A Class Of Single Death Q-process With Disaster

Abstract:The ordinary Markov branching processes is one of the most important branches of Markov processes and have quite extensive applications in biology, physics, queueing theory and so on. There has always been an increasing effort to generalize the ordinary Markov branching processes. In this thesis, we consider a class of single death Q-process with disaster based on the existent conclusions and the theory of continuous time Markov chains.There are five parts in this thesis. In chapter one, we mainly introduce the research background, the development of the branching processes and the work of this thesis. In chapter two, we give some basic knowledges about continuous-time Markov chains.Then we introduce the contents of this thesis in two different cases. First, we assume that the process will stop immediately when the system first becomes empty in chapter3and4. That is, the sate0is absorbing for the system. Thus the Q-process is transient. There is no normal stationary distribution, hence we turn our attention to its decay proprieties. Second, external particles are allowed to enter the system. Specifically, the contents are as follows:In chapter three, the equations that the decay parameter λt satisfies will be given. They are determined by the corresponding Q-matrix and the arrival rate of disaster. Then, we give the decay proprieties. And we also prove the Q-process is always λC-transient. In chapter four, we prove the absorbing probability is1when the process starts at any state except the absorbing state. Then the mean absorbing time is given. By the conclusions in chapter four, we find the process in chapter five, is always positive recurrent, and give its stationary distributions. Then we prove the process is strong ergodic and exponentially ergodic...