agh university of science and technologyjanowski/lucjanjanowskidissertation.pdf · acknowledgements...

AGH University of Science and TechnologyFaculty of Electrical Engineering, Automatics, Computer Science and Electronics

Ph.D Thesis

Lucjan Janowski

Analysis of a self-similar traffic

envelope for a trade-off of leaky

bucket algorithm parameters

Supervisor:

Prof. Dr Hab. Inż. Zdzisław Papir

AGH University of Science and Technology

Faculty of Electrical Engineering, Automatics, Computer Science and ElectronicsDepartment of Telecommunications

Al. Mickiewicza 30, 30-059 Kraków, Poland

tel. +48 12 6345582

fax +48 12 6342372

www.agh.edu.pl

www.kt.agh.edu.pl

Reviewers:

Prof. dr hab. inż. Tadeusz CzachórskiProf. dr hab. inż. Andrzej Jajszczyk

Copyright c© Lucjan Janowski, 2006All rights reserved

Printed in Poland

Acknowledgements

This dissertation has been written by me but it has been more than anything, acollaborative effort and a work of the heart of many.This dissertation could not have been written without the support of my

dissertation advisor Prof. Zdzisław Papir. I wish to thank Prof. Papir for hisbelief in me, for his encouragement and advice. He not only helped me withsolving and correcting my scientific and language mistakes but he supported methroughout all my research work even if my problems were not related to scientificor university work.He is my leader and exemplary of a high quality scientific work who always

distinguishes information from knowledge. I hope that as a young scientist Iwould be able to follow his path and at least this way show Prof. Papir howgrateful I am for all of his hard work and effort.I wish to thank Margo who helped me to improve my English grammar and

style. I must also thank Adam Ćmiel and Karol Życzkowski who would alwaysfind time for me and help me to solve my problems. Last but not least, I wantto thank Jacek Kmiecik for his help in exploring the world of LATEX.PS. The final version of the thesis is improved by numerous detail corrections

and valuable remarks about grammar and formatting made by Prof. AndrzejJajszczyk. I would like to thank Prof. Jajszczyk not only for correcting but alsofor giving me very interesting suggestions how to write better articles.

iv Acknowledgements

Abstract

Numerous investigations have revealed that the packet network traffic is burstyand highly correlated. If many large bursts meet in a single queue an overflow isunavoidable even for a slightly loaded network, therefore shaping packet trafficbecomes a crucial issue. One of the most commonly used traffic shaping algo-rithms is a leaky bucket. An important property of a leaky bucket algorithm isthat one can find different pairs of leaky bucket parameters (r, b) for the samedrop probability pd. The burstiness curve binds leaky bucket parameters (r, b)for the fixed drop probability pd. The basic goal of the analysis presented inthe dissertation is computing a burstiness curve for the FARIMA traffic modelwith support of original results obtained for its envelope analysis; an FGN modelhas been used as a reference. The motivation for finding the burstiness curve isthat trading r and b values for the same pd yields in changing traffic parame-ters without changing QoS. The reason for choosing the FARIMA model is thatthe FARIMA process can exhibit SRD and LRD structures independently andsimultaneously and a distribution of a FARIMA process is known. The obtainedresults have been confirmed by a simulation study. An additional achievementof this dissertation is derivation of an envelope process for the three types ofFARIMA models.

vi Abstract

Contents

Acknowledgements iii

Abstract v

List of figures xii

List of tables xiv

1 Introduction 11.1 Motivation and thesis . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Chapters review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Self-similar stochastic processes 72.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Long Memory Processes LRD . . . . . . . . . . . . . . . . . . . . . 112.3 Hurst parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Queue analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.1 Norros formula . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.2 Other results . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 FGN and FARIMA processes 233.1 FGN process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 ARIMA process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 FARIMA process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.1 FARIMA definition . . . . . . . . . . . . . . . . . . . . . . . 293.3.2 FARIMA properties . . . . . . . . . . . . . . . . . . . . . . 303.3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

viii CONTENTS

4 Envelope process 374.1 Envelope construction . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Envelope process for FARIMA process . . . . . . . . . . . . . . . . 46

4.3.1 Envelopes for selected FARIMA processes . . . . . . . . . . 464.3.2 Envelope process parameters identification . . . . . . . . . . 49

4.4 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 Queue length computation 535.1 Queue process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 Busy period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.3 Drop Tail parameter computation . . . . . . . . . . . . . . . . . . . 605.4 Leaky Bucket parameter computation . . . . . . . . . . . . . . . . 645.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6 Burstiness curve 676.1 Definition and application . . . . . . . . . . . . . . . . . . . . . . . 676.2 Burstiness curve for an FGN model . . . . . . . . . . . . . . . . . . 696.3 Burstiness curve for FARIMA models . . . . . . . . . . . . . . . . 736.4 Overestimation analysis . . . . . . . . . . . . . . . . . . . . . . . . 766.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7 Summary 81

Appendix A 83

Appendix B 89

Appendix C 95

Appendix D 101

Bibliography 105

Acronyms and symbols index 113

Index 116

List of Figures

2.1 An example of a time scale change for Ethernet measurements. . . 92.2 A comparison between SRD (AR(1)) process and LRD (FMA(1))

process with their autocorrelation functions. . . . . . . . . . . . . . 132.3 Examples illustrate how a time scale influences a process deviation. 142.4 H − ss, si process example (figure (a)) and increment process ob-

tained for the process from figure (a) (figure (b)). . . . . . . . . . . 162.5 SRD process (AR(1)) autocorrelation function and LRD process

(increments of H − ss, si process) autocorrelation function for dif-ferent Hurst parameter values. . . . . . . . . . . . . . . . . . . . . 18

2.6 An average queue length as a function of utilisation for differentHurst parameter values and M/M/1 queue model. . . . . . . . . . 19

2.7 An average queue length for the same measurements and differentpermutations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1 H − ss, si process example (figure (a)) and the increment processobtained for the process from figure (a) (figure (b)). . . . . . . . . 25

3.2 ARMA(1,1) autocorrelation functions for different φ and θ param-eters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 FAR(1,d) autocorrelation functions for different φ and d parame-ters. Note that for φ = 0 the FAR(1,d) process is the FARIMA(0,d,0)process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 FMA(d,1) autocorrelation functions for different θ and d parameters. 33

4.1 The example of a deterministic function with two different enve-lope processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 An example of a stochastic process Zn ∼ U(0, n) varies in boundedintervals with its envelope process . . . . . . . . . . . . . . . . . . 39

4.3 An example of a stochastic process Zn ∼ Exp( 1n ) and two envelope

processes obtained for different ǫ values . . . . . . . . . . . . . . . 40

x LIST OF FIGURES

4.4 An example of two paths of the FBM process and two envelopeprocesses obtained for different ǫ values. . . . . . . . . . . . . . . . 43

4.5 An example of two paths of the accumulated FARIMA(0,d,0) d=0.3and the envelopes for two different ǫ values. . . . . . . . . . . . . . 47

4.6 An example of two paths of the accumulated FMA(d,1) d=0.4,φ = 0.9 and the envelopes for two different ǫ values. . . . . . . . . 49

4.7 An example of two paths of the accumulated FMA(d,1) d=0.2,θ = −0.8 and the envelopes for two different ǫ values. . . . . . . . . 50

4.8 The 95% confidence intervals obtained from simulation of a prob-ability that an FMA(d,1) process exceeds the envelope process.d=0.1, θ = −0.9 and ǫ = 0.2. . . . . . . . . . . . . . . . . . . . . . 51

4.9 The 95% confidence intervals obtained from simulation of a prob-ability that an FAR(1,d) process exceeds the envelope process.d=0.4, φ = 0.5 and ǫ = 0.05. . . . . . . . . . . . . . . . . . . . . . . 52

5.1 Queue length, accumulated and served traffic processes. . . . . . . 555.2 An example of an accumulated workload process proposed by Nor-

ros for Zt with the FBM distribution with H = 0.8, a = 1, σ = 1. . 565.3 The queue process for the accumulated workload process presented

in Figure 5.2 and for the link capacity c = 1.5. . . . . . . . . . . . 575.4 The maximum delayDmax in ms as a function of the link utilisation

obtained for different Hurst parameter values. a = 138 bit/ms,σ = 420 bit/ms and ǫ = 0.05. . . . . . . . . . . . . . . . . . . . . . 62

5.5 The LB algorithm scheme and parameters. . . . . . . . . . . . . . 65

6.1 An example of two different LB parameters with the same quantityof dropped packets (red packets) value. . . . . . . . . . . . . . . . . 69

6.2 An illustration of the influence of the Hurst parameter value onthe burstiness curve shape. a = 138, η = 0.8 and ǫ = 0.05. . . . . . 71

6.3 An illustration of the difference between a theoretical drop proba-bility ǫ and a drop probability ps obtained by a simulation study;a = 60 kbit/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.4 The differences between the theoretical drop probability ǫ and thedrop probability ps obtained by a simulation study obtained forthe FAR(1,d) and FMA(d,1) models. Traffic parameters: a = 60kbit/s, η = 1.0 and H = 0.7. . . . . . . . . . . . . . . . . . . . . . . 75

D.1 The influence of t unit on Z1 distribution. . . . . . . . . . . . . . . 102D.2 An example of two different divisions of the time scale for the

discrete time process. . . . . . . . . . . . . . . . . . . . . . . . . . . 103

List of Tables

2.1 Autocorrelation functions decrease for different processes. Thefirst and second examples are SRD processes and the third andfourth examples are LRD processes. . . . . . . . . . . . . . . . . . 17

5.1 A busy period in milliseconds for different workload processes anddifferent correlation structure parameters; a = 138 bit/ms, σ =420 bit/ms and ǫ = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Qmax in kbit for different workload processes and different corre-lation structure parameters; a = 138 bit/ms, σ = 420 bit/ms andǫ = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.3 Link utilisation ρ for different workload processes and differentcorrelation structure parameters; a = 138 bit/ms, σ = 420 bit/msand ǫ = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.1 A comparison between a theoretical drop probability ǫ and a dropprobability ps obtained by simulation study for the FGN model;η = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72



6.4 A comparison between the theoretical drop probability ǫ and thedrop probability ps obtained by a simulation study for the FAR(1,d)model; η = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.5 A comparison between the theoretical drop probability ǫ and thedrop probability ps obtained by a simulation study for the FAR(1,d)model; η = 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

xii LIST OF TABLES

6.6 A comparison between the theoretical drop probability ǫ and thedrop probability ps obtained by a simulation study for the FMA(d,1)model; η = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.7 A comparison between the theoretical drop probability ǫ and thedrop probability ps obtained by a simulation study for the FMA(d,1)model; η = 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Chapter 1

Introduction

The main considered topic in the dissertation is a single leaky bucket algorithmfed by an LRD workload process and specifically a burstiness curve for tradingleaky bucket parameters. This chapter presents the motivation underlying thechoice of the topic and then the thesis of the dissertation as well as the tools usedto prove it. The last section offers a brief presentation of all the chapters.

1.1 Motivation and thesis

The Internet (packet) traffic is difficult to describe since each protocol from theprotocol stack influences the traffic parameters. Moreover, a user behaviour isdifferent from the one in the traditional telephone traffic. Additionally, Internetdesign is more complicated since not only the capacity of transmission links butseveral QoS parameters (like the delay and packet loss ratio) have to be takeninto account.The Internet is a complex system, however, at its very bottom queueing algo-

rithms are used. Therefore, considering a single queue is an important topic thathas been analysed by means of various methods [15, 69]. A possibility of usingqueueing analysis results strongly depends on a kind of the considered workloadprocess. If the workload process is far from the real traffic behaviour the resultscannot be taken into consideration.A key proposition of a proper workload process for a packet traffic for the

last fifteen years has been a self-similar process that exhibits an important prop-erty of the Internet traffic, i.e., the Long Range Dependence (LRD). Numerouspapers have presented that Internet traffic is self-similar for at least several timescales. Other works have shown that if a workload process is self-similar (LRD),an average queue length is much longer than for the traditional Short Range

2 Introduction

Dependence (SRD) models.Since a packet network cannot be controlled only by the duration of connection

time and the link capacity because other parameters like the packet delay areinfluencing QoS, one of the most important challenges of a packet network designis how to manage the quality of service and traffic control. Again, a solution thathas worked for the telephone traffic cannot be applied.For a packet network any traffic source has some fluctuations of the mean

value. The problem of most common traffic sources is that these fluctuationsare large and highly correlated which makes traffic multiplexing more difficult.The reason is that the bursts from different sources can appear more or lesssynchronously and, in consequence, cause an overflow even for a slightly loadednetwork. Therefore, a traffic shaping algorithm is needed and a commonly usedsolution is to apply a leaky bucket algorithm. This algorithm limits the meanvalue of the source while allowing to send some limited bursts of data. Such asolution makes it possible to manage traffic flows since the worst behaviour of atraffic source limited by a leaky bucket algorithm is known.In this dissertation, a burstiness curve has been computed for a FARIMA ac-

cumulated traffic using the envelope approach for bounding a stochastic process.A burstiness curve is a curve that relates a leaky bucket parameters, i.e. a tokenaccumulation rate r and a token bucket size b. A token bucket size is a functionof a token accumulation rate and vice verse. The function is computed with theassumption that for each pair (r, b(r)) the drop probability pd is the same. Sucha curve defines how to change leaky bucket parameters for the same quality ofservice, described by the drop probability. Therefore, knowing the shape of theburstiness curve results in a better resource allocation and link utilisation.One of the application of the burstiness curve is video traffic management.

Video traffic becomes more and more important in modern teletraffic networks.The high compression algorithms decrease the mean bit rate for a video sourcebut they increase bit rate fluctuations. With the higher bit rate fluctuations amanagement of a network delivering video traffic is getting more difficult.The obtained results can be used in the algorithms which help in determining

proper number of connections which a single video server can serve. For instance,if a network is not heavy loaded then some larger bursts, i.e. higher b values arenot a danger for other connections. However, if a network is heavily loaded thena burst size cannot be too large. The burstiness curve makes it possible to changeb in such a way that from the point of view of a source the drop probability isnot changed.

1.2 Methodology 3

As a summary of the presented motivation premises and a research work donethe following thesis is proposed:The envelope approach applied to a FARIMA accumulated work-

load specifies the leaky bucket burstiness curve (r, b) yielding in aflexible traffic shaping at the same drop probability.

1.2 Methodology

This dissertation is basically a theoretical work. The most important tool thathas been used in the dissertation is an envelope process. The concept of envelopeprocesses is described in Chapter 4, here only the most important properties aregiven.An envelope process is a deterministic function Z(t) computed for a stochastic

process Zt. The crucial property of an envelope process is

P (Zt > Z(t)) = ǫ (1.1)

where ǫ is a chosen probability.Note that the function Z(t) is 1− ǫ quantile of the process Zt and, therefore,

if resources are sufficient for an envelope Z(t) then with probability 1 − ǫ theyare sufficient for Zt as well.Such a definition of the envelope process makes it possible to substitute a

stochastic process which describes a workload process by a deterministic functionand in consequence considering an algebraic equation instead of a stochastic one.Such a substitution is valid with the chosen probability ǫ. The main example ofexpressing a queue process by a stochastic equation used in the dissertation is aleaky bucket algorithm.The other important result presented in the dissertation is derivation of an

envelope of an accumulated FARIMA process. The motivation for choosing sucha process is that a video traffic exhibits an SRD and LRD independently, thesame as a FARIMA process does. Derivation of an envelope process is basedon probabilistic properties of the underlying stochastic process. An analyticalform of the envelope has been computed with the help of a symbolic kernelof the Mathematica package. The obtained results have been proved by themathematical induction which is shown in Appendices A-C.The burstiness curve obtained by an envelope analysis has been validated

by a simulation study. The simulation has been conducted in the Mathematicapackage, too. All simulation results have been obtained from numerous repeatedsimulations and, therefore, their errors have been computed.

