| Queueing theory is definitely a great important application stochastic process, while the GI / G/1 queueing system is the most typical and important model among all of the systems. The research for waiting time distribution is of great importance in the analysis process of queueing systems. Thus, the study of the waiting time distribution in a GI / G/1 queueing system is valuable.This paper is organized as follow, the first chapter is the introduction of basic knowledge. Here we introduce the development history of queueing theory, the research presentation of this field, the frame of queueing models and the research presentation of GI / G/1 queueing system. In chapter 2, we present some typical queueing models, a few of important probability distributions and stochastic processes have been introduced, it also can be seen as the preparation of the whole paper. Then chapter 3 is the main part of this paper. We provide a new method for solving the waiting time distribution GI / G/1 in queueing system, which is with the start of Lindley's integral equation, using the properties of queueing system and the method of integrating by part to give the two other useful forms of distribution. Besides, we use the technique of transform and complex variable with the properties of convolution and analytic function and the method of spectrum factorization to give the Laplace transform of the waiting time distribution. Then, we can solve the waiting time distribution with the inverting transform of the Laplace transform. In the last chapter we transform the problem of waiting time distribution which is Weiner-Hopf type into searching for the answer of linear equations, at the same time we discuss the convergence and the complexity by using iterative. Finally, as an example, we provide several typical models to check this method, and the result shows that our method is available with better accuracy, efficiency and case of implementation while the approximating error can be controlled in demand.
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