全文分为三章, 主要研究状态空间为有限的一族连续时间生灭链的可加泛函满足cutoff的充要条件. 第一章针对具有吸收态的生灭过程, 通过数学归纳法以及Laplace变换求出随机泛函一阶矩和二阶矩的具体表达和性质, 得出生灭过程的可加随机泛函满足cutoff的充要条件. 第二章针对遍历的生灭过程, 通过对偶以及最快强平稳时用各个状态首次击中时表示的方法, 得出遍历的生灭过程的可加随机泛函满足cutoff的充要条件. 第三章将本文的结论运用到实际例子中, 得出了很好的验证.

	This article is divided into three chapters, and the main research is that the additive functional of a family of
continuous time birth and death chains with finite state space have necessary and sufficient condition  to satisfy cutoff. In the first chapter, for the birth and death process with absorbing
states, The specific expressions and properties of the first and second moments of the random functional are obtained through mathematical induction and
Laplace transform. and the sufficient and necessary conditions for the additive functional of the birth and death process to satisfy the cutoff are obtained.
In the second chapter, for the ergodic birth and death process, the sufficient and necessary conditions for the ergodic birth and death process to satisfy the
cutoff are obtained by the method of duality and representing the first hitting time of each state in the fastest strong stationary time.  In the third chapter, the conclusion of this
article is applied to practical examples, which are well verified.