4 Introduction

1.3 Chapters review

All figures and tables presented in this dissertation were plotted with usage ofprocedures written in Matlab or Mathematica package. The procedures are basedon results presented in the literature or obtained by the author. Figures 2.6 and2.7 are based on the figures presented in [77] and [23], respectively. The restfigures are concepts of the author and/or the supervisor Prof. Papir.This dissertation is organised as follows:In Chapter 2 the self-similar processes have been presented. At the be-

ginning of the chapter a theoretical introduction and the notation of self-similarprocesses are given. The difference between the change of a time scale for discreteand continuous time processes has been presented. Then an LRD property as aconsequence of a correlation structure for the special case of the self-similar pro-cesses that is the self-similar processes with independent increments have beenshown. The consequences of existing LRD in a network traffic are presented,too. Then the Hurst parameter as well as the relationship with β parameterdescribing the LRD property are given. All these results make it possible to un-derstand queueing consequences of a self-similar workload process. As an exampleof the influence of an LRD traffic on a queue process, an interesting experimentaccording to Willinger’s work has been shown. The last section of Chapter 2describes briefly different solutions of the queue distribution for self-similar orapproximately self-similar processes obtained using a different methodology.In Chapter 3 two self-similar processes have been described. The first is

a basic and strictly self-similar process called Fractional Gaussian Noise (FGN).FGN is a generalisation of a Gaussian process where the subsequent variablesare not independent but given as a function of the Hurst parameter. The mostimportant properties of FGN and closely related Fractional Brownian Motionhave been presented. The next section describes an ARIMA process. The ARIMAprocess is a linear process describing its values as a function of random noise andprevious values of the process. Such a description easily determines the shapeof an SRD correlation. The next section describes a more general form of theARIMA process called a Fractional ARIMA (FARIMA) process. The FARIMAprocess can describe LRD and, similarly as the ARIMA process, different SRDstructures. A short description of applications of the FARIMA process has beenshown in the last section.In Chapter 4 an envelope process has been described. In the beginning, a

simple idea of an envelope process built for deterministic and simple stochasticprocesses is shown. Then an envelope process for FGN and α-stable processesare shown. Both results have been presented in literature. Additionally, a shortdescription of α-stable processes is given. In Section 4.3 an envelope process forthree different FARIMA processes has been presented. This result is one of the

1.3 Chapters review 5

results obtained by the author of this dissertation. In this section, a simulationconfirmation has been presented. The last subsection presents a description ofmodel parameters computations.In Chapter 5 a queue process is described. In the beginning, an idea of

analysing a queue process by a deterministic equation is described and justified.According to this assumption interesting queue properties have been computedfor FGN, α-stable and FARIMA processes. The first of them is a busy period.The result shows clearly the influence of an LRD structure on the mean busy pe-riod length. The next result is a drop tail queue computation where the influencesof an LRD and SRD structure on the maximum delay and the link utilisation havebeen shown. The last considered queueing algorithm is a leaky bucket algorithmfor computing of the drop probability for different leaky bucket parameters. Theresults obtained for FARIMA processes could be achieved because of the envelopeprocess found by the author and presented in the previous section.In Chapter 6 an idea of a burstiness curve is presented and the references

to a heuristic algorithm are given. In Section 6.2 and 6.3 an analytic form of theburstiness curve has been presented. Each result has been validated by a simula-tion study where an assumption of the overestimation of the drop probability hasbeen checked. Some limitations of the traffic parameters for which the obtainedresults are valid have been presented, too. An example of such parameters ob-tained for different video traffic sources has been shown and compared with thelimitations.In Appendices A-C proofs of the equations of the envelope processes given

in Section 4.3 are presented.In Appendix D a problem of unit computation for self-similar processes has

been described in details.

6 Introduction

Chapter 2

Self-similar stochastic

processes

Self-similarity derives itself from the fractal theory, which can be found as oneof the most interesting and widely applied mathematical theories developed inthe last century. Fractals are applied to numerous topics belonging to arts, com-puter graphics, animation, physics, and mathematics. The self-similar stochasticprocesses (SS), which may be named as stochastic fractals, have numerous appli-cations from engineering to financial market models.In this chapter the self-similar stochastic processes are brought closer, a no-

tation is introduced and the most important theorems and properties of SS pro-cesses are recalled.

2.1 Definition

The formal definition of an SS process can be found in [21]. However, in thedissertation the SS definition given in [31] is used which is a theorem proved in[21].

Definition 1 A stochastic process {Zt, t ≥ 0} is said to be self-similar with Hparameter (H − ss) if

{Zat} d= aH{Zt} (2.1)

where d= denotes the equality in distribution, i.e. for all functions f that arecontinuous and limited Ef(Zat) = Ef(aHZt). H parameter is called the Hurstparameter passing assumption H ≥ 0 [21]. In telecommunication applicationsthe Hurst parameter varies in the range H ∈ [12 , 1) [8].

8 Self-similar stochastic processes

The Hurst parameter is a self-similarity measure and it is described in thefollowing part of the chapter after the introduction of LRD processes. The aparameter can be interpreted as a stochastic process time scale change. Accordingto the a parameter interpretation Definition 1 means that the process on a timescale at has the same distribution as the process on a time scale t exact to aconstant parameter aH . The interpretation explains the connection between SSprocesses and geometric (deterministic) fractals which shape repeats on differentobservation’s scales.Data transmission measurements are of discrete nature and only for some

special cases they can be approximated by time continuous processes. Datatransmission measurements are counting processes of events that occur in adja-cent time intervals. Therefore, Definition 1 has to be interpreted in the case ofdiscrete time random processes.To define the discrete time SS process, a time scale change has to be adapted

for the discrete time processes.

Definition 2 A stochastic process {zn}∞n=1 (averaged) in a time scale m denoted

by {z(m)n }∞n=1 is given by

z(m)n =

1

m

nm∑

i=(n−1)m+1

zi (2.2)

Definition 2 has a simple technical interpretation. If a measured data series{zn} represents a workload measured in Mbit/s observed during a one millisecond,than a sequence {z(1000)

n } represents an average workload measured in Mbit/soffered during 1 second.In Figure 2.1 the time scale change for Ethernet measurements is shown. The

vertical arrows show which samples measured on a base time scale constitute sum(2.2). In the example, the time scale change is m = 5. A base time scale is saidto be the time scale that comes from the measurements precision. A higher timescale is obtained by the time scale change from a base time scale by equation(2.2) (a time scale m1 is higher than a time scale m2 if m1 > m2).It is essential for the telecommunication applications to find properties of an

increment process related to the SS process defined by equation (2.1). Since theSS process is not stationary the increment process of the SS process is not nec-essarily stationary. On the other hand, many teletraffic measurements are donein the case that the increment process of the total observed traffic is stationary1.There are two main reasons. The first is that an analysis of non-stationary pro-cesses is more difficult. The second reason results from considering, in most cases,

1A process Zt is said to have the stationary increments, if any joint distribution of {Zt+h −Zh, t ≥ 0} is independent of h ≥ 0 [21]

2.1 Definition 9

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4x 10

5

time [s]

wor

kloa

d [M

bit/s

]

(a) Seconds time scale

0 1 2 3 4 5 6 7 8 9 101

1.5

2

2.5x 10

5

time [5s]

wor

kloa

d [M

bit/s

]

(b) Time scale m=5 with reference to (a)

Figure 2.1: An example of a time scale change for Ethernet measurements.

a busy hour. During a busy hour, where network load is the highest, the assump-tion of stationary of the increment process is often true. That is why a specialclass of H − ss processes is considered here. The class is called H-self-similarwith stationary increments process (H − ss, si process).An important property of H − ss, si processes is an equation corresponding

to the self-similar definition that describes a distribution of a sum of incrementprocess realisations. The equation is important from the application point ofview since it allows describing an increment process on a different time scales.The equation is as follows

1

m

m∑

i=1

zid= mH−1z1 (2.3)

Proof:

Assume that Zt is an H−ss, si process, then Zm−Z0 can be rewritten in the

form Zm −Z0d= Zm since Z0 = 0 with probability 1 [21]. As Zm is an H − ss, si


process, from Definition 1 we have

Zmd= mHZ1 (2.4)

and moreover, Z1d= Z1 −Z0

d= z1. On the other hand, Zm −Z0 can be rewritten

in the following form

Zm − Z0 = (Zm − Zm−1) + (Zm−1 − Z0) = · · · =

m∑

i=1

zi (2.5)

by combining equations (2.4) and (2.5) and the equality Z1d= z1 we obtain

m∑

i=1

zid= mHz1 (2.6)

Dividing the last equation by 1m concludes the proof. �

Telecommunication measurements are often discrete in time and forpractical reasons we have to consider time intervals where measure-ments data are stationary. It follows that we have to considerH−ss, si

processes in a greater detail. That is the reason why equation (2.3) isso useful because it describes the distribution of an increment processof an H − ss, si process.From an application point of view, the H − ss, si processes are important

and, therefore, some properties of those processes are presented. The H − ss, siprocesses definition can be found in [8, 21]. Only selected properties of the H −ss, si processes are described in the dissertation. More properties and theoremscan be found in [8, 21].

Conclusion 1 An H − ss, si process with H ∈ (0, 1) has the expected value, ifit exists, equal to 0.

Proof: {E(Zt − Zt−1) = Ez1E(Z1 − Z0) = Ez1 (2.7)

because Z0 = 0 [8, 21], EZ1 = Ez1. From Definition 1 and a linearity of theexpected value operator we have

∀t > 0 ∀H ∈ (0, 1) Ez1 =(tH − (t− 1)H

)EZ1 (2.8)

The last equation is true (in the non trivial case) only for EZ1 = 0. Therefore,an H − ss, si process has the expected value equal to 0. �

2.2 Long Memory Processes LRD 11

Conclusion 2 An autocorrelation function of Zt, if it exists, has the followingform [8]

E(ZtZs) = cov(Zt, Zs) =1

2σ2(t2H − |t− s|2H + s2H) (2.9)

Conclusion 3 An autocorrelation function of the increment process {zi}∞i=1 hasthe following form [8]

ρ(k) =1

2[(k + 1)2H − 2k2H + (k − 1)2H ] (2.10)

Conclusion 4 An asymptotic behaviour of an autocorrelation function of incre-ment process for H ∈ (1

2 , 1) has form [8]

ρ(k)

H(2H − 1)k2H−2

k→∞−→ 1 (2.11)

Conclusion 5 If an H − ss, si autocorrelation function exists, the Hurst pa-rameter value is limited to the interval H ∈ (0, 1) [8]. In the dissertation onlyH ∈ [12 , 1) is considered according to the telecommunication applications [8].

Data transmission measurements, for example, a workload volume in a timeinterval, are some random variables with positive values. From Conclusion 1 it isevident that anH−ss, si process cannot model telecommunication measurementsdirectly. It follows that H − ss, si processes can be used for modellingof a deviation from the mean value of a measured telecommunicationtraffic.Conclusions 2-4 are related to a “long memory” property which is presented

in the next section.

2.2 Long Memory Processes LRD

For numerous reasons the most important properties of a stochastic processare hidden in an autocorrelation function. Firstly, if a current process valueis strongly dependent on its previous values, then we can predict a future processvalue with a high probability and accuracy. Secondly, from the autocorrelationdecay rate we can conclude how long the process memory is, i.e. how many pre-vious process values have a critical influence on the current process value. Thethird important process property related to its autocorrelation function resultsfrom the equation for the variance of the sum of random values [40]

D2

(n∑

i=1

zi

)

=

n∑

i=1

D2zi + 2∑

1≤i<j≤n

cov(zi, zj) (2.12)


where cov(zi, zj) is the process zi autocovariance function related to the autocor-relation function ρ given by

ρ(zi, zj) =cov(zi, zj)√D2ziD2zj

(2.13)

Note that we can conclude from equation (2.12) that for sufficiently largecov(zi, zj) values the autocovariance values have larger influence on variation ofthe sum of random values than the sum of all variances of single elements of thesum.For the stationary processes, according to stationarity definition, we may

writecov(zk, zk+i) = cov(z1, zi+1) = γ(i) = ρ(i)D2z (2.14)

where γ(i) is an autocovariance function and ρ(i) is an autocorrelation func-tion. In the dissertation, γ(i) and ρ(i) function are used interchangeably if it isnecessary.An autocorrelation function deeply influences properties of stochastic pro-

cesses. Because of that stochastic processes are divided in two classes [69]. Firstof those are short range dependence processes (SRD), for which the autocorrela-tion function is summable, that is

∞∑

i=1

ρ(i) <∞ (2.15)

The second class of stochastic processes are long range dependence processes(LRD). The LRD processes are the most important for the dissertation context.The LRD processes pass the following assumption

limn→∞

n∑

i=1

ρ(i) = ∞ (2.16)

Note that the series ρ(i) is divergent if

ρ(i) ∼ 1

iβfor i→ ∞ and β ≤ 1; (2.17)

Additional justification of splitting stochastic processes to the SRD and LRDclasses is that the SRD process behaviour on higher time scales can be modelledby the Gaussian process because the SRD processes pass Central Limit Theorem(CLT) assumptions. LRD processes do not pass CLT assumptions [40], therefore,their behaviour on higher time scales cannot be approximated by the Gaussian

2.2 Long Memory Processes LRD 13

distribution. A detailed description of limit properties of self-similar processesand related LRD processes can be found in [21].On the base of Conclusion 4 and condition (2.17) we can deduce that for

H > 12 the H − ss, si increment process is LRD [8, 21].Examples of SRD and LRD processes with their autocorrelation functions are

presented in Figure 2.2.

0 20 40 60 80 100−10

−5

0

5

number of observation

valu

e of

obs

erva

tion

(c) Fractional Moving Average process FMA(1)

0 20 40 60 80 100−5

0

5

number of observation

valu

e of

obs

erva

tion

(a) Autoregressive process AR(1)

0 20 40 60 80 1000

0.5

1

lag k

ρ(k)

(b) autocorrelation function for the process (a)

0 20 40 60 80 1000

0.5

1

lag k

ρ(k)

(d) Autocorrelation function for the process (c)

Figure 2.2: A comparison between SRD (AR(1)) process and LRD (FMA(1))process with their autocorrelation functions.

In Figure 2.2.c we can observe a relatively long period where the observationstend to decrease (first 60 values). The observation can suggest that the processis not stationary, however, the observed trend is due to a strong correlation. Astrong correlation occurs in long periods where the observation tends to havevalues higher or a lower then the process mean value, at short time periods, thereseem to be cycles or local trends [8]. The properties make difficult to distinguishbetween LRD and non-stationarity.Another important property of LRD processes is the variance of the process

on a different time scale {z(m)i }∞i=1 decreasing with m increasing, however, slower

than for the SRD processes

D2z(m) =D2z

m+

2D2z

m2

m∑

i=1

(m− i)ρ(i) (2.18)

where z has a stationary distribution.What is the practical consequence of a slowly decreasing variance value? Fig-

ure 2.3 illustrates the difference in behaviour for different speeds of the correlation


decrease.

