Fundamentals of Queueing Networks
Volume 46 of the series Stochastic Modelling and Applied Probability pp 125158
SingleStation Queues
 Hong ChenAffiliated withFaculty of Commerce and Business Administration, University of British Columbia
 , David D. YaoAffiliated withDepartment of Operations Research and Industrial Engineering, Columbia University
Abstract
The main subject of this chapter is limit theorems for the queuelength and the workload processes in the G/G/1 queue. The limit theorems include the functional strong law of large numbers (FSLLN), the functional law of the iterated logarithm (FLIL), the functional central limit theorem (FCLT) and the strong approximation. In addition, we also establish an exponential rate of convergence result for the fluid approximation. The limit of the FSLLN and the FLIL is a single station fluid model. Because of this, the FSLLN is often known as the fluid approximation. Similarly, the limit of the FCLT and the strong approximation takes the form of a onedimensional reflected Brownian motion. Since this limit is a diffusion process, the FCLT has been conventionally known as diffusion approximation. Throughout this and the following chapters we shall follow the convention to use the terms “fluid approximation” and “diffusion approximation” interchangeably with their underlying limit theorems. (In contrast, in Chapter 10, where the proposed approximations are not necessarily supported by limit theorems, we shall use the term “Brownian approximation,” in keeping with the Brownian network models in the research literature as approximations for queueing networks.)
 Title
 SingleStation Queues
 Book Title
 Fundamentals of Queueing Networks
 Book Subtitle
 Performance, Asymptotics, and Optimization
 Pages
 pp 125158
 Copyright
 2001
 DOI
 10.1007/9781475753011_6
 Print ISBN
 9781441928962
 Online ISBN
 9781475753011
 Series Title
 Stochastic Modelling and Applied Probability
 Series Volume
 46
 Series ISSN
 01724568
 Publisher
 Springer New York
 Copyright Holder
 Springer Science+Business Media New York
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 Authors

 Hong Chen ^{(5)}
 David D. Yao ^{(6)}
 Author Affiliations

 5. Faculty of Commerce and Business Administration, University of British Columbia, V6T 1Z2, Vancouver, British Columbia, Canada
 6. Department of Operations Research and Industrial Engineering, Columbia University, 10027, New York, NY, USA
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