organized as follows.In Chapter1, we introduce the background where the problems are produced, list the recent development and the main research contents of Markov branching processes, review the main results and the innovative contributions of this dissertation.In Chapter2, we outline the main contents of the dissertation and give some preliminary tools used in proving the main results at the end of this chapter.In Chapter3, we consider a kind of Collision Branching Processes with Immigration{CBIP). Some important properties of the generating functions for CBI q-matrix are firstly investigated in detail. Then it is shown that there exists exactly one CBIP for any given CBI q-matrix, and sufficient and easily checked conditions for the CBIP to be recurrent are given. Moreover, the exact value of the decay parameter λz is obtained and expressed explicitly for the communicating class Z+in the case that the immigration is independent of states. It is shown that this λz can be directly obtained from the generating functions of the corresponding q-matrix. Finally, the invariant vectors and invariant measures are considered.In Chapter4, we consider the uniqueness, recurrence and decay properties of the Interacting Branching Collision Processes with Immigration and Resurrection(BCP-IR) and some important properties of the generating functions for BC-IR q-matrix are firstly investigated in detail. We establish that there is a unique BCP-IR, and derive sufficient conditions for it to be recurrence that are easily to be checked. Moreover, the exact value of the decay parameter λz is obtained and expressed explicitly for the communicating class Z+. It is shown that this λz can be directly obtained from the generating functions of the corresponding q-matrix. The invariant vectors are then presented.In Chapter5, we consider decay properties including the decay parameter, invariant measures, invariant vectors, and quasi-stationary distributions for n-type Markov branching processes on the basis of the ordinary Markov branching processes and2-type Markov branching processes. Investigating such behavior is crucial in realizing life period of branching models. In this chapter, some important properties of the generating functions for n-type Markov branching q-matrix are firstly investigated in detail. The exact value of the decay parameter λc of such model is given for the communicating class C=Z+n\0. It is shown that this λC can be directly obtained from the generating functions of the corresponding q-matrix. Moreover, the λC-invariant measures/vectors and quasi-distributions of such processes are deeply considered. A λC-invariant measures and quasi-stationary distributions for the process on C are presented.In Chapter6, we consider a kind of n-type Markov branching processes with immigration and resurrection. First we obtain the uniqueness criteria of such model. Moreover, we find a new method and hence get the explicit expression of the extinction probability in the case of the absorbing. On one hand, if the state0is not absorbing, we give the recurrence property and the ergodicity property of the process. On the other hand, if the resurrection is equal to the immigration rate, we consider the decay properties of such process in detail, and the exact value of the decay parameter λz of such model is given and expressed explicitly for the communicating class Z+n. Furthermore, the corresponding λz-invariant measures/vectors and quasi-distributions are considered and the structure of all the λz-invariant measures and quasi-stationary distributions for the process on Z+n are presented.124references...