Research Of The M/G/1 Queueing Model With Additional Optional Service And No Waiting Capacity

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, first introduce supplementary variable technique, then we put forward the problems that we study in this thesis. Chapter 2 consists of three sections, last we introduce the results of other researchers about this modle. In Section 1, first we introduce the mathematical model of the M/G/1 queueing system with additional optional service and no waiting capacity, then we convert the model into an abstract Cauchy problem in a Banach space by introducing state space, operators and their domains. In Section 2, we study well-posedness of the queueing model, that is, prove existence and uniqueness of a positive time-dependent solution of the queueing model by using the Hille-Yosida theorem, the Philips theorem and the Fattorini theorem in functional analysis. In Section 3, we study asymptotic property of the time-dependent solution of this model. First by discussing properties of the C₀—semigroup generated by the operator corresponding to this model we obtain that the C₀—semigroup is a quasi-compact operator, thus we deduce the C₀—semigroup exponentially to a project operator. Second when the service rates are constants, we study expression of this project operator and therefore we obtain that the time-dependent solution of this model converges exponentially to the steady-state solution of this model.