0 1 2 3 40

500

1000

1500

time

(a) Ethernet measurements, time scale 0.01s

0 1 2 3 40

500

1000

1500

time

(d) Gamma distributed values

0 10 20 30 400

500

1000

1500

time

(b) Ethernet measurements, time scale 0.1s

0 100 200 300 4000

500

1000

1500

time

(c) Ethernet measurements, time scale 1s

0 10 20 30 400

500

1000

1500

time

(e) Gamma distributed values, m=10

0 100 200 300 400

500

1000

1500

time

(f) Gamma distributed values, m=100

Figure 2.3: Examples illustrate how a time scale influences a process deviation.

Figure 2.3.a and Figure 2.3.d (first row) are similar to each other. We mayconclude wrongly that the process presented in Figure 2.3.d is a proper model ofEthernet traffic presented in Figure 2.3.a. The similarity is “delusive” becausethe processes obtained for the time scale change ofm = 100 (presented in Figures2.3.c and 2.3.f) are visibly different. The reason why we can observe such an effectis that the variance of the SRD process decreases faster with the time scale changegrowth than for real traffic measurements. A simple graphical test presenting thesame process on a different time scales as presented in Figure 2.3 proves that itis not possible to model measured traffic by SRD processes, especially if multipletime scales have to be considered.

2.3 Hurst parameter

The only parameter describing SS processes is the Hurst parameter H . We willdiscuss its relation to the autocorrelation function of the increment process of theH − ss, si process in a greater detail.As it was mentioned before, the Hurst parameter is a self-similarity measure

2.3 Hurst parameter 15

and its value indicates how the process on a time scale is different from theprocess on a different time scale. For an H − ss, si process the Hurst parametervalue decides if the increment process is SRD or LRD. From equations (2.11) and(2.17) a relationship between the autocorrelation function decay rate β and theHurst parameter H can be found:

{β = 2H − 2 for H ∈ (1

2 , 1)β is undefined for H = 1

2

(2.19)

Note that relationship (2.19) is true for H − ss, si processes not for all self-similar processes. Additionally, we have to remember that if a process isH−ss, siit implies that relationship (2.19) is true. The opposite theorem is not true [85],however, the assumption that an accumulated process for an LRD process is anSS one is commonly used. The assumption leads to coexistence in literaturetwo Hurst parameters. The first is “real” Hurst parameter coming from the SSprocess definition. The second is the Hurst parameter obtained from relationship(2.19) where β is known (mostly estimated from a stationary data traffic).We have to note that the self-similar process is non-stationary even if the

increment process is stationary what makes SS processes hard to analyse. Theself-similarity estimations cannot (apart from the Abry-Veitch test [1]) be usedfor Hurst parameter estimation if the data are non-stationary.In the dissertation, accumulated processes and their increment processes are

considered, but the Hurst parameter is related only to stationary increment pro-cesses. The self-similarity of an accumulated process is not considered in thedissertation. Therefore, an LRD process is denoted by its Hurst parameter H ,instead of the autocorrelation parameter β (equation (2.17)). The interpretationof the Hurst parameter is based on the assumption that the process described bythe Hurst parameter is LRD, not self-similar H − ss or even H − ss, si.Before the interpretation of the Hurst parameter is given, the difference be-

tween the self-similar (exactly H−ss, si process) and increment process is shownin Figure 2.4. The H − ss, si process shown in Figure 2.4 a is the FractionalBrown Motion (FBM) process proposed by Mandelbrot (the description can befound in [8, 21, 68]).FBM has the mean 0 and its variance is as |t|2H , in opposition to the increment

process (Fractional Gaussian Noise (FGN)) which has a constant variance value.

In telecommunication applications the Hurst parameter varies in the rangeH ∈ [1/2, 1). The 1

2 value describes a non self-similar (non LRD) process. It canbe caused by independence of values of the process or a sufficiently fast decreaseof the autocorrelation function. Examples of non self-similar processes, wherethe autocorrelation function decreases exponentially, are Markov chains [40] orAutoRegressive Moving Average (ARMA) processes [9].


0 100 200 300 400 500 600 700 800 900 1000−100

−50

0

50

100

time

(a) Fractional Brownian Motion H=0.7

0 100 200 300 400 500 600 700 800 900 1000−4

−2

0

2

4

time

(b) Fractional Gaussian Noise H=0.7

Figure 2.4: H−ss, si process example (figure (a)) and increment process obtainedfor the process from figure (a) (figure (b)).

Stochastic processes with the Hurst parameter values H ∈ (12 , 1) reveal the

long range dependence. The autocorrelation function decreases slower for a higherHurst parameter value. The conclusion comes from equation (2.11).Table 2.1 shows how the Hurst parameter impacts an autocorrelation function

and points out at a difference between SRD and LRD processes.Figure 2.5 illustrates a comparison of the autocorrelation decay rate inten-

sity for an SRD process (AR(1), which is discussed in the third chapter) andincrements of an H − ss, si process. The SRD process is characterised by astrong autocorrelation function value for small lag k values but the autocorrela-tion function decreases exponentially, therefore, for lag values greater than 50 theautocorrelation of the SRD process is smaller than the autocorrelation functionof any self-similar process presented in Figure 2.5. The observation is due to Con-clusion 4, that an LRD process autocorrelation function decreases hyperbolically.In Figure 2.5 we can also observe how the Hurst parameter value influences the

2.4 Queue analysis 17

lag Markov chain ARMA(1,1) FGN FARIMA(0,d,1)

k P =

0.8 0.1 0.10.2 0.7 0.10.1 0.2 0.7

φ = 0.5,θ = −0.7

H = 0.7d = 0.4(H = 0.9),θ = 0.7

1 0.646 0.740 0.320 -0.2072 0.415 0.370 0.189 0.0463 0.265 0.185 0.146 0.0684 0.169 0.092 0.122 0.0725 0.107 0.046 0.107 0.07310 0.011 0.001 0.070 0.06720 9.2E-5 1.4E-6 0.046 0.059100 0 0 0.018 0.043

Table 2.1: Autocorrelation functions decrease for different processes. The firstand second examples are SRD processes and the third and fourth examples areLRD processes.

autocorrelation function decrease intensity.

2.4 Queue analysis

One of the most important questions related to an SS or LRD traffic model ishow the model influences a queue behaviour? Just after the self-similarity wasdiscovered the topic was considered and the first results were obtained. The mostimportant conclusion is that the mean queue length for the workload processmodelled by a self-similar process is much different than the mean queue lengthobtained from Markov process analysis. The section presents a short descriptionof the results presented in literature.

2.4.1 Norros formula

The first result of a tail queue length distribution with a self-similar workloadprocess was obtained by Norros in [68] and Duffield and Connell in [20]. Themost important result coming from the equation derived by Norros is that thequeue tail has a Weibullian’s distribution instead of an exponential one (obtainedfor Markov models [15]). The exact equation is [63]

P (Q > b) ∼ exp(−δb2−2H) (2.20)

where δ is a positive constant that depends on the service rate of the queue[20, 68], Q is the queue length and P (Q > b) is the probability that the queue


0 10 20 30 40 50 60 70 80 90 100

10−4

10−3

10−2

10−1

100

lag k

ρ(k)

H=0.5 (AR(1), φ=0.9

H=0.6

H=0.7

H=0.8

H=0.9

Figure 2.5: SRD process (AR(1)) autocorrelation function and LRD process (in-crements of H − ss, si process) autocorrelation function for different Hurst pa-rameter values.

length exceeds a constant b, which can be interpreted as a buffer size.The result obtained by Norros assumes FBM workload process distribution

and the result is a lower bound of a queue length distribution. More resultsassuming FBM workload process distribution can be found in [20] (large buffer)[65] (exact expression) and [59] (upper bound). All these papers are cited andshortly discussed in [35].The results obtained in literature are summarised in [77] where approximation

of the average queue length as a function of a Hurst parameter H and utilisationρ (i.e. ratio of a workload volume to a service rate) is proposed. The equation isas follows

EQ ∼ ρ1

2(1−H)

(1 − ρ)H

1−H

(2.21)

Figure 2.6 presents a comparison between the classic M/M/1 and self-similarmodels with different H values. As we can see, the average queue lengths arehigher for the same ρ value for a higher Hurst parameter (except small utilisationvalues that are not interesting). Note that if an average queue length has to be

2.4 Queue analysis 19

shorter than a value (for example 10) then for a workload process of a higherHurst parameter, the link utilisation has to be smaller and resources are used inless effectively.

Figure 2.6: An average queue length as a function of utilisation for different Hurstparameter values and M/M/1 queue model.

In order to get a deeper insight into a queue length distribution we quote someinteresting simulation results to be found in [23]. The most important conclusioncoming from [23] is that the central impact on an average queue length has along range correlations structure not short range correlations. That improves theimportance of considering LRD models in queue length computation.Figure 2.7 presents the same reasoning as presented in [23]. The www traffic

is used as a workload process. The considered measurement data consist of onemillion measured values of the web traffic. The curve labelled as “original data”presents an average queue length obtained in simulation with unprocessed inputdata. The next two curves are obtained by dividing the whole traffic into 1000blocks each having 1000 values. The curve “internal block permutation” is ob-tained by permutation of the values inside each block but the order of the blocksis the same. Note that the effect of such a permutation is that the short rangecorrelation structure is removed form the input data but the long range correla-tion structure is almost unchanged. As we can realise for such input data average


queue lengths are almost not changed (sic!). The curve “block permutation” isobtained by permutation of blocks, where all values inside block are unchanged.Note that this time the short range correlation structure is almost unchangedbut the long range correlation structure is removed. As we can see the obtainedcurve (block permutation) has essentially lesser values than the curve obtainedfor original data. The last curve presents values obtained for random ordering ofall values, i.e. for input traffic where correlation is 0. The last example shows ushow critical is to take into consideration the autocorrelation function.

Figure 2.7: An average queue length for the same measurements and differentpermutations.

2.4.2 Other results

In literature different methods of a queue behaviour analysis can be found. Threedifferent analysis methods are shortly described here just to note that the methodproposed in the dissertation is not the only one. The first group of models is basedon diffusion analysis coming directly form physics theories. The second one is aspecial Markov chain or Markov process, and the third one is M/Pareto models.A description of the diffusion model can be found in [15]. The diffusion model

allows to compute G/G/1 and more complex queues or even queue systems. In

2.5 Chapter summary 21

[20] the generalisation of the diffusion model is proposed and the solution for ageneral correlated process is obtained. The result obtained in [20] confirms theresult obtained in [68], that the queue tail has the Weibullian’s distribution. Thegeneralisation of the diffusion analysis makes possible to compute more compli-cated than a simple single queue server system. A queue occupancy analysisfor different classes of services is an example of results obtained for the generaldiffusion model [16].As it was mentioned, the Markov models are SRD and cannot model an

SS process, but according to [25] we can find engineering time scales that havethe main influence on a queue length distribution. It is known that Markovmodels can match an autocorrelation function for numerous time scales. Suchmodels can be found in [41, 71, 72]. The average queue length for the modelproposed by Robert is described in [51]. Other models based on Markov processare Markov Modulated Poisson Processes (MMPP) described in [64] in detail.The Markov models are getting more interest because of development of newmatrix computation methods and, therefore, possibility to consider large Markovmodels, as presented in [10] and papers cited there.The models based on the assumption that the traffic consists of ON/OFF

sources are described in [5], where the queue length as a function of traffic pa-rameters is presented. Once again we are provided with formulae for a queuetail distribution. The result obtained for ON/OFF sources is interesting fromthe application point of view. The assumption is realistic and, therefore, can beeasier understood and analysed in the context of real traffic behaviour.The last cited result describes the asymptotic tail distribution in a network

of queues with the self-similar cross traffic [55]. The results allow analysing theinfluence of single nodes on delay in networks of queues. The delay distributionis the same as for single queue, i.e. Weibullian’s. The most important from theapplication point of view is that a single queue with traffic with the highest Hurstparameters value has the main influence on the traffic delay. The result confirmsthe importance of single queue analysis that is the topic of the dissertation.

2.5 Chapter summary

The chapter specifies notation and presents main definitions and theorems relatedto self-similar processes. The special case of H− ss, si processes is described andjustification for using these processes is presented. The long range dependenceprocesses strongly related to the self-similarity concept are defined and describedas well. The difference between LRD and increment of H − ss, si processes isshortly discussed. Section 2.3 describes and gives some intuition how the Hurstparameter value influence H − ss, si process properties and gives some intuitionhow the value can be understood. The last section shortly presents some results


of analysis of a queue with the self-similar workload process that are describedin literature.

Chapter 3

FGN and FARIMA

processes

One of the most important factors influencing the accuracy of a queue lengthcomputation is the workload process model. As it was mentioned in the pre-vious chapter, a workload process has to be self-similar. An FGN (FractionalGaussian Noise) process is the simplest and the most frequently used SS process.The process being an accumulated sum of FGN values is called FBM (FractionalBrownian Motion) that is an H − ss, si process. Since the FGN process showssome limitations, numerous different SS processes are proposed in literature. Inthe dissertation, the FARIMA process is considered. This chapter presents thedefinitions and the most important properties of the FGN and FARIMA pro-cesses.

3.1 FGN process

An FGN process is an increment process of an FBM process. The FBM processhas been proposed by Mandelbrot who originated the fractal theory. Accordingto [21] the FBM process is defined as

Definition 3 Let 0 < H ≤ 1. A real-valued Gaussian process {BH(t), t ≥ 0} iscalled “Fractional Brownian Motion” if EBH(t) = 0 and

E(BH(t)BH(s)) =1

2

{t2H + s2H − |t− s|2H

}EB2

H(1) (3.1)

Note that the Gaussian process is determined by its mean value and covari-ance structure. Therefore, two conditions set in Definition 3 determine a uniqueGaussian process.

24 FGN and FARIMA processes

If EB2H(1) = 1 an FBM process is called a normalised FBM process [68].

Selected properties of a normalised FBM process concluded from Definition 3 arepresented in [68].

Conclusion 6 BH(t) has stationary increments.

Conclusion 7 BH(0) = 0, and EBH(t) = 0 for all t > 0.

Conclusion 8 EB2H(t) = D2BH(t) = t2H for all t > 0.

Conclusion 9 BH(t) has continuous paths.

Conclusion 10 BH(t) is Gaussian, i.e. its finite-dimensional distributions aremultivariate Gaussian distributions.

Conclusion 6 indicates that BH(t) is an H − ss, si process. As it has beenmentioned in Chapter 2, a model of real traffic has to be stationary and thatis why to model telecommunication measurements properly an FGN process isused.An FGN process is obtained from an FBM process by the following equation

[21]zi = BH(i∆t) −BH((i− 1)∆t) (3.2)

Note that if D2BH(t) and a time interval ∆t are normalised to one E((BH(i+1) − BH(i))2) = EB2

H(1) = 1, i.e. the corresponding FGN process is also nor-malised.The examples of FBM and FGN processes are shown in Figure 3.1.An FGN process is an LRD process for H > 1

2 since its autocorrelationfunction has the following form [8]

ρ(k) =1

2((k + 1)2H − 2k2H + |k − 1|2H) (3.3)

and ρ(k) for k → ∞ is approaching [21, page 22]

ρ(k) ∼ H(2H − 1)k2H−2EB2H(1), k → ∞ and H 6= 1

2(3.4)

3.1 FGN process 25

0 100 200 300 400 500 600 700 800 900 1000−100

−50

0

50

100

time

(a) Fractional Brownian Motion H=0.7

0 100 200 300 400 500 600 700 800 900 1000−4

−2

0

2

4

time

(b) Fractional Gaussian Noise H=0.7

Figure 3.1: H − ss, si process example (figure (a)) and the increment processobtained for the process from figure (a) (figure (b)).

