Dynamic Analysis Of Several Queueing Models

By using C₀semigroup theory in functional analysis we do dynamic analysis for theM^[X]/G/1retrial queueing model with server breakdowns and constant rate of repeated at-tempts, the batch arrival retrial queueing model with two phases of service and service inter-ruption, the M/G/1queueing model with second optional service and the M/G/1queueingmodel with optional second service and obtain that the M^[X]/G/1retrial queueing modelwith server breakdowns and constant rate of repeated attempts and the batch arrival retrialqueueing model with two phases of service and service interruption have a unique nonnegativetime-dependent solution which satisfies probability condition; the time-dependent solutions ofthe M^[X]/G/1retrial queueing model with server breakdowns and constant rate of repeatedattempts, the M/G/1queueing model with second optional service and the M/G/1queueingmodel with optional second service strongly converge to their steady-state solutions. In thisthesis, we introduce our research idea, methods and main results by taking the M^[X]/G/1retrial queueing model with server breakdowns and constant rate of repeated attempts as anexample.This thesis consists of3chapters. The first chapter introduces some basic concepts,lemmas and theorems that will be used in later chapters. Chapter2is split into two sections.In Section1, we introduce briefly the history of queueing theory. In Section2, we firstintroduce the supplementary variable technique, then we state the problems that we willstudy in this thesis. Chapter3is the main part of this thesis and consists of four sections. InSection1, firstly we introduce the M^[X]/G/1retrial queueing model with server breakdownsand constant rate of repeated attempts, then we convert the model into an abstract Cauchyproblem in a Banach space by introducing a state space, operators and their domains. InSection2, by using the Hille-Yosida theorem, the Phillips theorem and the Fattorini theoremwe prove that the model has a unique nonnegative time-dependent solution which satisfiesprobability condition. In Section3, we study spectral properties of the M^[X]/G/1operatorwhich corresponds to the model. Firstly, by introducing the probability generating function and using the result in section2we prove that0is an eigenvalue of the M^[X]/G/1operatorwith geometric multiplicity one. Next, by using Greiner’s idea we derive that all points on theimaginary axis except zero belong to the resolvent set of the operator. Lastly, we determinethe adjoint operator of the operator and verify that0is an eigenvalue of the adjoint operator.Therefore, by combining these results with previous results we deduce that the time-dependentsolution of this model strongly converges to its steady-state solution. In Section4, we statesome further research problems.