Negative Eigenvalues Of The Operator Which Corresponds To The M/M/1 Queueing Model With Second Optional Service

This thesis is divided into two chapters.Chapter 1 is split into two sections.In Section 1,we introduce briefly the history of queueing theory.In Section 2,we first introduce the supplementary variable technique,then we state the problem that we will study in this thesis.Chapter 2 consists of two sections.In Section 1,firstly we introduce the mathematical model of the M/M/1 queueing system with second optional service,then we convert the model into an abstract Cauchy problem in a Banach space by introducing a state space,operators and their domains,lastly we introduce the main results obtained by other researchers.In Section 2,we prove that if the arrival rate of customers?,the service rate of the first essential service ?1 and the service rate of the second optional service?2 satisfy ?/?1 + ?/?2<1,then y(-?)+(1-y)? are eigenvalues of the operator,which corresponds to the M/M/1 queueing model with second optional service,with geometric multiplicity 1 for all y ?(0,1),herer represents the probability that customers opt the second optional service.Our result shows that the operator has uncountable eigenvalues on the left real line of-?.Hence,we deduce that the C0-semigroup generated by the operator is not compact,even not eventually compact.