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QUEUE THEORY EXPLANATION MG1 - A Short Introduction to...

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A Short Introduction to Queueing Theory Andreas Willig Technical University Berlin, Telecommunication Networks Group Sekr. FT 5-2, Einsteinufer 25, 10587 Berlin email: [email protected] July 21, 1999
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Contents 1 Introduction 3 1.1 Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Scope of Queueing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Basic Model and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Little’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Markovian Systems 9 2.1 The M/M/1-Queue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Steady-State Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 Some Performance Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 The M/M/m-Queue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.1 Steady-State Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.2 Some Performance Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 The M/M/1/K-Queue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 Steady-State Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.2 Some Performance Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 A comparison of different Queueing Systems . . . . . . . . . . . . . . . . . . . . . . . . 17 3 The M/G/1-System 19 3.1 Some Renewal Theory Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 The PASTA Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 The Mean Response Time and Mean Number of Customers in the System / Pollaczek- Khintchine Mean Value Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4 Distribution of the Number of Customers in the System . . . . . . . . . . . . . . . . . . 23 3.5 Two Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.5.1 The M/M/1 Queue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.5.2 The M/D/1 Queue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.6 Distribution of the Customer Response Times . . . . . . . . . . . . . . . . . . . . . . . . 27 4 Queueing Networks 28 4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Product Form Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2.1 The Jackson Theorem for Open Queueing Networks . . . . . . . . . . . . . . . 32 4.2.2 The Gordon-Newell Theorem for Closed Queueing Networks . . . . . . . . . . 33 1
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4.3 Mean Value Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 A Probability Generating Functions and Linear Transforms 36 A.1 Probability Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 A.2 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2
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Chapter 1 Introduction 1.1 Disclaimer This script is intended to be a short introduction to the field of queueing theory, serving as a mod- ule within the lecture “Leistungsbewertung von Kommunikationsnetzen” of Prof. Adam Wolisz from the Telecommunication Networks Group at Technical University Berlin. It covers the most im- portant queueing systems with a single service center, for queueing networks only some basics are mentioned. This script is neither complete nor error free. However, we are interested in improving this script and we would appreciate any kind of (constructive) comment or “bug reports”. Please send all suggestions to [email protected] . In this script most of the mathematical details are omitted, instead often “intuitive” (or better: prosaic) arguments are used. Most of the formulas are only used during a derivation and have no numbers, however, the important formulas are numbered. The author was too lazy to annotate all statements with a reference, since most of the material can be found in the standard literature. 1.2 Scope of Queueing Theory Queueing Theory is mainly seen as a branch of applied probability theory. Its applications are in different fields, e.g. communication networks, computer systems, machine plants and so forth. For this area there exists a huge body of publications, a list of introductory or more advanced texts on queueing theory is found in the bibliography. Some good introductory books are [9], [2], [11], [16].
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