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Technical Desiderata

  • Hong Chen
  • David D. Yao
Part of the Stochastic Modelling and Applied Probability book series (SMAP, volume 46)

Abstract

This chapter collects background materials for the many limit theorems in queues and queueing networks that will appear in later chapters. We start with presenting some preliminaries in basic probability theory, such as almost sure convergence and weak convergence, Donsker’s theorem and Brownian motion, in Sections 5.1–5.3. Then in Sections 5.4–5.6, we focus on a pair of fundamental processes: the partial sum of i.i.d. random variables and the associated renewal counting process. The pair serves as a building block for modeling many queueing systems. We show that under different time—space scaling the pair converges differently, leading to functional versions of the strong law of large numbers and the central limit theorem. Furthermore, with additional moment conditions (on the i.i.d. random variables), we can refine these limits via functional versions of the law of iterated logarithms and strong approximations. When the generating function of the i.i.d. random variables exist, we can further characterize the convergence rate via exponential bounds.

Keywords

Brownian Motion Weak Convergence Sample Path Wiener Process Iterate Logarithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Hong Chen
    • 1
  • David D. Yao
    • 2
  1. 1.Faculty of Commerce and Business AdministrationUniversity of British ColumbiaVancouverCanada
  2. 2.Department of Operations Research and Industrial EngineeringColumbia UniversityNew YorkUSA

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