Stochastic Network Calculus

240 Pages·2008·1.586 MB·English

by  Yong Liu, Yuming Jiang (auth.)

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Stochastic
Network
Calculus
Stochastic Network Calculus
·
Yuming Jiang Yong Liu
Stochastic Network Calculus
ABC
YumingJiang,BSc,MEng,PhD YongLiu,BEng,MEng,PhD
DepartmentofTelematics OpticalNetworksLab
NorwegianUniversityofScience DepartmentofElectricalandComputer
andTechnology Engineering
Trondheim NationalUniversityofSingapore
Norway Singapore
ISBN:978-1-84800-126-8 e-ISBN:978-1-84800-127-5
DOI:10.1007/978-1-84800-127-5
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Preface
Networkcalculusisatheorydealingwithqueuingsystemsfoundincomputer
networks. Its focus is on performance guarantees. Central to the theory is the
use of alternate algebras such as the min-plus algebra to transform complex
network systems into analytically tractable systems. To simplify the analy-
sis, another idea is to characterize traffic and service processes using various
bounds. Since its introduction in the early 1990s, network calculus has devel-
oped along two tracks—deterministic and stochastic. This book is devoted to
summarizing results for stochastic network calculus that can be employed in
the design of computer networks to provide stochastic service guarantees.
Overview and Goal
Like conventional queuing theory, stochastic network calculus is based on
properly defined traffic models and service models. However, while in con-
ventional queuing theory an arrival process is typically characterized by the
inter-arrival times of customers and a service process by the service times of
customers, the arrival process and the service process are modeled in net-
work calculus respectively by some arrival curve that (maybe probabilisti-
cally) upper-bounds the cumulative arrival and by some service curve that
(maybe probabilistically) lower-bounds the cumulative service. The idea of
usingboundstocharacterizetrafficandservicewasinitiallyintroducedforde-
terministic network calculus. It has also been extended to stochastic network
calculus by exploiting the stochastic nature of arrival and service processes.
While stochastic network calculus can be considered as generalized from de-
terministic network calculus, this generalization is not straightforward. De-
terministic network calculus is based on a worst-case analysis that usually
does not need to worry about the stochastic behavior of traffic and service.
On the other hand, stochastic network calculus must take into consideration
in the analysis the stochastic characteristics of traffic and service processes
to better make use of their statistical multiplexing gains. For example, while
independentcaseanalysisisnotconsideredindeterministicnetworkcalculus,
it is critical in stochastic network calculus since it can usually provide much
better results.
viii Preface
The main goal of this book is to summarize results for stochastic service
guaranteeanalysisusingthetheoryofstochasticnetworkcalculus.Sincemany
networks (such as wireless networks) provide only stochastic service guaran-
tees, and many applications (such as multimedia applications) perform well
with stochastic service guarantees, the results in this book will be useful for
analysis and provision of service guarantees in such network scenarios.
Structure
This book is structured as follows. Chapter 1 gives an introduction to the
basic properties required from a theory for tractable performance analysis of
computer networks. Also in this chapter, the background knowledge useful
for helping understand the rest of the book is presented. Chapter 2 reviews
fundamental concepts and results of deterministic network calculus. They in-
clude the arrival curve traffic model, the service curve server model, and the
basic properties supported by deterministic network calculus. From Chapter
3toChapter9,wefocusonintroducingimportantconceptsandresultsinthe
contextofstochasticnetworkcalculus.Specifically,Chapter3introducesvari-
ousstochastictrafficmodelsandtheirrelationshipswitheachotheraswellas
with some well-known traffic models such as the effective bandwidth model.
Chapter4definesstochasticservermodelsforstochasticnetworkcalculusand
introduces their relationships with each other. Chapter 5 summarizes the ba-
sic properties of stochastic network calculus under different combinations of
traffic and server models introduced in Chapters 3 and 4, that are obtained
withoutconsideringthepossibleindependenceofarrivalandserviceprocesses.
Chapter 6 focuses on independent case analysis, under which the basic prop-
erties of stochastic network calculus under different combinations of traffic
and server models are presented. In Chapter 7, the analysis is on stochastic
service guarantees under different scheduling disciplines. Also in this chapter,
an application using the analysis results in admission control is presented. In
Chapter 8, stochastic network calculus is extended and applied to study to
the extent to which a flow becomes non-conformant with respect to its initial
traffic characterization after it passes through a network. Finally, in Chapter
9, the theory of stochastic network calculus is applied to study generalized
processor sharing systems with long-range dependent traffic inputs. In order
toprovideamorecompletepictureofthefield,anappendixispresentedthat
summarizes the book and discusses open research challenges in the area of
stochastic network calculus.
Use of the Book
This book presents an overall picture of the state of the art of stochastic
service guarantee analysis and a comprehensive treatment of this active re-
search area. The content of the book can be divided into two parts. The first
part,consistingofChapters1to6,providesthemainsetofresults.Thesecond
