Mathematical Methods in the Theory of Queuing

by A. Y. Khinchin, D. M. Andrews (Translator), M. H. Quenouille (Translator)

Overview

Written by a prominent Russian mathematician, this concise monograph examines aspects of queuing theory as an application of probability. The three-part treatment begins with a study of the stream of incoming demands (or "calls," in the author's terminology). Subsequent sections explore systems with losses and systems allowing delay. Prerequisites include a familiarity with the theory of probability and mathematical analysis.
A. Y. Khinchin made significant contributions to probability theory, statistical physics, and several other fields. His elegant, groundbreaking work will prove of substantial interest to advanced undergraduates, graduate students, and professionals in the fields of statistics, probability, and operations research.

Product Details

ISBN-13:	9780486316314
Publisher:	Dover Publications
Publication date:	07/04/2013
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	128
File size:	13 MB
Note:	This product may take a few minutes to download.

Table of Contents

Preface

Part I The Incoming Stream of Calls

1 Theory of the Simple Stream

1 Definition and statement of the problem 11

2 Elementary solution 12

3 Method of differential equations 15

4 Intensity of a simple stream 17

5 A stream with a variable parameter 18

2 General Properties of Stationary Streams

6 The stream of demands as a random process 23

7 Fundamental properties of stationary streams 26

8 General form of a stationary stream without after-effects 30

3 Palm's Functions

9 Definition and proof of existence 37

10 Palm's formulae 39

11 The intensity of a stationary stream - Korolyook's theorem 41

4 Streams with Limited After-Effects

12 Another method of describing a stream of calls 43

13 Streams with limited after-effects 44

5 Limit Theorem

14 Statement of the problem-Palm's theorem 49

15 Limiting behaviour of functions 51

16 Limit theorem 55

Part II Systems with Losses

17 Introductory remarks 57

6 Erlang's Problem for a Finite Collection

18 Statement of the problem 60

19 Markoff's theorem 63

20 Erlang's formulae and equations 65

21 The ergodic theorem 69

7 Erlang's Problem for an Infinite Collection

22 Equations for the generating function 73

23 Solution of the problem 75

24 Stream with a variable parameter 77

25 The infinite collection with an arbitrary distribution function for the lengths of conversation 79

8 Palm's Problem

26 Statement of the problem 82

27 Elementary calculations 84

28 Palm's fundamental theorem 87

29 Deduction from the basic system of equations 88

30 Laplace's transformation 90

31 Determination of functions Ψ _r(t) 91

32 Expansion of functions Ψ_r(t) in partial fractions 93

33 Conclusion 94

Part III Systems Allowing Delay

9 The Case of an Exponential Distribution for the Length of Conversations

34 Probabilities of different states 96

35 The distribution function of the time of waiting 99

10 Single-Line Systems with a Fixed Length of Conversation

36 Difference-differential equations for the problem 102

37 Distribution function of the waiting time 104

11 General Theory of Single-Line Systems

38 Statement of the problem and definitions 106

39 Subsidiary hypotheses 107

40 The characteristic function of the time of waiting 114

Section Notes and References 119

Bibliography 120