Study Of The Exhaustive-Service M/D/1 Queueing Model With Optional Server Vactions

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 sup-plementary variable technique, then we state the problem that we will study in this thesis. Chapter 2 consists of three sections. In Section 1, firstly we introduce the mathematical model of the exhaustive-service M/D/1 queueing model with optional server vacations, 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, by using the Hille-Yosida theorem, Phillips theorem and the Fattorini theorem we prove that the queueing model has a unique positive time-dependent solution which satisfies probability condition. In Section 3, under a certain condition, by studying the spectral properties of the underlying operator we deduce that its time-dependent solution strongly converges to its steady-state solution.