Queues and Point Processes

P. Franken; D. König; U. Arndt; V. Schmidt

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Queues and Point Processes

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Language: German

Published/Copyright: 1981

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This book is in the series

Volume 53 | Mathematische Lehrbücher und Monographien / Abteilung 2. Mathematische Monographien

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Business and Economics Business Management Management, Organization, Corporate Governance

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Frontmatter
1

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Preface
5

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Contents
9

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1. Random marked point processes and processes with an embedded marked point process
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1.0. Introduction
13

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1.1. Random marked point processes
14

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1.2. Stationary RMPP's. The Palm distribution
21

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1.3. Characterizations of the Palm distribution
29

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1.4. The distribution P⁰_L
39

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1.5. Processes with embedded marked point process
44

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1.6. Intensity conservation principle
47

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1.7. Remarks and references
53

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2. Time- and customer-stationary processes for queueing systems. Existence and ergodic theorems
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2.0. Introduction
58

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2.1. Notations
60

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2.2. General approach to existence, uniqueness and ergodic theorems First we shall make
62

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2.3. Systems without delay
68

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2.4. Systems with delay
77

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2.5. Remarks and references
86

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3. Continuity theorems for time- and customer-stationary quantities
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3.0. Introduction
88

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3.1. Systems without delay
89

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3.2. Systems with delay
95

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3.3. Remarks and references
101

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4. Relationships between time- and customer-stationary quantities. I: Basic systems
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4.0. Introduction
104

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4.1. Systems with a Poisson process of arrival instants
105

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4.2. Little's formulae
106

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4.3. The number of customers
111

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4.4. Some formulae concerning busy cycles
124

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4.5. Extensions of Takács' formulae for G/G/1/∞
126

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4.6. The work load in G/G/s/r
130

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4.7. Repairman problem
138

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4.8. Remarks and references
139

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5. Relationships between time- and customer-stationary quantities. II: Further systems
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5.0. Introduction
143

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5.1. Several customer types
143

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5.2. Group arrivals
150

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5.3. Varying service rate
152

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5.4. Uniformly bounded work load; warming up
155

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5.5. Remarks and references
157

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6. Insensitivity of stationary state probabilities for a class of queueing systems
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6.0. Introduction
160

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6.1. General queueing systems
161

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6.2. Construction of a suitable queueing process
163

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6.3. The equation system Z and the property E((I_j)_j€J)
166

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6.4. The main result
167

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6.5. Embedded probabilities and mean sojourn times of insensitive systems
171

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6.6. Poisson output of an insensitive queueing system
172

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6.7. Examples
174

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6.8. The structure of insensitive queueing systems
183

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6.9. Remarks and references
186

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Bibliography
190

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Result Index
200

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Symbol Index
202

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Index
206

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Publishing information

Pages and Images/Illustrations in book

eBook published on:

December 31, 1981

eBook ISBN:

9783112780497

Hardcover published on:

December 31, 1981

Hardcover ISBN:

9783112780480

Original publisher:

Akademie-Verlag Berlin

Edition:

Reprint 2024

Pages and Images/Illustrations in book

Main content:

208

https://doi.org/10.1515/9783112780497

eBook ISBN: 9783112780497

Hardcover ISBN: 9783112780480

Audience(s) for this book

College/higher education;

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Queues and Point Processes

Overview

Topics

Table of contents

Frontmatter

Preface

Contents

1.0. Introduction

1.1. Random marked point processes

1.2. Stationary RMPP's. The Palm distribution

1.3. Characterizations of the Palm distribution

1.4. The distribution P0L

1.5. Processes with embedded marked point process

1.6. Intensity conservation principle

1.7. Remarks and references

2.0. Introduction

2.1. Notations

2.2. General approach to existence, uniqueness and ergodic theorems First we shall make

2.3. Systems without delay

2.4. Systems with delay

2.5. Remarks and references

3.0. Introduction

3.1. Systems without delay

3.2. Systems with delay

3.3. Remarks and references

4.0. Introduction

4.1. Systems with a Poisson process of arrival instants

4.2. Little's formulae

4.3. The number of customers

4.4. Some formulae concerning busy cycles

4.5. Extensions of Takács' formulae for G/G/1/∞

4.6. The work load in G/G/s/r

4.7. Repairman problem

4.8. Remarks and references

5.0. Introduction

5.1. Several customer types

5.2. Group arrivals

5.3. Varying service rate

5.4. Uniformly bounded work load; warming up

5.5. Remarks and references

6.0. Introduction

6.1. General queueing systems

6.2. Construction of a suitable queueing process

6.3. The equation system Z and the property E((Ij)j€J)

6.4. The main result

6.5. Embedded probabilities and mean sojourn times of insensitive systems

6.6. Poisson output of an insensitive queueing system

6.7. Examples

6.8. The structure of insensitive queueing systems

6.9. Remarks and references

Bibliography

Result Index

Symbol Index

Index

Bibliographic data

1.4. The distribution P⁰_L

6.3. The equation system Z and the property E((I_j)_j€J)