part, Chapters 7 to 9, extends the first part with more results and example
applicationsofstochasticnetworkcalculus.Intheappendix,acomprehensive
discussionofopenresearchchallengesintheareaofstochasticnetworkcalcu-
lusisprovided.Thebookcanbeusedinadvancedundergraduateorgraduate
Preface ix
coursesonperformanceevaluationofcomputernetworks.Suchacourseshould
coverthefullfirstpartofthebook,whichprovidesafirstcourseonstochastic
network calculus. The second part can be used flexibly, where each chapter
is self-contained. In addition, researchers in the area of performance evalua-
tion of computer networks and provision and analysis of service guarantees
in computer networks can also benefit from the comprehensive discussion of
stochastic service guarantee issues in this book.
Notes and Acknowledgments
Part of content of this book has been lectured by the first author in two
graduate courses related to traffic analysis of communication networks at the
Norwegian University of Science and Technology since 2006. Many results in
the book are based on works authored or co-authored by the two authors.
We would like to thank other people who have contributed to these works.
They include Dr. Peder J. Emstad, Dr. Markus Fidler, Dr. Shengming Jiang,
Ms. Anne Nevin, Dr. Victor F. Nicola, Dr. Chunming Qiao, Dr. Chen-Khong
Tham, Dr. Ian L.-J. Thng, Dr. Qinghe Yin, and Dr. Xiang Yu. In addition,
wewouldliketothankthepublisherandMs.CatherineBrettandMr.Wayne
Wheeler there for their patience, support, and help.
Trondheim, April 2008 Yuming Jiang
Singapore, April 2008 Yong Liu
Contents
Preface ........................................................ vii
List of Figures.................................................xvii
List of Tables .................................................. xix
1 Introduction............................................... 1
1.1 Quality of Service Guarantees............................. 2
1.2 Basic Properties for Network Analysis...................... 3
1.3 Notation and Mathematical Background.................... 6
1.3.1 Notation ......................................... 6
1.3.2 Min-Plus Algebra Basics ........................... 8
1.3.3 Maximum Horizontal Distance and Maximum
Vertical Distance .................................. 10
1.4 Random Variable and Stochastic Process Basics ............. 10
1.4.1 Random Variables................................. 10
1.4.2 Stochastic Processes ............................... 13
1.4.3 Stochastic Ordering................................ 13
1.5 Min-Plus Linearity of Queuing Systems .................... 15
1.6 Summary and Bibliographic Comments .................... 16
Problems ................................................... 17
2 Deterministic Network Calculus ........................... 19
2.1 Traffic Models .......................................... 19
2.1.1 (σ,ρ) Traffic Characterization....................... 19
2.1.2 Arrival Curve ..................................... 20
2.1.3 Envelope Process.................................. 22
2.2 Server Models........................................... 23
2.2.1 Service Curve..................................... 23
2.2.2 Network Elements Offering Service Curves............ 24
2.2.3 Strict Service Curve ............................... 26
xii Contents
2.2.4 Service Envelope Process ........................... 26
2.3 Basic Results ........................................... 27
2.3.1 Service Guarantees ................................ 28
2.3.2 Output Characterization ........................... 30
2.3.3 Concatenation Property............................ 31
2.3.4 Leftover Service ................................... 33
2.3.5 Superposition ..................................... 34
2.3.6 Example: Analysis of a Network
of Arbitrary Topology.............................. 34
2.4 Summary and Bibliographic Comments .................... 36
Problems ................................................... 37
3 Traffic Models for Stochastic Network Calculus............ 41
3.1 (σ(θ),ρ(θ)) Traffic Characterization........................ 41
3.2 t.a.c. Stochastic Arrival Curve ............................ 42
3.2.1 Difficulties and Additional Constraints ............... 45
3.3 v.b.c. Stochastic Arrival Curve ............................ 48
3.4 m.b.c. Stochastic Arrival Curve ........................... 53
3.4.1 θ-m.b.c. Stochastic Arrival Curve.................... 56
3.5 Stochastic Envelope Process .............................. 57
3.6 Summary and Bibliographic Comments .................... 60
Problems ................................................... 63
4 Server Models for Stochastic Network Calculus............ 65
4.1 Weak Stochastic Service Curve............................ 65
4.2 Stochastic Service Curve ................................. 66
4.2.1 θ-Stochastic Service Curve.......................... 68
4.3 Stochastic Strict Service Curve............................ 69
4.3.1 Stochastic Strict Server due to Impairment ........... 71
4.4 Service Envelope Process ................................. 74
4.4.1 Latency Rate Characterization ...................... 76
4.4.2 Relationship with (σ(θ),ρ(θ)) Traffic Characterization . 77
4.5 Summary and Bibliographic Comments .................... 78
Problems ................................................... 79
5 Basic Properties of Stochastic Network Calculus........... 83
5.1 Service Guarantees ...................................... 83
5.1.1 Backlog Bound.................................... 83
5.1.2 Delay Bound...................................... 85
5.2 Output Characterization ................................. 88
5.2.1 Output t.a.c Stochastic Arrival Curve................ 88
5.2.2 Output v.b.c. Stochastic Arrival Curve............... 90
5.2.3 Output m.b.c Stochastic Arrival Curve............... 93
5.2.4 Output θ-m.b.c. Stochastic Arrival Curve............. 96