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.
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