From equation (3.3) the relation between the Hurst parameter and the esti-mated LRD process parameter is

β = 2H − 2 (3.5)

Since the Hurst parameter is fully dependent on the type of LRD behaviourthe autocorrelation function of an FGN process is fully dependent on the Hurstparameter. That is why only the LRD property can be modelled by anFGN process. If the estimated data have a different type of SRD be-haviour than the behaviour coming from the LRD property an FGNprocess will not be able to fully express the data autocorrelation struc-ture. This disadvantage does not occur for the FARIMA process sinceit can model the SRD and LRD properties simultaneously.


3.2 ARIMA process

An Integrated Autoregression and Moving Average (ARIMA) process is describedin detail in [9]. The construction of the ARIMA process makes it flexible andallows modelling different autocorrelation structures. As it has been mentioned inChapter 2 an autocorrelation structure has a critical impact on the properties ofa stochastic process and that is the reason why the ARIMA process has numerousapplications.Before the ARIMA process is defined a useful notation is introduced. Since

an ARIMA process value is defined as a function of its previous values, backshiftoperator B defined by the equation Bzn = zn−1 simplifies the notation of theARIMA process.The ARIMA process includes three different processes. The first one is an

AutoRegression (AR) process. The AR(p) process, i.e. the autoregression processof the order p, assumes that the process value zn depends on the p previous valuesand a random iid (independent identically distributed) variable an. It is given by

zn = φ1zn−1 + φ2zn−2 + · · · + φpzn−p + an (3.6)

where an are iid random variables with a common zero mean and a variance σ2a.

Equation (3.6) is replaced by

zn − φ1zn−1 − φ2zn−2 − · · · − φpzn−p = Φ(B)zn = an (3.7)

where Φ(B) = 1 − φ1B − φ2B2 − · · · − φpB

p is a polynomial of an order p.The second process included in the ARIMA process is a Moving Average

(MA) process. The MA(q) process, i.e. the moving average process of the orderq, assumes that the process value zn depends on the q previously known iidrandom values and a new iid random value. It is given by

zn = an − θ1an−1 − θ2an−2 − · · · − θqan−q = Θ(B)an (3.8)

where Θ(B) = 1 − θ1B − θ2B2 − · · · − θqB

q is a polynomial of an order q.The last constituent of the ARIMA process makes the modelling of a non-

stationary process possible. In order to analyse a non-stationary process, Boxand Jenkins proposed to compute d differences until the stationary process wn

is obtained. The differences are computed as follows (an example, for d = 1 andd = 2)

d = 1 wn = zn − zn−1 =(1 −B)zn

d = 2 wn = (zn − zn−1) − (zn−1 − zn−2)=(1 −B)2zn(3.9)

The differences in equation (3.9) are denoted for short by ∇d, therefore, wn =∇dzn, where d denotes the number of difference operations and ∇ = (1 −B).

3.2 ARIMA process 27

From equations (3.6), (3.8) and ∇d = (1 −B)d the ARIMA(p,d,q) process isdefined by equation

Φ(B)(1 −B)dzn = Θ(B)an (3.10)

where an are iid random variables with a common zero mean and a variance σ2a,

Φ(B) and Θ(B) are polynomials of the order p and q, respectively.The ARIMA process is the general process with p+q+1 parameters. The first

premise is that the ARIMA autocorrelation function depends on two polynomialsΦ(B) and Θ(B). The second premise is that for d ≥ 1 ARIMA models non-stationary data. However, non-stationary processes are out of the scope of thisdissertation so the ARMA process is the most interesting one for the dissertation.The ARMA(p,q) process is the ARIMA(p,d,q) process given by

Φ(B)zn = Θ(B)an (3.11)

The properties of the ARMA process are dependent on the Φ(B) and Θ(B)polynomials. If both polynomials Φ(x) and Θ(x) are non-zero for all complexx such that |x| ≤ 1 then the ARMA process is a stationary and an invertibleprocess. If the ARMA process is stationary then

zn =

∞∑

i=0

ψian−i (3.12)

which is the AR(∞) representation. The ψi (weights) sequence is absolutelysummable (i.e.

∑∞

i=0 |ψi| <∞).The invertible feature means that the ARMA process could be expressed as

an =

∞∑

i=0

πizn−i (3.13)

which is the MA(∞) representation. The πi sequence is absolutely summable.The AM(∞) representation makes it possible to find a general expression of

the autocorrelation function

ρk =

∑∞

i=0 ψiψi+k∑∞

i=0 ψ2i

(3.14)

for the ARMA(p,q) process.As an example, the autocorrelation function of the ARMA(1,1) process is

presented. The process is given by

zn − φzn−1 = an − θan−1 (3.15)


The ARMA(1,1) process is stationary if |φ| < 1 and |θ| < 1 [9] and theautocorrelation function is then

ρ1= (1−φθ)(φ−θ)1−2φθ+θ2

ρk=ρ1φk−1

(3.16)

0 5 10 15 20−1

−0.5

0

0.5

1φ=−0.75, θ=0.75

lag k

ρ k

0 5 10 15 20−0.4

−0.3

−0.2

−0.1

0φ=0.25, θ=0.75

lag k

ρ k

0 5 10 15 20−0.2

−0.1

0

0.1

0.2φ=−0.65, θ=−0.95

lag k

ρ k

0 5 10 15 200

0.5

1φ=0.75, θ=−0.75

lag k

ρ k

Figure 3.2: ARMA(1,1) autocorrelation functions for different φ and θ parame-ters.

In Figure 3.2 the autocorrelation functions for different φ and θ values arepresented. The ARMA process is able to model numerous different autocorrela-tion functions. The first value, i.e. the value for lag k = 1, may be of any signand value. The consecutive values can be of the same sign or alternately negativeand positive.The flexibility of the ARMA process when modelling autocorrelation struc-

tures provides numerous applications of the ARMA process including predictionof process values.The autocorrelation function may be easily modelled but it always

decreases exponentially (for finite q and p). Therefore, all ARMA

3.3 FARIMA process 29

processes are SRD according to equation (2.15). The ARIMA processes for apositive and integer d value are non-stationary so it is not possible to classify themas SRD or LRD processes. Restrictions of the ARMA process does notconcern the Fractional Integrated Autoregression and Moving Average(FARIMA) process described in the next section.

3.3 FARIMA process

In Section 3.2 the ARMA process has been shortly described. The ARMA processis SRD which is a strong disadvantage because, as it has been described in Chap-ter 2, the Internet traffic is LRD. A generalisation of the ARIMA process, calleda Fractional ARIMA (FARIMA) process, solves the problem since the FARIMAprocess is stationary for a proper value of d parameter (as the ARMA process is)and an LRD process.

3.3.1 FARIMA definition

The generalisation of the ARIMA process has been proposed by Hosking anddescribed in [38]. The ARIMA(p, d, q) process is characterised by p + q + 1parameters. d is a natural number and for values larger than 0 the obtainedprocess is non-stationary. Since non-stationary processes are not as useful tomodel telecommunication traffic as stationary processes d is always set to zero.Hosking’s generalisation is based on permitting d to take any real value. In [38]

a deep analysis of a stable and invertible cases of the FARIMA process is shown.It is the process where d ∈ (− 1

2 ,12 ); however, in telecommunication applications

d variability is limited to the interval d ∈ [0, 12 ). The FARIMA process with

d ∈ [0, 12 ) is a stable and invertible process. As mentioned in Section 3.2, such

a process can be described in the useful forms of the AR(∞) and MA(∞) sums(3.12) and (3.13) [38].Hosking’s generalisation needs ∇d to be defined for real values of d. A general

binomial series describes a general form of ∇d

∇d = (1 −B)d =∞∑

k=0

(d

k

)

(−B)k =∞∑

k=0

Γ(k − d)

Γ(−d)Γ(k + 1)Bk (3.17)

where Γ(·) denotes the gamma function.The ∇ definition allows one to define the FARIMA(p,d,q) process by the

equation which has defined the ARIMA process (3.10)

Φ(B)

∞∑

k=0

Γ(k − d)

Γ(−d)Γ(k + 1)Bkzn = Θ(B)an (3.18)


Note that for the FARIMA process an infinite number of zn−k values is needed.Since (1−B)d is a polynomial of an order d and Φ(B) is a polynomial of an order ptheir product is a polynomial of an order d+p. Therefore, only a limited numberof zn−k and an−j values is needed to compute the zn value for the ARIMAprocess. Nevertheless, for the FARIMA process the infinite number of the zn−k

values is needed according to equation (3.18).

3.3.2 FARIMA properties

From the LRD processes analysis point of view the most important property ofthe FARIMA process is that for d ∈ (0, 1

2 ) the autocorrelation function for k → ∞behaves like k2d−1 since [38]

limk→∞

k1−2dρk = a (3.19)

where a is a finite constant.From (3.19) we can deduce that ρk ∼ k2d−1 for sufficiently large k. There-

fore, according to equation (2.17) the FARIMA process with d ∈ (0, 12 ) is an LRD

process. Hence, the FARIMA process allows modelling the LRD property andbecause Φ(B) and Θ(B) polynomials influence the autocorrelation function forsmall lags the FARIMA process can model the LRD and SRD properties indepen-dently. It is essential to understand that the LRD property determineshow the autocorrelation function behaves at large lags k but it doesnot characterise the autocorrelation function for small lags. Therefore,the autocorrelation function can be changed in a short range in orderto fit the real traffic behaviour better than a model that suits only theLRD behaviour.We know the behaviour of the autocorrelation function for k → ∞, however,

in order to model traffic fluctuations and to obtain some results wehave to know an exact form of the autocorrelation function. In [38]the exact form of the autocorrelation function, for three special casesof FARIMA processes, is presented and this is why in this dissertationonly these three processes have been analysed.The first process is the FARIMA(0,d,0) process. The process is given by

∇dzn = an (3.20)

For the process described by equation (3.20) the weights ψi are given by [38]

ψi =(i+ d− 1)!

i!(d− 1)!(3.21)


The autocorrelation function is given by [38]

ρk =(−d)!(k + d− 1)!

(k − d)!(d − 1)!(3.22)

As k → ∞ρk ∼ (−d)!

(d− 1)!k2d−1 (3.23)

The FARIMA(0,d,0) process for an ∼ N (0, σ2a) is similar to an FGN process

[38, 56, 89]. Like for an FGN process we have only one parameter that is relatedto the Hurst parameter by

d = H − 1

2(3.24)

The FARIMA(0,d,0) process is presented here just for a reference as FARIMAprocesses with more parameters are more interesting. The reason is that theautocorrelation function of the FARIMA(0,d,0) process is fully dependent on theHurst parameter. More general types of FARIMA processes allow changing ashort range behaviour of the autocorrelation function without changing the LRDproperty.The second process is the FARIMA(1,d,0) process described for short as the

FAR(1,d) process. The process is given by

(1 − φB)∇dzn = an (3.25)

The ψ weights and the autocorrelation function are expressed in terms of thehypergeometric function defined by

2F1(a, b; c; z) = 1 +ab

c · 1z +a(a+ 1)b(b+ 1)

c(c+ 1) · 1 · 2 z2 + . . . (3.26)

The ψ weights and the autocorrelation function are given by [38]

ψi =(i+ d− 1)!

i!(d− 1)!2F1(1,−i; 1− d− i;φ) (3.27)

ρk = ρxk

2F1(1, d+ k; 1 − d+ k;φ) + 2F1(1, d− k; 1 − d− k;φ) − 1

(1 − φ)2F1(1, 1 + d; 1 − d;φ)(3.28)

where ρxk is the autocorrelation function of the FARIMA(0,d,0) process and is

given by equation (3.22).In this case the autocorrelation function is complicated and cannot be anal-

ysed easily. To give some idea about the possible forms of the autocorrelation


10 20 30 40 50−1

0

1(a) FAR(1,0.05) for φ=−0.9

lag k

ρ k

10 20 30 40 50−1

0

1(b) FAR(1,0.45) for φ=−0.9

lag k

ρ k10 20 30 40 50

0

0.05

0.1(c) FARIMA(0,0.05,0) e.i. φ=0

lag k

ρ k

10 20 30 40 50

0.6

0.8

1(d) FARIMA(0,0.45,0) e.i. φ=0

lag kρ k

10 20 30 40 500

0.5

1(e) FAR(1,0.05) for φ=0.9

lag k

ρ k

10 20 30 40 50

0.70.80.9

(f) FAR(1,0.45) for φ=0.9

lag k

ρ k

Figure 3.3: FAR(1,d) autocorrelation functions for different φ and d parameters.Note that for φ = 0 the FAR(1,d) process is the FARIMA(0,d,0) process.

function the examples of the autocorrelation function for different d and φ pa-rameters are presented in Figure 3.3.The third process is the FARIMA(0,d,1) process described for short as the

FMA(d,1) process. The process is given by

∇dzn = (1 − θB)an = an − θan−1 (3.29)

The ψ weights are equal to ψ weights of the FAR(1,−d) process where φparameter is θ. The autocorrelation function is given by [38]

ρk = ρxk

(1−θ)2

1+θ2−(2θd)/(1−d)k2 − (1 − d)2

k2 − (1 − d)2(3.30)

where ρxk is the autocorrelation function of the FARIMA(0,d,0) process and is

given by equation (3.22).


In this case the autocorrelation function is less complicated than for theFAR(1,d) process. In Figure 3.4 the examples of the autocorrelation functionfor different d and θ are presented.

10 20 30 40 500

0.2

0.4

0.6

0.8

lag k

ρ k

(a) FMA(0.05,1) for θ=−0.9

10 20 30 40 50

0.7

0.8

0.9

1

lag kρ k

(b) FMA(0.45,1) for θ=−0.9

10 20 30 40 50−0.6

−0.4

−0.2

0

0.2

lag k

ρ k

(c) FMA(0.05,1) for θ=0.9

10 20 30 40 50−0.4

−0.2

0

0.2

lag k

ρ k(d) FMA(0.45,1) for θ=0.9

Figure 3.4: FMA(d,1) autocorrelation functions for different θ and d parameters.

In Figure 3.4 the case of θ = 0 has been omitted because for θ = 0 theFMA(d,1) process is the FARIMA(0,d,0) process and, therefore, if the case θ = 0was presented the obtained result would be the same as in Figure 3.3.c and 3.3.d.The FARIMA(p,d,q) autocorrelation structure for large lags is described by

equation (3.19) where d determines the LRD property of the FARIMA process.According to equations (3.28) and (3.30) φ and θ changes the autocorrelationfunction for small lags k. The only two parameters left to fully describe theFARIMA process are σa and the an distribution.In the dissertation the an process has the Gaussian distribution what allows

computing the distribution of an accumulated sum of zn values being a processvital for the dissertation. The limitation that the considered process has theGaussian distribution can be overcome by the following transformation proposed


in [32, 56]h(zn) = F−1

Y (FX(zn)) (3.31)

where FX is the cumulative distribution function of the original process and F−1Y

is the inverse cumulative distribution function of the target distribution. Thetransformation does not change the long-range correlation structure [56]. As thistransformation influences the short-range dependence structure it has not beenconsidered in this dissertation but it constitutes an interesting topic for furtherresearch.The standard variation σa has no influence on the autocorrelation structure

and it is estimated from real data to obtain a more efficient model. Accordingly,we assume that σa = 1.

