Study On The Quasi-stationary Distributions And Domain Of Attraction Of M/M/1 With Killing

For a special class of birth and death processes with killing on the state space E={0}∪C,where 0 is an absorbing state and C=｛1,2,…} is an irreducible set,bi=b,di=d,ki=k,bi,di,ki,i≥1 are birth rate,death rate and killing rate respectively and we call the processes M/M/1 queue with killing.This paper first transforms it into M/M/1 queue through appropriate transformation,and then the relationships between the processes before and after transformation are given.Finally,we study the related problems of quasi-stationary distributions of the processes according to the relationships.In terms of research results,this paper gives the representation of the transition probability function of the class of birth and death processes and another calculation method of decay parameters λC,and then studies the existence and uniqueness of quasi-stationary distributions.In the study of domain of attraction,for general Markov processes,if the Quasi-stationary distribution V exists,there must be a positive real number λ(V)corresponding to V.This paper discusses the relationship between the initial distribution M of its attraction domain and λ(V)when the limit condition distribution V exists,and a necessary condition for the existence of the limit condition distribution is given.