Further Research On The M/M/1 Queueing Model With Compulsory Server Vacations

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 two sections. In Section 1, firstly we introduce the M/M/1 queueing model with compulsory server vacations, 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 λ, the service rate of the server μ, the vacation rate of the server b and the maximum number of customers who can receive service at the same time M satisfy is an eigenvalue of the operator, which corresponds to the M/M/1 queueing model with compulsory server vacations, with geometric multiplicity one. Our result shows that the operator has at least one eigenvalue in the left real line.