3.3.3 Applications

A basic description of the FARIMA process and selected applications have beenpresented in [4, 11, 36] together with other SS models. There are a few cru-cial advantages of the FARIMA processes in comparison to other SS processes.Firstly, the FARIMA processes have an analytic form, i.e. we know the zn dis-tribution. For example, an ON/OFF process has an FGN distribution, however,for a number of ON/OFF sources tending up to infinity. Secondly, some resultsobtained for the ARIMA processes are valid for the FARIMA processes, too. Agood example here is the problem of the prediction with the minimum meansquare error which is already solved for the AR(∞) process [9]; the only task isto find ψ values in terms of FARIMA parameters. The values with the FARIMAdistribution can be generated on the basis of the algorithm presented in [8]. Mostimportantly the FARIMA processes can display different short-range dependencestructures for the same long-range dependence structure.The FARIMA process is applied to different types of data. In the paper where

the FARIMA process is defined [38] the author has proposed the application ofthis process to model economic and hydrology processes. As one of the applica-tions of the ARIMA is process prediction; the FARIMA process can be used insuch a case as well [19, 75].The FARIMA process can be applied as a network traffic model for all mea-

surements which reveal the LRD property [89], but because an FGN model isLRD and it is simpler and more efficient from the analytic point of view, theFARIMA processes are used in such cases where the short-range autocorrela-tion structure is essentially different from the one obtained for an FGN process.Therefore, the main application of the FARIMA process in teletraffic engineeringis modelling of video traffic.Applying a FARIMA process to video modelling traffic is presented in [32].

Numerous papers have been based on the Garrett and Willinger publication and

3.4 Chapter summary 35

have studied improving the video traffic modelling. Models based on the FARIMAprocesses can be found in [6] where the FARIMA process is used as a self-similartraffic generator that models the autocorrelation structure for each frame I, P,and B in the MPEG stream. Another example is [75] where the FARIMA processis used to predict the one-step-ahead traffic level. Another example is [91] wherethe FARIMA process is used to model multimedia traffic allowing analysis ofcomplicated buffer management algorithms.In [34] a generalisation of the FARIMA process, called stable FARIMA, is

presented. The generalisation is based on an assumption that an is an α-stableprocess. One of the properties of the α-stable processes is that E|X |p = ∞ forp ≥ α and α ∈ (0, 2). Therefore, for α < 2 we get the an process with infinitevariance [31]. Note, that α = 2 generates a special case with a finite variance.The generalisation is interesting but the obtained process is difficult to analyseas all the processes with infinite variance. Therefore, the stable FARIMA processis much beyond the scope of the dissertation.

3.4 Chapter summary

In this chapter two important processes have been described. The first of them isan FGN process that is an exact self-similar process. Its autocorrelation charac-teristics are fully described by the Hurst parameter. The second process describedin this chapter is the FARIMA process. The process can model the LRD as wellas SRD property. Three special cases of the general FARIMA process have beenpresented with their autocorrelation functions. In the last subsection, the appli-cation and generalisation of the FARIMA processes have been presented.

Chapter 4

Envelope process

Since a queue process is a stochastic process, therefore, to obtain the maximumqueueing delay the queue process distribution has to be found. Since the processdistribution is not easy to find in general it has been proposed to change thisproblem to an easier one of finding a maximum of a special deterministic function[60]. The function is called an envelope process. In this chapter the constructionof an envelope process for different stochastic processes is presented.

4.1 Envelope construction

In order to facilitate the understanding of the envelope process an envelope pro-cess for a deterministic function is going to be shown. An envelope process fora set S and a deterministic function f(t) is any function f(t) that passes thefollowing condition

∀t ∈ S f(t) ≥ f(t) (4.1)

The envelope process can be useful as an approximation in a case when thefunction f(t) is difficult to obtain in an analytic form or an analytic form cannotbe found at all. On the other hand, f(t) is a sufficient approximation from anapplication point of view.A sample function and two envelope processes f(t) are sketched in Figure 4.1.

Note that for the same function f(t) we can find infinite different envelopefunctions, therefore, in order to choose the best envelope function some way ofcomparing envelope functions has to be found. An example comparison algorithmis: f1(t) is better than f2(t) if ∀t ∈ S f1(t) ≤ f2(t). In Figure 4.1, the functionf1(t) is better than f2(t) according to the above condition.

38 Envelope process

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

t

f(t

),f(t

)

f(t)

f2(t)

f1(t)

Figure 4.1: The example of a deterministic function with two different envelopeprocesses

Envelope processes for stochastic processes cannot be defined like for deter-ministic functions since stochastic processes do not have a single value for aparticular t. Firstly, we focus on stochastic processes Zt with finite values, i.e.Zt varies in a bounded set Et for each time t. Then, the envelope process Z(t) is

Z(t) = supx∈Et

x (4.2)

The important property of definition (4.2) of the envelope process is that the

4.1 Envelope construction 39

stationarity of a stochastic process Zt is not involved. This property of definition(4.2) seems to be crucial as accumulative processes that will be used to computea queue length in Chapter 5 are non-stationary.As a simple example of an envelope process obtained form definition (4.2)

let consider a discrete time process Zn ∼ U(0,n) where U(0,n) means a uniformdistribution in an interval (0,n). In Figure 4.2, a single realisation of Zn and itsenvelope Z(n) are presented.

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

n

Zn,Z

(n)

Zn

Z(n)

Figure 4.2: An example of a stochastic process Zn ∼ U(0, n) varies in boundedintervals with its envelope process

Equation (4.2) is valid only for a random process that is defined in a boundedset. For a stochastic process that is defined in an unbounded set it is impossibleto find a function that is exceeded by a stochastic process with zero probability.That is why the probability of exceeding an envelope process has to be involved.Equation (4.2) for these processes gets the following form

P (Zt > Z(t)) = ǫ (4.3)

40 Envelope process

where ǫ is called an exceed probability.It is important to understand that the envelope process defined by equation

(4.3) depends on the ǫ value. Therefore, notation Zǫ(t) should be used, however,it is common to abbreviate it to Z(t).The second important property is that equation (4.2) is a special case of

equation (4.3) where ǫ = 0. It means that it is sufficient to consider definition(4.3) and envelope properties within its content.A simple example illustrating definition (4.3) is presented in Figure 4.3. In

this case Zn ∼ Exp( 1n ), where Exp(λ) is the exponential distribution with λ

parameter. In Figure 4.3 an example of Zn realisation and two envelope processesfor different ǫ values are presented.

0 20 40 60 80 1000

50

100

150

200

250

300

n

Zn,Z

(n)

Zn

Z(n) for ǫ = 0.05

Z(n) for ǫ = 0.1

Figure 4.3: An example of a stochastic process Zn ∼ Exp( 1n ) and two envelope

processes obtained for different ǫ values

4.2 Related work 41

An envelope process can be described as a function of a cumulative distribu-tion function because

P (Zt > Z(t)) = 1 − FZt(Z(t)) = ǫ

⇓Z(t) = F−1

Zt(1 − ǫ)

(4.4)

where FZt(·) is a cumulative distribution function of a process Zt.

According to the properties of the cumulative distribution functions, Z(t)takes on greater values for a lower ǫ. It leads to the conclusion that for smaller ǫvalues the accuracy of the envelope process is lower as its values are farther fromthe process values.An envelope process allows replacing a stochastic process by a de-

terministic function and considering the upper bound instead of thedistribution. Most of the results do not depend on the exact form ofthe process for which the envelope functions are found. On the otherhand, an envelope process can be found for any process for which thecumulative distribution function is known or at least it is numericallycomputable. Therefore, the envelope analysis allows obtaining generalresults and extending them to some new processes.

4.2 Related work

Network measurements prove that network traffic reveals the LRD property [23,56, 77, 80]. According to the LRD definition (2.16) only stationary processes canbe LRD. Therefore, to model network traffic properly, a stationary process has tobe applied. On the other hand queue models which are needed to compute QoSparameters use accumulated processes. An accumulated process is non-stationarysince its increments are stationary.Norros proposed in [68] a stationary stochastic storage process Vt given by

Vt = sups≤t

(At −As − c(t− s)) (4.5)

where c is a link capacity and At and As both constitute a workload input fromtime 0 till time t and s, respectively, given by

At = at+ σZt (4.6)

where a is the mean of the workload process, σ is its standard deviation and Zt

is a normalised FBM process.In [60] an envelope analysis and an envelope process of the workload given

by equation (4.6) have been proposed. The envelope process of the workload

42 Envelope process

described by equation (4.6) is

A(t) = at+ σF−1(1 − ǫ)tH (4.7)

where F is the cumulative distribution function of the normalised Gaussian dis-tribution.The model proposed by Norros displays an other important prop-

erty, namely if the envelope process of Zt is known then the envelopeprocess of At is given by

A(t) = at+ σZ(t) (4.8)

Proof:The envelope process of Zt is Z(t), i.e.

P (Zt > Z(t)) = ǫ (4.9)

On the other hand, the envelope process of At, obtained for the same proba-bility ǫ, is given by

P (at+ σZt > A(t)) = ǫ (4.10)

By simple algebraic transformations we obtain

P

(

Zt >A(t) − at

σ

)

= ǫ (4.11)

Since the right sides of inequalities (4.9) and (4.11) are equal, therefore, A(t) =at+ σZ(t). �

Equation (4.8) makes it possible to find an envelope process for the Zt processonly instead of finding an envelope process of the At process. Therefore, the restof this dissertation describes only envelope processes of Zt processes.A detailed derivation of an envelope process of a normalised FBM process Zt

is as followsP (Zt > Z(t)) = P (Z1 > t−H Z(t)) = ǫ (4.12)

since Zt is an H − ss process.Thus it can be written that P (Z1 ≤ t−H Z(t)) = 1 − ǫ. If F (·) is a cu-

mulative distribution function of the normalised Gaussian distribution we haveF (t−H Z(t)) = 1 − ǫ and, finally, the normalised FBM envelope process is givenby

Z(t) = F−1(1 − ǫ)tH (4.13)

In Figure 4.4 two Zt paths and two Z(t) envelopes obtained for different ǫvalues have been outlined.

4.2 Related work 43

0 20 40 60 80 1000

10

20

30

40

50

60

70

t

Zt,Z

(t)

Z1

t

Z2

t

Z(t) for ǫ = 0.05

Z(t) for ǫ = 0.2

Figure 4.4: An example of two paths of the FBM process and two envelopeprocesses obtained for different ǫ values.

The second process for which an envelope process has been described in litera-ture is anH−ss, si process with a symmetric α-stable (SαS) marginal distribution[33]. The α-stable distributions are important distributions since a normalisedsum of iid random variables tends to an α-stable distribution1 [26]. The α-stabledistributions are given by their characteristic function [31]

ΦX(θ) = Eexp(jθX) =

{exp

[jµθ − |σθ|α

(1 + jβ sign(θ) tan

(πα2

))]if α 6=1

exp[jµθ − |σθ|

(1 + jβ 2

πsign(θ) ln |θ|

)]if α=1

(4.14)

The probability density function is known for three special cases of α-stable1A special case of α-stable distribution is a normal distribution obtained for α = 2.

44 Envelope process

distribution only. The distributions are: Gaussian distribution for α = 2, Cauchydistribution for α = 1 and Le´vy distribution for α = 1

2 .A general α-stable distribution is denoted by Sα(σ, β, µ), where α ∈ (0, 2] is

an index determining the weight of a distribution tail, σ ≥ 0 is a scale parameter,β ∈ [−1, 1] is a skewness parameter and µ is a shift (location) parameter [31].Since the α-stable distribution is the limit of the normalised sum of random

variables and in network traffic an aggregation of different sources is common, theα-stable distribution can be useful to model network traffic. Here some propertiesof the α-stable distribution are given [47].Property 1: When 1 < α ≤ 2, the location parameter µ is equal to the

expected value of the random α-stable variable X ∼ Sα(σ, β, µ)

EX = µ (4.15)

Property 2: When 0 < α < 2 and the random α-stable variable X ∼Sα(σ, β, µ) then

E|X |p < ∞ if 0 < p < αE|X |p = ∞ if p ≥ α

(4.16)

Property 3 (Stability): A random variable X is strictly stable if for any posi-tive constants b1 and b2, there is a positive number bX , such that

b1X1 + b2X2d= bXX (4.17)

where X1, X2, and X are iid random variables. Furthermore, if X1 and X2 havethe α-stable distribution bX is given by

bX = (bα1 + bα2 )1α (4.18)

Note that the consequence of equation (4.18) is that an iid α-stable randomsequence passes equation (2.3) where H = 1

α . Equation (2.3) is obtained directlyfrom the self-similarity definition and this is why the iid α-stable random sequencecan be estimated wrongly as an H − ss, si process.A special case of the α-stable distribution, the symmetric α-stable distribu-

tion (SαS), has been considered in [33]. The SαS distribution is the α-stabledistribution for β = 0 and µ = 0, i.e. Sα(σ, 0, 0). The SαS characteristic functionhas the form

ΦX(θ) = exp (−|σθ|α) (4.19)

For a random variable with the SαS distribution X ∼ Sα(σ, 0, 0) the followingequation are true [33]

X + a ∼ Sα(σ, 0, a) (4.20)

aX ∼ Sα(|a|σ, 0, 0) (4.21)

4.2 Related work 45

Equations (4.20) and (4.21) allow better understanding of σ and µ influence.The above description shows the marginal distribution of a workload process

based on the α-stable distribution only. In order to define its evolution withtime the stochastic integral is used. There are two main approaches to define anH − ss, si process with the marginal α-stable distribution; more details are givenin [31, 47]. The first approach is based on the Linear Fractional Stable Motion(LFSM) concept given the following integral

Lα,H(t) =

∫∞

−∞

(

a

[

(t − x)H−

1α

+− (−x)

H−1α

+

]

+ b

[

(t − x)H−

1α

−− (−x)

H−1α

−

])

M(dx)

(4.22)

where M(dx) is the α stable random measure, a and b are real constants, and

y+ =

{y y ≥ 00 y < 0

y− =

{0 y ≥ 0−y y < 0

(4.23)

For an LFSM process the Hurst parameter is bounded by the interval definedas follows

H ∈{

(0, 1] α ≥ 1(0, 1

α ] α < 1(4.24)

The second approach, the Log-Fractional Stable Motion (Log-FSM), is givenby the following integral

Λα(t) =

∫ ∞

−∞

(ln |t− x| − ln |x|)M(dx) (4.25)

where M(dx) is the α stable random measure.For the Log-FSM process the Hurst parameter is H = 1

α for α ∈ (1, 2) [33].An envelope process for LFSM and Log-FSM processes with the SαS marginal

distribution has been presented in [33]. For both of these processes the envelopeprocess has the same form given by

Lα,H(t) = Λα(t) = K(ǫ)σ1tH (4.26)

where K(ǫ) = F−1α (1 − ǫ), where Fα is the cumulative distribution function

of Sα(1, 0, 0) distribution and σ1 has to be computed for LFSM and Log-FSMseparately. The details are given in [33] and references therein.For LFSM σ1 is given by [33]

σα1 =(|a|α + |b|α)

∫ ∞

0

(

(1 + x)H−1/α − xH−1/α)α

dx+

+

∫ 1

0

∣∣∣a(1 − x)H−1/α − bxH−1/α

∣∣∣

α

dx

(4.27)

46 Envelope process

For Log-FSM σ1 is given by [33]

σα1 = 2

∫ ∞

0

|ln(1 + x) − ln(x)|α dx+

∫ 1

0

|ln(1 − x) − ln(x)|α dx (4.28)

Unfortunately, K(ǫ) has to be determined numerically; it can be based ondata and computer programs presented in [67].

