Study On Quasi-Stationary Distributions Of A Class Of Birth-Death Processes With Killing

The birth-death process with killing is an important markov process,its quasi-stationary distribution theory has strong theoretical support for the research of life science and financial insurance.In this paper,we transform a class of birth-death processes with killing into the birth-death processes,and we mainly study the relationship between the two processes regarding the decay parameter and quasi-stationary distribution.And we discuss decay parameter and quasistationary distributions problem of the birth-death processes with killing.First of all,we mainly introduce the research background and significance of birth-death processes with killing,and resume the research history and the present situation of continuous-time and discrete-state markov processes and the birth-death processes with killing.We further elaborate the problems and results of this paper.Secondly,we introduce some definitions and theories of continuous time Markov chains,birth-death processes,birth-death processes with killing and quasistationary distributions.Then,we present a class of birth-death processes with killing model studied in this paper,and transform the processes into the birth-death processes.We prove that the two processes have same decay parameter,and we obtain the relationship of quasi-stationary distributions for two processes.We calculate the decay parameter of birth-death processes with killing,where rates are constant,we analyze the existence of the quasi-stationary distribution,and we further give an expression for the quasi-stationary distribution.Finally,we summarize the conclusions and present some of the relevant issues to be studied.