With The State - The Recession Of The Independent Immigration Branching Processes

The ordinary Markov branching processes (MBPs) form one of the most important branches of Markov processes and has quite extensive applications in queueing theory, biology, physics and so on. The ordinary Markov branching model has been studied extensively. The basic property which governs the evolution of an MBP is the branching property. That is that different particles act independently when giving birth or death. In most realistic situation however, the above independence property is unlikely to be appropriate. Indeed, in practical cases, particles usually interact with each other. This may explain the reason why there always been an increasing effort to generalize the ordinary Markov branching processes to more general branching models for many scholars, such as studying the Markov branching processes with immigration(no matter state-independent or state-dependent).There have been some papers about this. In this thesis, we study the decay property of MBPs with state-independent immigration based on the existent conclusions and the theory of continuous time Markov chains.In this thesis, we consider decay properties including decay parameter, invariant measures, invariant vectors and quasi-stationary distributions of Markov branching processes with state-independent immigration. In the case that the process is transient, it is important to investigate the deca y property of the process. In this thesis, we firstly consider the uniqueness of the process. Secondly, the exact value of the decay parameterλ_Z is obtained and expressed explicitly. Moreover, the criteria forλ_Z-recurrence and positivelyλ_Z -recurrence are revealed. Finally, theλ_Z -invariant measures,λ_Z -invariant vectors and quasi-stationary distributions are presented. The generating functions of these invariant measures and quasi-stationary are explicitly presented.