4.3 Envelope process for FARIMA process

Both results presented in references [33, 60] describes modelling the LRD propertybut not the SRD property. The FARIMA process can capture both the LRDand SRD properties for better modelling of network behaviour. Since, the knownenvelope process can be use to determine some network parameters in this sectionthe envelope processes for selected FARIMA processes have been determined.

4.3.1 Envelopes for selected FARIMA processes

Modelling a workload process by the FARIMA process means that thenumber of bits/bytes that arrival to the queue during a time interval∆t is modelled by the FARIMA process zn. In order to compute the queuelength the accumulated process Zn has to be determined, i.e. Zn distribution hasto be found. Note that Zn is given by

Zn =

n∑

i=1

zi (4.29)

According to equation (3.18) the FARIMA process is defined as a sum ofGaussian random variables multiplied by weight coefficients. The weights aresummable since the considered FARIMA processes are stationary and, therefore,the obtained value has the Gaussian distribution as well [9].The distribution of the sum of FARIMA variables has to be computed. Since

a single FARIMA variable has a Gaussian distribution the sum has a Gaussiandistribution as well. As the Gaussian distribution is fully described by its meanand variance values and moreover, an from equation (3.10) has a mean zero theonly value to be found is the Zn variance.The variance of a sum of correlated and stationary random values is given

by equation (2.18). In all three cases for which the autocorrelation functionof the FARIMA process has been presented in the previous chapter (equations(3.22),(3.28) and (3.30)) the envelope process has been determined by the authorof this thesis [43].

4.3 Envelope process for FARIMA process 47

The first considered process is the simplest FARIMA(0,d,0). Ac-cording to results developed in [43] the envelope process has the fol-lowing form

Z(n) = F−1(1 − ǫ)

√

2Γ(−1 − 2d)

(1

Γ2(−d) − (−1)n

Γ(−d− n)Γ(−d+ n)

)

(4.30)

where F (·) is a cumulative distribution function of the normalised Gaussian dis-tribution and it is assumed that D2an = 1.

Equation (4.30) has been proved in Appendix A.

In Figure 4.5 an example of FARIMA(0,d,0) and its envelope is presented.

0 10 20 30 40 50 60 70 80 90 100−10

0

10

20

30

40

50

60

70

80

n

Zn,Z

(n)

Z(n) for ǫ = 0.05

Z(n) for ǫ = 0.2

Z1

n

Z2

n

Figure 4.5: An example of two paths of the accumulated FARIMA(0,d,0) d=0.3and the envelopes for two different ǫ values.

The second considered process is FAR(1,d). Its covariance valuesare described by a hypergeometric function (equation (3.26)), so its

48 Envelope process

envelope is determined by the same function and is given by

Z(n) = F−1(1 − ǫ)√c1 · (4.31)

·√

c2 + (−1)n1 −2 F1(1, 1 + d− n;−d− n;φ) −2 F1(1, 1 + d+ n;−d+ n;φ)

Γ(−d− n)Γ(−d+ n)

where F (·) is the cumulative distribution function of a normalised Gaussian dis-tribution. It is assumed that D2an = 1 and c1 and c2 are given by

c1 =2Γ(−1 − 2d)

1 − φ2(4.32)

c2 =22F1(1, 1 + d;−d;φ) − 1

Γ2(−d) (4.33)

Equation (4.31) has been proved in Appendix B.In Figure 4.6 an example of FAR(1,d) and its envelope is presented.The last considered process is FMA(d,1) described by equation

(3.29). According to results developed in [43] the envelope processhas the following form

Z(n) = F−1(1 − ǫ)√

2Γ(−1 − 2d)·√

2θ + d(1 + θ)2

dΓ2(−d) +(−1)n((n2 − d2)(θ − 1)2 − 2θd(1 + 2d))

Γ(1 − d− n)Γ(1 − d+ n)

(4.34)

where F (·) is the cumulative distribution function of the normalised Gaussiandistribution and it is assumed that D2an = 1.Equation (4.34) has been proved in Appendix C.In Figure 4.7 an example of FMA(d,1) and its envelope is presented.As it should be expected equations (4.31) and (4.34) are equivalent to equation

(4.30) for θ = 0 and φ = 0 respectively since in these cases FMA(d,1) andFAR(1,d) processes reduce to a FARIMA(0,d,0) process.The first equality holds because 2F1(a, b; c; 0) = 1 (see equation(3.26)).The second equality holds as [86]

Γ(1 − n− d)Γ(1 − d+ n) = (d2 − n2)Γ(−n− d)Γ(−d+ n) (4.35)

The accuracy of the obtained results was analysed by simulation studies. Thesimulation scenario was based on generatingN different paths for each consideredtime point n. For all values of the accumulated process for a considered n ithas been found whether its value has been greater than the envelope process ornot. Next, the probability that the envelope process has been exceeded by the


0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

n

Zn,Z

(t)

Z(n) for ǫ = 0.05

Z(n) for ǫ = 0.2

Z1

n

Z2

n

Figure 4.6: An example of two paths of the accumulated FMA(d,1) d=0.4, φ = 0.9and the envelopes for two different ǫ values.

simulated paths has been computed with a 95% confidence interval. Since thenumber of independent paths that have to be generated in order to obtain anaccurate confidence interval is ǫN > 5 [3] the minimum considered ǫ value is0.05.Since the obtained results are very similar, only two examples of results ob-

tained from the simulation study are presented in Figures 4.8 and 4.9.The theoretical probability for both presented cases lies within the confidence

intervals for almost all n values, which is expected.

4.3.2 Envelope process parameters identification

The envelope process as proposed by equation (4.8) has some parameters, two ofthem are a and σ, the other parameters depend on a type of Zn process.In equation (4.6) it has been assumed that Zt is normalised. In such a case

parameters estimation is as follows; a is the mean value; σ is the standard devi-ation of the At process on time scale 1, i.e. D2A1 = σ2. The Zt parameters are

50 Envelope process

0 10 20 30 40 50 60 70 80 90 100−20

−10

0

10

20

30

40

50

60

70

80

n

Zn,Z

(n)

Z(n) for ǫ = 0.05

Z(n) for ǫ = 0.2

Z1

n

Z2

n

Figure 4.7: An example of two paths of the accumulated FMA(d,1) d=0.2, θ =−0.8 and the envelopes for two different ǫ values.

estimated according to the type of Zt.In case where Zt is an accumulated sum of FARIMA random variables Zt is

not a normalised process. Therefore, σ is not a simple standard deviation of theA1 distribution. In each case of the three considered FARIMA processes, σ isevaluated in terms of the d, φ, θ and D2A1 values.The first case is the FARIMA(0,d,0) distribution for which [38]

D2Z1 =(−2d)!

((−d)!)2 (4.36)

and, therefore, σ is given by

σ2 =D2A1((−d)!)2

(−2d)!(4.37)

where D2A1 is the workload variance.


Figure 4.8: The 95% confidence intervals obtained from simulation of a proba-bility that an FMA(d,1) process exceeds the envelope process. d=0.1, θ = −0.9and ǫ = 0.2.

The second case is the FAR(1,d) process for which [38]

D2Z1 =(−2d)!

((−d)!)22F1(1, 1 + d; 1 − d;φ)

1 + φ(4.38)


σ2 =D2A1((−d)!)2

(−2d)!

1 + φ

2F1(1, 1 + d; 1 − d;φ)(4.39)

where D2A1 is the workload variance.The last case is the FMA(d,1) process for which [38]

D2Z1 =(−2d)!

((−d)!)2(

1 + θ2 − 2θd

1 − d

)

(4.40)


σ2 =D2A1((−d)!)2

(−2d)!

1 − d

(1 − d)(1 + θ2) − 2θd(4.41)

where D2A1 is the workload variance.

52 Envelope process

Figure 4.9: The 95% confidence intervals obtained from simulation of a probabil-ity that an FAR(1,d) process exceeds the envelope process. d=0.4, φ = 0.5 andǫ = 0.05.

4.4 Chapter summary

In this chapter the envelope process has been presented. The first section de-scribes the envelope process in general terms without assuming what kind ofprocess the envelope process is computed for. We have started from an envelopeprocess obtained for a deterministic function in order to define an envelope pro-cess of stochastic processes. The second section describes the envelope processesobtained for the FBM process and the H − ss, si process with SαS marginaldistribution presented in literature. Moreover, in this section a traffic model ofa queueing process proposed by Norros has been referred to. The last sectionpresents the results obtained by the author of the thesis. An envelope process forthree special cases of FARIMA(p,d,q) has been obtained. The presented equa-tions of the envelope processes are proved in appendixes and validated by thesimulation study. The obtained results are functions valid for discrete values ofthe n argument (describing time).

Chapter 5

Queue length computation

A computer network is built of different devices connected with each other bydifferent technologies. Nevertheless, there is one common problem for all tech-nologies, namely the follow-up effects of implementing some queueing algorithms.The investigation of the queue behaviour has a critical impact on enforcing thequality of service (QoS) offered by the network. In this chapter an envelope pro-cess presented in Chapter 4 is used to compute some QoS and queue parameters.

5.1 Queue process

The understanding of queueing processes is crucial since they influence QoS pa-rameters. The first QoS parameter is a drop probability pd, i.e. a probability ofdropping a particular packet coming to a queue; packet dropping is caused eitherby a finite buffer space (Drop Tail) or by congestion countermeasures (RandomEarly Detection).The second QoS parameter is a queueing delay D. However, it is worth noting

that different networking applications are sensitive to different delay parameters.The real time applications are sensitive to the maximum delay Dmax. Mean delaycharacterises a TCP connection efficiency. Delay jitter (fluctuations around itsmean value) influences greatly time synchronous applications which rely on aregular data transfer.In this dissertation the envelope process is described and used to compute

queue properties. However, not each queue property can be computed by usingthe envelope analysis. The delay jitter cannot be computed by using the envelopeanalysis since to obtain a jitter value a correlation structure of the queue processhas to be known. The envelope analysis is based on changing a queue processto a deterministic function, and a correlation structure of the queue cannot be

54 Queue length computation

analysed.The mean queue length can be computed according to the following equation

[40]

EQt =

∫ ∞

0

P (Qt > x)dx (5.1)

where Qt is a queue process and Qt ≥ 0.The computation of integral (5.1) is difficult therefore, this task will be con-

sidered in future research.The maximum delay and the drop probability are QoS parameters that can

be computed from the envelope analysis. Note that both these parameters relyon each other. If the queue is longer then the drop probability is smaller forthe same workload process and the same link capacity. On the other hand, alonger queue influences a higher Dmax since Dmax is simply a delay that occursfor the last packet in a full queue buffer. The relation between Dmax and pd,that can be computed by the envelope analysis, is important since from the QoSparameters point of view Dmax and pd should have low values at the same time.The relationship between Dmax and pd helps to find the optimum for a particularapplication or a network technology.A common queue control algorithm derived is the Leaky Bucket (LB) algo-

rithm. The LB algorithm does not store packets but it stores tokens. A packet issent if there are enough tokens, otherwise it is dropped or marked to be dropped.Therefore, for the LB algorithm there is no Dmax but instead there is the max-imum burst size b. The envelope analysis makes it possible to compute themaximum burst size (in fact, it is the bucket size b) as a function of a tokenaccumulation rate r. This function is called the burstiness curve and is describedin the next chapter.Another interesting queue parameter that can be computed via the envelope

analysis is a busy period length. The busy period length is [60] the length of thetime while the queue is never empty. The busy period is an important parametersince the maximum delay of a packet which can occur in a queue is bounded bythe maximum busy period length [60]. Note that If there are two traffic classesthe lower class busy period is longer or equals the busy period of higher trafficclasses.In order to build a queue process an accurate workload model has to be

found. Since a workload process can be characterised by its mean, variance andautocorrelation function, the model proposed by Norros and defined by equation(4.6) is accurate and simple.The Norros model concerns an accumulated process. Here a short explanation

is given why an accumulated workload process instead of an increment processis discussed. Assume that ai is an increment process of a workload process.Therefore, according to equation (4.6) the amount of traffic which enters a queue

5.1 Queue process 55

during i-th time interval of length ∆t is given by

ai = a∆t+ σzi (5.2)

where zi is a process with a particular correlation structure (Norros used an FGNprocess) and zi depends on ∆t value.In the rest of this dissertation an accumulated workload process is called

the FGN model if the autocorrelation function of ai is modelled by an FGNprocess. Similarly, an α-stable model or a FARIMA model are accumulatedworkload processes if an autocorrelation structure is modelled by increments ofanH−ss, si process with an α-stable marginal distribution or a FARIMA process,respectively.Let us denote a discrete time queue length process by Qi. If ai +Qi−1 ≥ c∆t

then after an i-th time interval the queue length process is described by

ai +Qi−1 ≥ c∆t ⇒ Qi = Qi−1 + ai − c∆t (5.3)

Based on the implication (5.3) and the assumption that Q0 = 0 we canconclude that

∀ 1 < i < n ai +Qi−1 ≥ c∆t⇒ Qn = Qn−1 + an − c∆t = Qn−2++an−1 − c∆t+ an − c∆t = · · · =

∑ni=1 ai − cn∆t = An − cn∆t

(5.4)

In brief, a queue length Qn at time instant n equals the accumulated trafficAn entering the queue, decreased by the served traffic cn∆t.Since ai = a∆t+ σzi and Zn =

∑ni=1 zi we have

Qn = an∆t+ σZn − cn∆t (5.5)

An explanation of equation (5.5) is showed in Figure 5.1

An =∑n

i=1 ai

Qn =∑n

i=1(ai − c∆t) = An − cn∆t

ct

Figure 5.1: Queue length, accumulated and served traffic processes.

If ∆t is normalised, i.e. ∆t = 1, we obtain the final equation for the queuelength after n observations

Qn = an+ σZn − cn = An − cn (5.6)


where An is the accumulated workload process.If a queue process is time continuous then a queue length is the amount of

traffic that comes to a server and has not been served till time t and is given by

Qt = At − ct (5.7)

Like for a discrete time process, equation (5.7) is valid if during all timeinterval (0, t) the queue is never empty, i.e. Qt = At − ct⇔ ∀ s ∈ (0, t) Qs > 0.The queue process can be described by equations (5.6) or (5.7)

depending on the type of the process which is either continuous ordiscrete in time. Both these equations hold on condition that for allobserved time instants the queue length is nonzero.

In Figure 5.2 an example of an accumulated workload process with a corre-lation structure modelled by the FGN process with H = 0.8 (FGN model) ispresented.

0 20 40 60 80 1000

20

40

60

80

100

120

t

At

Figure 5.2: An example of an accumulated workload process proposed by Norrosfor Zt with the FBM distribution with H = 0.8, a = 1, σ = 1.

5.1 Queue process 57

For the FGN model presented in Figure 5.2 the queue process for c = 1.5 ispresented in Figure 5.3.

