In Section 1,first we introduce the M/M~2/1 queueing model,then we convert this model into an abstract Cauchy problem in a Banach space by introducing state space,operators and their domains,last we introduce the main results obtained by other researchers.

The M/M/1 queuing model of the elevator service system is presented in this paper.

Considering the characteristics of the AGV convey system, we setup a M/M/1 queuing model and adopt average queue length and remain time as performance indexes, and then estimate whether the results are reasonable according to system demands and the simulation curves of the M/M/1 model.

Existence and uniqueness of solution for M/M/1 queueing model;

A special M/G/1 queue is studied which the server plans to spend finite number of vacationswhen the sever has served all customers.

It is firstly proved by using C0 -semigroup theory that the M/Ek/1 queueing model has a unique positive time dependent solution which satisfies probability condition, then the spectral properties of the operator corresponding to this model is studied.