Application Of Functional Analysis To Research On Dynamic Analysis Of Several Queueing Models

Queueing systems are widely used in transportation,inventory management,manufacturing management,computer network and telecommunications network.In this thesis,by using functional analysis theory and methods we do dynamic analysis for several queueing systems,so our results are important in view of theory and practice.In this thesis,by taking the M/G/1 queueing system with vacations and multiple phases of operations as an example,I introduce my research works about dynamic analysis of queueing models.First of all,by taking service time as a supplementary variable and using Markov process and total probability law we establish the dynamic mathematical model to describe the M/G/1 queueing system with vacations and multiple phases of operations.It is an equation system which consists of infinitely many first order partial differential equations with integral boundary conditions.By choosing a suitable state space,operators and their domains we convert the M/G/1 queueing model with vacations and multiple phases of operations into an abstract Cauchy problem in a Banach space,next under the condition that all service rates of n phases are bounded functions,by using the Hille-Yosida theorem,Phillips theorem and Fattorini theorem we prove that the underlying operator,which corresponds to the M/G/1 queueing system with vacations and multiple phases of operations,generates a positive contraction C0-semigroup that is isometric for a set containing the initial value.Therefore,we deduce that the M/G/1 queueing model with vacations and multiple phases of operations has a unique positive time-dependent solution which satisfies the probability condition.Fourthly,if the arrival rate of customers ?m and service rate of the server in m phase satisfy then by using the Greiner idea to perturb boundary condition and probability generating function we obtain the spectral distribution of the underlying operator,which corresponds to the M/G/1 queueing model with vacations and multiple phases of operations,on the imaginary axis,that all points on the imaginary axis except 0 belong to its resolvent set and 0 is its eigenvalue with geometric multiplicity one.By determining the adjoint operator of the underlying operator and applying ordinary differential equation theory and functional analysis theory we verify that 0 is an eigenvalue of the adjoint operator.Thus,we derive that if then the time-dependent solution of the M/G/1 queueing model with vacations and multiple phases of operations strongly converges to its steady-state solution.When the service rates of the server in all phases are constants,the M/G/1 queueing system with vacations and multiple phases of operations is called the M/M/1 queueing system with vacations and multiple phases of operations.We study spectral properties of the underlying operator which corresponds to the M/M/1 queueing model with vacations and multiple phases of operations.Firstly,we describe its point spectrum,thus we deduce that the C0-semigroup generated by the underlying operator is not compact,not eventually compact,essential growth bound of the C0-semigroup is 0,the C0-semigroup is not quasi-compact,it is impossible that the time-dependent solution of the M/M/1 queueing model with vacations and multiple phases of operations exponentially(uniformly)converges to its steady-state solution.This implies that it is impossible that the timedependent solution of the M/G/1 queueing model with vacations and multiple phases of operations exponentially(uniformly)converges to its steady-state solution.In other words,the convergence result "strong convergence" of the time-dependent solution of the M/G/1 queueing model with vacations and multiple phases of operations that we obtain is optimal.Moreover,we discuss other spectrum of the underlying operator which corresponds to the M/M/1 queueing model with vacations and multiple phases of operations.By using cone theory and monotone operator theory we prove that if then the dynamic queueing length of the M/G/1 queueing system with vacations and multiple phases of operations converges to its steady-state queueing length.