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14

16

18

t

Qt

t1 t2 t3

Figure 5.3: The queue process for the accumulated workload process presentedin Figure 5.2 and for the link capacity c = 1.5.

In Figure 5.3 time intervals where the assumptions of equations (5.7) are validhave been marked by t1, t2 and t3.Since Qt is strongly autocorrelated there could be a dependency between the

drop probability for t1, t2 and t3 intervals. The intriguing topic whether queueingprocesses obtained for different busy periods are correlated has been omitted inthe dissertation.Note that if an envelope process of a workload process is known

(discrete or continuous time) the queue envelope process is given by

Q(t) = A(t) − ct = (a− c)t+ σZ(t) (5.8)

orQ(n) = A(n) − cn = (a− c)n+ σZ(n) (5.9)

for ∆t = 1. The justification is the same as the justification for equation (4.8).In order to use the equations (5.8) and (5.9) properly it is crucial to under-

stand the unit computation that is described in Appendix D. To improve the


practical usage of the results obtained in this dissertation and to com-pare different envelope processes for different workload processes themean and variance values have been taken from the real measurements.The Ethernet measurements available in file BC-pAug89.TL (it can be down-

loaded from http://www.ist-mome.org/database/MeasurementData/?cmd=datadetail&id=1) have been considered as the reference data. The measurements havebeen taken by Willinger and described in [54]. Since the time precision of theEthernet data described in [54] is 5 ms here we have used the values that arescaled to ∆t = 1 ms. The scaling has been conducted in the same way as for anH − ss, si process; the details can be found in Appendix D. In the rest of thischapter a = 138 bit/ms and σ = 420 bit/ms.

5.2 Busy period

A busy period τb is an amount of time during which a queue system is neverempty. According to the definition presented in [60] τb is given by

τb = t: P (At > ct) = pb (5.10)

where At is a workload process, pb is a probability that a busy period is τb longand c is a link capacity.Equation (5.10) means that a probability that an accumulated workload pro-

cess at time τb is higher than all the traffic having been served till τb (i.e. thereis not an empty queue) equals pb.Note that according to the envelope formalism we know that P (At > A(t)) =

ǫ. Therefore, if for some tb we have A(tb) = ctb, then pb = ǫ. That is thereason why the envelope analysis makes it possible to compute τb by means ofthe equation

τb = tb: A(tb) = ctb (5.11)

FGN modelA busy period for the FGN model is given by [60]

τb =

(kσ

c− a

) 11−H

(5.12)

where k = F−1(1 − ǫ), F (·) is the normalised Gaussian distribution and ǫ = pb.α-stable modelSince an envelope process for a workload process with α-stable marginal dis-

tribution is similar to an envelope process obtained for the FGN model (seeequations (4.13) and (4.26)), the obtained result is almost the same as in theequation (5.12). The only difference is that k is given by

k = K(ǫ) (5.13)

5.2 Busy period 59

where K(ǫ) = F−1(1− ǫ) where F (·) is a cumulative distribution function of theSα(1, 0, 0) distribution. In equation (4.26) a factor σ1 occurs but since σ1 is thescale parameter for time t = 1 and σ in equation (5.7) is the scale parameter,too. Since σ1 and σ have exactly the same meaning, it is assumed that the σ1

factor is included in the σ factor.FARIMA modelIt is difficult to give a function of τb for the FARIMA model since Z(n) is

given in terms of the gamma function. Nevertheless, a numerical solution can befound and these results are presented.ComparisonTable 5.1 shows the values of a busy period τb in milliseconds obtained for dif-

ferent models. Each time the mean a = 138 bit/ms, standard deviation σ = 420bit/ms and pb = ǫ = 0.05. The link capacity is measured by the link utilisationρ = a

c . Since considering the influence of correlation parameters is the mostinteresting, these parameters are taken into consideration.

α-Stable FBM FAR(1,d)α ρ = 0.7 ρ = 0.3 H ρ = 0.7 ρ = 0.3 φ ρ = 0.7 ρ = 0.3

2 136 4.60 0.5 136 4.600.6 549 16−0.6 36 ≤ 2

1.5 3.6 · 103 22.3 0.7 3.6 · 103 12.70.6 15 · 103 52−0.6 458 ≤ 2

1.2 1.1 · 108 4.3 · 103 0.9 4.7·1010 2.1 · 103 0.6 2.1 · 1011 9.2 · 103

−0.6 1.4 · 109 61

Table 5.1: A busy period in milliseconds for different workload processes anddifferent correlation structure parameters; a = 138 bit/ms, σ = 420 bit/ms andǫ = 0.05.

In Table 5.1, a symbol ≤ 2 denotes the results for which the consideredprecision makes it possible to obtain only an upper bound result. An explanationwhy more precise results cannot be obtained are given in Appendix D.One can conclude from Table 5.1 that a long term correlation structure has a

critical impact on the busy period length as the busy period extends for all themodels for higher H or lower α values.As it should be expected, τb for the α-stable distribution and the FBM process

are the same for α = 2 and H = 0.5. For a higher H and a lower α the obtainedvalues are different. The FBM process is more dependent on the link utilisationsince for the link utilisation ρ = 0.7 it has higher dH values but for the linkutilisation ρ = 0.3 the situation is opposite.The most interesting conclusion for this dissertation is that an SRD struc-

ture influences the busy period length for the FARIMA model. This


effect cannot be analysed using α-stable and FGN models which do notexhibit the SRD structure. A positive φ value results in a positive SRD, anegative φ value results in a negative SRD. It can be seen that φ value has a largeinfluence on τb. Note that for the FGN model with H = 0.7 we get τb = 3.6 · 103.For the same Hurst parameter H = 0.7 and the FARIMA model with a positiveSRD (φ = 0.6) the busy period τb extends more than 4 times. Similarly, for anegative SRD (φ = −0.6) the busy period curtails 7 times. The same behaviourcan be observed for a lower link utilisation. These results indicate that the SRDhas a large influence τb and it should be taken into consideration while analysingan LRD traffic. An envelope process of the FARIMA process provides means toconsider both SRD and LRD structures at the same time.

5.3 Drop Tail parameter computation

The Drop Tail (DT) is the simplest queue algorithm where each coming packetis sent immediately and each packet which cannot be sent is stored in a queue ifthere is enough free space or otherwise it is dropped.An important parameter of the DT algorithm is the drop probability pd (the

probability of packet dropping). The drop probability pd depends on the linkcapacity c, the buffer size B and the workload process. As Qt is the queuelength, the drop probability pd equals the probability that the queue processexceeds the buffer size B and it is given by

pd = P (Qt > B) (5.14)

Since Q(t) is a queue envelope process it is true that

ǫ = P (Qt > Q(t)) (5.15)

where ǫ is the exceed probability.From equations (5.14) and (5.15) we can conclude that if B = Q(t) then

pd = ǫ. If the buffer size could follow the workload envelope process,the drop probability would equal the exceed probability.Since the buffer size is a constant value and it is better to use the overesti-

mation of the drop probability parameter the maximum of the queue envelopeprocess is taken as the buffer size because

pd = P (Qt > Q(t)) ≤ P(

Qt > maxt≥0

Q(t))

= P (Qt > Qmax) (5.16)

On the basis of equation (5.16) two parameters of the DT algorithm can befound: the drop probability pd is approximated by the exceed probability ǫ andthe buffer size B = Qmax.

5.3 Drop Tail parameter computation 61

The other important QoS parameter is the maximum delay Dmax. Note thatfor the FIFO scheduling algorithm the maximum delay can be easily computedfrom the following equation

Dmax =B

c=Qmax

c(5.17)

where c is the link capacity.The Last equation holds under assumption that the workload process is ap-

proximated by an envelope process.FGN modelThe DT queue algorithm parameters have been computed in [60] where the

workload process is described by equation (4.6) where Zt is an FBM process.Since the queue envelope process is described by equation (5.8) and the FBMenvelope process is given by equation (4.13), Qmax is

Qmax = (a− c)t∗ + kσ(t∗)H (5.18)

where k = F−1(1 − ǫ) where F (·) is a cumulative distribution function of thenormalised Gaussian distribution and t∗ is given by

t∗ =( kσH

c− a

) 11−H

(5.19)

and is obtained by solving the equation

∂Q(t)

∂t= 0 (5.20)

Equation (5.18) can be rewritten in a convenient form [60]

Qmax = (c− a)H

H−1 (kσ)1

1−H HH

1−H (1 −H) (5.21)

The maximum delay Dmax, which is a QoS parameter, is strongly dependenton the buffer size B as it is described by equation (5.17). The link capacity cthat is needed to obtain the particular maximum delay Dmax with a particulardrop probability ǫ can easily be derived from the envelope analysis by a numericcomputation from the following equation

Dmax =(c− a)

H

H−1 (kσ)1

1−H HH

1−H (1 −H)

c(5.22)

An example of Dmax as a function of the link utilisation ρ is given in Figure5.4.


Figure 5.4: The maximum delay Dmax in ms as a function of the link utilisationobtained for different Hurst parameter values. a = 138 bit/ms, σ = 420 bit/msand ǫ = 0.05.

α-stable model

For an α-stable model the equation of Qmax is similar to the result obtainedfrom the FGN model analysis since the only difference between these envelopeprocesses is the k value. For the FGN model k is a quantile of the normalisedGaussian distribution and for the α-stable model k is a quantile of the normalisedSαS distribution.FARIMA model

In this case it is impossible to derive a function describing Qmax since Q(n) isa discrete process and the maximum value of Q(n) is not easy to find. The Q(n)function cannot be approximated by a continuous function where n is replacedby t since the gamma function cannot be computed for all real values.Similarly to the previous chapter the numerical results can be obtained. Both

Qmax and Dmax have been obtained by the numerical algorithms.ComparisonTables 5.2 and 5.3 show Qmax values in kbit and the link utilisation ρ obtained

for different models and different correlation parameters. “≤ 0.4” and “≤ 0.6”denote a lack of precision if the obtained results cannot be computed more pre-

5.3 Drop Tail parameter computation 63

cisely and only upper bound results can be found. All the results presented inthe tables have been obtained by the author of this dissertation on the basis ofthe presented equations.

α-Stable FBM FAR(1,d)α ρ = 0.7 ρ = 0.3 H ρ = 0.7 ρ = 0.3 φ ρ = 0.7 ρ = 0.3

2 2.02 0.37 0.5 2.02 0.370.6 8.02 0.91−0.6 ≤ 0.6 ≤ 0.4

1.5 31.5 1.06 0.7 27.9 0.540.6 116 2.03−0.6 3.55 ≤ 0.4

1.2 4.4 · 108 92.3 0.9 1.1·1011 25.80.6 4.8 · 1011 115−0.6 3.2 · 109 0.78

Table 5.2: Qmax in kbit for different workload processes and different correlationstructure parameters; a = 138 bit/ms, σ = 420 bit/ms and ǫ = 0.05.

The values from Table 5.2 present a type of behaviour similar to the oneobserved for the busy period values (Table 5.1). Again, the most important isthat the maximum queue length is not influenced by an LRD structure only.The queue length obtained for the FGN model with Hurst parameter H = 0.7is almost 7 times longer than the value obtained for H = 0.5. It confirms theresults obtained in references [8, 68]. The comparison of the results obtained forthe same Hurst parameter value but for different SRD structure reveals that anSRD structure can also have a large influence on QoS parameters.Table 5.3 shows the link utilisation that is obtained for particular traffic pa-

rameters, the drop probability and two different delays.

α-Stable FBM FMA(d,1)

αDmax = Dmax =

HDmax = Dmax =

θDmax = Dmax =

50 500 50 500 50 500

2 0.90 0.99 0.5 0.90 0.990.6 0.99 1.00−0.6 0.83 0.98

1.5 0.59 0.79 0.7 0.61 0.790.6 0.85 0.94−0.6 0.56 0.75

1.2 0.25 0.33 0.9 0.30 0.350.6 0.46 0.52−0.6 0.24 0.34

Table 5.3: Link utilisation ρ for different workload processes and different corre-lation structure parameters; a = 138 bit/ms, σ = 420 bit/ms and ǫ = 0.05.

Generally, the trends are similar, the only difference is that for the FMA(d,1)process a positive SRD correlation structure is obtained for a negative θ value. As


it can be observed, the positive SRD correlation structure for FMA(d,1) modelhas less influence on a ρ than FAR(1,d) on Dmax.Another conclusion which can be drawn from Table 5.3 is that the Gaussian

distribution, even with a strong and positive LRD and SRD correlation structure,provides a higher link utilisation than the α-stable distribution. It is caused bya high variability of the α-stable distribution for low α values.

5.4 Leaky Bucket parameter computation

Network traffic as a random process has a mean and a variation of the amount oftraffic sent during a time interval. The research of the traffic structure revealedthat in a network traffic a commonly observed phenomenon is so called burstsof traffic [28, 54]. Burst means that during some period of time a source sendsmuch more data that an average value and after that it sends much less or evenit does not send any data. On the other hand, a user traffic has to be limitedin order to avoid congestion. Note that if the TD algorithm is used to control abursty traffic the delay or drop probability would be high. One of the solutionsis to use the Leaky Bucket (LB) algorithm which enables one to limit the usertraffic and send bursts.The LB algorithm is described in terms of two parameters. The first is the

bucket size b and the second is the token accumulation rate r (see Figure 5.5). Apacket is sent if there are enough tokens; a single token can be associated with asingle packet or even a single bit. The second case (a single token is associatedwith a single bit) is considered in this dissertation. If there are not enough tokensthe packet is dropped or marked for dropping. The token bucket is filled by thetokens with a constant rate r.Note that a packet can be sent if there are enough tokens. Therefore, if one

token equals one bit then a packet is dropped if

rt + b−At = b− (At − rt) < 0 (5.23)

where At is an accumulated workload process, b and r are the LB parameters.Equation (5.23) is valid under similar assumptions as equation (5.7), i.e. ∀ s ∈

(0, t) 0 < As − rs < b.Note that the term At − rt can be understood as an amount of tokens that

are needed at time t. In this notation the drop probability is given by

pd = P (At − rt > b) (5.24)

where pd is the drop probability.

5.4 Leaky Bucket parameter computation 65

Tokenaccumulationrate r

tokenbucketsize b

An =∑n

i=1 aienoughtokens?

yes

yes

no

no

tagging?

packetdropped

packettagged

Figure 5.5: The LB algorithm scheme and parameters.

Note that equation (5.24) is similar to equation (5.14) obtained for the DTalgorithm where c = r and B = b. Therefore,

b = A(t) − rt⇓

pd = ǫ(5.25)

On the basis of the assumption (5.25) in [29] a new LB algorithm namedFractional Leaky Bucket (FLB) has been proposed. The FLB algorithm allows abucket size b to be a time function. The details can be found in [29].Here the classic LB algorithm is considered. In order to obtain an upper

bound of pd, similarly to the DT algorithm, b is given by

b = maxt

(A(t) − rt) (5.26)

Note that the buffer size obtained for the DT algorithm is the same as thebucket size obtained for the LB algorithm. Since Qmax = b, the table presentingthe bucket size as a function of correlation parameters for different processes willbe exactly the same as Table 5.2. As there is no delay for the LB algorithm amaximum delay has got no equivalent in the LB parameters.


In terms of the LB properties Table 5.2 is interpreted in the following manner:b = Qmax. ρ is the LB utilisation but not the link utilisation. The LB utilisationis given by ρ = a

r .A more extensive description of using the envelope process in computing the

LB parameters is given in Chapter 6 where a burstiness curve is analysed ingreater detail.

