Asymptotic Behavior Of The Time-dependent Solution Of The M/M/1Queueing Model With Optional Second Service

This thesis is divided into two chapters. Chapter1is divided into two sections. In thesection1, we introduce briefly the history of queueing theory. In Section2, we first introducethe supplementary variable technique, then we put forward the problem that we study inthis thesis. Chapter2is split into two sections. In Section1, first we introduce the M/M/1queueing model with optional second service, next we convert the model into an abstractCauchy problem in a Banach space by introducing a state space, operators and their domains.In the section2, first we study the spectrum of the underlying operator which correspondentsto the M/M/1queueing model with optional second service and obtain that all points onthe imaginary axis except zero belong to the resolvent set of the underlying operator, zero isan eigenvalue of the underlying operator and its adjoint operator with geometric multiplicityone. Thus we conclude that the time-dependent solution of the model strongly converges toits steady-state solution.