The main subject of this chapter is limit theorems for the queue-length 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 one-dimensional 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.)
KeywordsBrownian Motion Queue Length Diffusion Approximation Heavy Traffic Strong Approximation
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