The Birth And Death Processes With Geometric Catastrophes

The birth and death processes with geometric catastrophes is an important model of Markov processes, it has great practical significance. In this paper, we are devoted to study the birth and death processes with geometric catastrophes and mainly discuss the problems about the stochastic monotonicity, duality, extinction probability and the mean extinction time.The first chapter is the introduction of the paper, which mainly introduces the research background, research status and the main work of what I have done in this paper.The second chapter is about the basic theory of the continuous time Markov chains, the birth and death processes and the quasi-stationary distributions.The third chapter, we mainly study the the stochastic monotonicity and duality in the birth and death processes with geometric catastrophes. We conclude the expression of the dual matrix.In the last chapter, we mainly study the problems about the extinction proba-bility and the mean extinction time in the birth and death processes with geometric catastrophes, and we get the sufficient and necessary conditions for this model.