5.5 Chapter summary

In this chapter the applications of an envelope process have been presented. In thefirst section assumptions for a queue envelope process were discussed. The secondis devoted to the busy period analysis; the busy period is the time length whilethe queue is not empty. The busy period computation is presented for the threeprocesses described in Chapter 4. The third section describes the computation ofQoS parameters with the assumption that the DT queue with the FIFO queueingfashion is used. The last section discusses an LB algorithm and the mappingresults obtained for the DT algorithm for LB parameters. The presented resultsshow that it is not only an LRD structure which has a large influence on the QoSand network parameters. It can be concluded that also an SRD structure has animportant influence on these parameters. Therefore the FARIMA process, whichcan model SRD and LRD behaviour independently, models network traffic betterthan the processes that estimate only the LRD property. Since in Chapter 4 anenvelope of the FARIMA process is found the QoS and the network parametersfor this process can be found as well.

Chapter 6

Burstiness curve

This chapter presents the estimation of a burstiness curve (BC) by using theenvelope analysis. The burstiness curve has been determined separately for theFGN and FARIMA models. The obtained results are useful when modelling anddesigning video servers.

6.1 Definition and application

On the basis of the general knowledge of queue processes it can be inferred thatthe mean and the standard deviation of the traffic is not sufficient to determine thelink utilisation needed to send the traffic with particular drop probability. Whatis more, it can be observed that a packet traffic is bursty (numerous packets aresent one by one [28, 57]). Bursty traffic can be characterised by link utilisationand a burst size. In order to describe such a traffic the burstiness curve has beenproposed by Low and Varaiya in [57].The traffic characteristics are important since admission control (AC) rules are

based on traffic description. AC rules make it possible to take a proper decisionabout the acceptance or rejection of a new connection. Examples of using ACrules are controlling the amount of resources needed for the connections of theVoD or other streaming servers where decision about serving or rejecting a newcustomer has to be made. The connections which are served have to be restrictedto the declared values. A useful and easy way of traffic policing is an LB algorithmdescribed in Section 5.4 and related to the BC description of traffic by limitingthe maximum burst size and mean source capacity.The BC defined in [57] is obtained for a particular traffic trace. The BC are

pairs of the buffer size and the accumulated rate for which all packets are served(there is no dropping). Note that such a definition of the BC makes it impossible

68 Burstiness curve

to find a BC for the unbounded traffic model like the FGN or FARIMA models. Itis caused by the positive, but very small, probability of receiving a large amountof data.In literature three different terms for the same concept of modification of the

BC definition can be found. The first term is a probabilistic burstiness curve(PBC) given in [14]. The second term is an ǫ-weak burstiness curve given in [83].The third term is an effective burstiness curve (EBC) given in [82]. Since allthese terms describe the same concept, in this dissertation the burstiness curvefor a probability p and an accumulated rate r is defined as

b(r) = b: P (At − rt > b) = p (6.1)

where At is an accumulated workload process.A BC defined in a manner proposed by Low and Varaiya is obtained from

equation (6.1) for p = 0. In this dissertation the BC is represented by b(r), theprobability p for which the BC is computed is omitted in the BC notation.Note that if b is a bucket size and r is an accumulated rate, the BC defines

pairs of LB parameters for which the drop probability p has the same value.Given equation (6.1) and the interpretation of b and r values we can concludethat the BC is a decreasing function of r because if the bucket size b is increasedthe accumulated rate r has to be lower or equal; vice verse, if the bucket size bis decreased the accumulated rate r has to be higher or equal in order to obtainthe same probability p as for the previous LB parameters. An example of twodifferent LB parameters that give the same drop probability value is shown inFigure 6.1.The drop probability obtained for both LB parameters described in Figure

6.1 is the same but if the traffic shape changed, for example, the first burst ofdata does not contain 2 packets but 3 then the r and b values will not be propersince p obtained in the first case (b = 0.2Mbit and r = 0.5 Mbit/s) will be higherthan in the second case.Derivation of an exact burstiness curve for a particular traffic trace (like a

video stream) is a computation costly algorithm. Some approximations of anexact burstiness curve have been proposed in [17, 79]. In [79] the computationtime have been shortened from 6 hours to 10 seconds for about a 2 hour long videostream. The algorithm for computation of the BC for particular traffic traces orstreams is accurate and, therefore, can be used as a final solution. In order tobuild, plan or globally manage VoD or other streaming servers the theoreticalvalues of the BC have to be found.In [14] a BC has been computed with an assumption that a workload process

is modelled by an MMPP process. The BC makes it possible to establish areal-time multimedia connection with the guaranteed QoS. The model has beenpresented with an application to MPEG and JPEG video sequences. Based on

6.2 Burstiness curve for an FGN model 69

0

0

0

1

1

1

2

2

2

3

3

3

4

4

4

t[s]

t[s]

t[s]

Incoming traffic 1 Mbit/s

b = 0.2 Mbit, r = 0.5 Mbit/s

b = 0.6 Mbit, r = 0.25 Mbit/s

Figure 6.1: An example of two different LB parameters with the same quantityof dropped packets (red packets) value.

the BC parameters an AC mechanism selects a proper resource allocation. Theinfluence of an SRD structure on the BC shape has been shown as a consequenceof the screen to screen autocorrelation structure.A BC has been used to resource provisioning in a wireless environment [83].

An algorithm for finding an empirical BC has been shown and applied to anMPEG-4 video trace. The multiplexing gain has been shown since the BC of themultiplexed traffic is smaller than a sum of BCs of individual sources. The sameconsideration has been presented in [82].

6.2 Burstiness curve for an FGN model

A BC is given by equation (6.1) similarly to the equation describing a queue pro-cess (see equation (5.14)). The same as for the DT algorithm, an overestimationof the p value is an exceed probability ǫ since

P (At − rt > Q(t)) = p⇒ P (At − rt > Qmax) ≤ p (6.2)

as Qmax is the maximum of the envelope process Qt.Since p and ǫ values in equations (4.3) and (6.2) play the same role, the

probability p for which the BC is computed equals exceed probability ǫ. In therest of the dissertation p is replaced by ǫ.According to equation (6.2) an upper bound of a BC for the FGN model is

70 Burstiness curve

given by [43]b(r) = (r − a)

H

H−1 (kσ)1

1−HHH

1−H (1 −H) (6.3)

where k equals F−1(1 − ǫ).Equation (6.3) is rewritten to a more convenient form

b(a

ρ) = aHη

11−H (

1

ρ− 1)

H

H−1 k1

1−H HH

1−H (1 −H) (6.4)

where only the a parameter is unit dependent; η = σa is the index of dispersion

[69] and ρ = ar is the LB utilisation.

While analysing the right side of equation (6.4) parameter by parameter itcan be noticed that if the rest of the parameters is constant then:

• increasing a or η results in increasing b since 0 < H < 1 and terms H and1

1−H are positive;

• increasing LB utilisation ρ results in obtaining higher b values becauseH

H−1 < 0 and for ρ→ 1 the term 1ρ − 1 → 0;

• decreasing probability ǫ results in increasing b since k is a decreasing func-tion of ǫ and b is an increasing function of k.

The last parameter is the Hurst parameter. Equation (6.3) is a complicatedfunction of the Hurst parameter value, therefore, it is not evident how it influencesthe BC values. In Figure 6.2 BCs obtained for different H values have beenpresented.As one can see, an influence of the Hurst parameter is difficult to predict and

depends on other parameters like LB utilisation ρ. For a smaller ρ the possible bvalue is smaller for a higher H , in case of a larger ρ the behaviour is reverse.Note that equation (6.3) has been obtained according to the envelope analysis

described in Chapter 5. In this chapter a queue process has been approximated byequations (5.6) and (5.7) for discrete and continuous time processes, respectively.Both these equations hold with assumption that queue is not empty, therefore,the BC has been obtained with the same assumptions. Since these assumptionsare not easy to analyse a simulation study has been proposed.The goal of the simulation studies is to find ranges of the parameters for

which the equation ǫ > ps (where ǫ is the model drop probability and ps is thedrop probability obtained by a simulation study) is satisfied. Since it is difficultto consider small probability values by means of a simulation study ρ has beenchosen in such that the drop probability epsilon was higher than 0.01. The bucketsize b is not less than 3r where r is the token accumulated rate; such limitationcauses a small probability of overflow for a single time slot and, therefore, enablesto consider a long term LB behaviour. Moreover, the smallest possible value of the

6.2 Burstiness curve for an FGN model 71

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

50

100

150

200

250

ρ

b

H = 0.8

H = 0.65

H = 0.5

Figure 6.2: An illustration of the influence of the Hurst parameter value on theburstiness curve shape. a = 138, η = 0.8 and ǫ = 0.05.

bucket size has to be larger than the maximum packet size. The last parameter,an index of dispersion η, is restricted to η ∈ (0.4, 1.5) since values from such aninterval have been obtained by the analysis of different video traces available in[27].Tables 6.1-6.3 present the accuracy described by ps/ǫ of an approximation of a

drop probability ps by ǫ. The accuracy is a function of traffic and LB parameters(η, H , ρ and b) but it is independent of the chosen a value.The first observation is that the results obtained for a higher Hurst parameter

value have much greater variation and, therefore, the confidence interval obtainedfor a higher Hurst parameter value is wider. The last column in Tables 6.1-6.3 compares the accuracy obtained for different traffic parameters. It can beconcluded that for a longer buffer size the accuracy is lower and that the accuracyfor a higher Hurst parameter decreases much slower than for lower values. Fora higher index of dispersion η the accuracy is higher; a deeper simulation studyshowed that for η ≥ 8 the obtained drop probability can exceed ǫ. Since the

72 Burstiness curve

η H ρ b/r ǫ95% confidence interval

for psps/ǫ

0.5

0.50 0.99

3 0.242 (0.080, 0.095) 0.365 0.183 (0.042, 0.056) 0.2710 0.101 (0.018, 0.027) 0.2225 0.022 (0.003, 0.006) 0.19

0.65 0.95

3 0.200 (0.078, 0.111) 0.475 0.158 (0.059, 0.080) 0.4410 0.099 (0.035, 0.053) 0.4425 0.038 (0.006, 0.013) 0.25

0.85 0.80

3 0.126 (0.045, 0.150) 0.775 0.108 (0.025, 0.079) 0.4810 0.085 (0.018, 0.062) 0.4725 0.058 (0.009, 0.030) 0.34

Table 6.1: A comparison between a theoretical drop probability ǫ and a dropprobability ps obtained by simulation study for the FGN model; η = 0.5.


for psps/ǫ

1.0

0.50 0.95

3 0.208 (0.127, 0.138) 0.645 0.146 (0.075, 0.087) 0.5510 0.068 (0.031, 0.037) 0.5025 0.009 (0.003, 0.005) 0.47

0.65 0.85

3 0.168 (0.099, 0.120) 0.655 0.125 (0.067, 0.086) 0.6110 0.071 (0.034, 0.049) 0.5825 0.022 (0.007, 0.012) 0.44

0.85 0.60

3 0.084 (0.028, 0.107) 0.805 0.069 (0.014, 0.043) 0.4110 0.050 (0.009, 0.043) 0.5325 0.029 (0.008, 0.019) 0.46


video traces reveals η < 1.5 it has not been considered in this dissertation.For the two cases of traffic parameters, the BCs have been computed and

6.3 Burstiness curve for FARIMA models 73


for psps/ǫ

1.5

0.50 0.9

3 0.209 (0.165, 0.176) 0.825 0.148 (0.104, 0.111) 0.7310 0.069 (0.041, 0.047) 0.6425 0.010 (0.005, 0.007) 0.65

0.65 0.75

3 0.155 (0.108, 0.133) 0.775 0.113 (0.077, 0.096) 0.7710 0.061 (0.038, 0.049) 0.7125 0.017 (0.006, 0.010) 0.47

0.85 0.50

3 0.092 (0.041, 0.085) 0.695 0.075 (0.031, 0.070) 0.6710 0.055 (0.011, 0.031) 0.3825 0.034 (0.005, 0.024) 0.44


then by a simulation study the stability of ps for different r and b values has beenconsidered. Figure 6.3 shows the obtained results; note that b/r > 3. The resultsobtained for a smaller buffer size can exceed ǫ, but since very small buffer size isnot the most important from the application point of view such a limitation doesnot influence usability of results.When moving to the left side of BC (i.e. lower b and greater r) ps gets closer

to ǫ. Since the envelope analysis provides an upper bound of the drop probabilityps a relation b/r ≥ 3 is an important assumption for obtained results.On the basis of a simulation study one can conclude that ǫ is an upper bound

of the drop probability for b/r ≥ 3. Another limitation of the obtained results isan index of dispersion but since this limitation has been observed for large valuesof η ≥ 8 and considered video traces have much lower values of this parameter,the assumption that η < 8 does not influence possible applications of the envelopeanalysis. The next chapter presents similar consideration for FARIMA models.

6.3 Burstiness curve for FARIMA models

The BCs for the FARIMA model have been computed on the basis of an envelopeprocess obtained for the FARIMA model. As an envelope process of the FARIMAmodel is time discrete, an equation similar to equation (6.3) cannot be foundbecause the maximum of a discrete time process cannot be found in the general

74 Burstiness curve

(71, 3123) (77, 1355) (84, 698)0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

(r, b)

p s

η = 1.5 and H = 0.65

(80, 50014) (88, 7730) (97, 1463)0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055η = 1.0 and H = 0.85

(r, b)

p s

ǫ

Figure 6.3: An illustration of the difference between a theoretical drop probabilityǫ and a drop probability ps obtained by a simulation study; a = 60 kbit/s.

case. Nevertheless, the numeric results have been obtained with the usage of theMathematica package.The number of possible values of different parameters is huge and, therefore,

only the extreme values have been presented in the tables. The traffic parametershave been limited to specific intervals, i.e. η ∈ (0.4, 1.5), b/r > 3, similarly tothe FGN model. Since the motivation for such limitations is the same as for theFGN model, a detailed description has been not repeated (for details see Section6.2).Tables 6.4 and 6.5 present ps/ǫ values obtained for FAR(1,d) models; Tables

6.6 and 6.7 present ps/ǫ values obtained for FMA(d,1) models. The changes inaccuracy measured as ps/ǫ are the same as for the FGN model, i.e. the accuracyis higher for higher Hurst and η parameters. Moreover, a higher SRD correlationstructure increases the accuracy, too. Note that for the FAR(1,d) model an SRDstructure is stronger for a higher φ contrary to the FMA(d,1) model for which anSRD structure is stronger for a lower θ.Similarly to the FGN model, the overestimation values obtained for a larger

bucket size (higher b) are greater (lower ps/ǫ). The effect is not so evident for astronger LRD correlation structure (higher H), a stronger SRD structure (higherφ or lower θ) and a higher variation (higher η).The influence of both SRD parameters φ and θ on a leaky bucket utilisation ρ

is similar, i.e. values obtained for φ = 0.6 are similar to θ = −0.6. Note that the

6.3 Burstiness curve for FARIMA models 75

SRD structure influences strongly the leaky bucket utilisation ρ for all differentLRD structures (different H).Another observation described in numerous papers is that the traffic variance

influences the leaky bucket utilisation strongly since for a higher η the obtainedρ values are lower. For example, if one uses the bucket size b/r = 20 for a trafficwith η = 1.5, H = 0.85 and θ = −0.6, then the leaky bucket utilisation ρ = 0.5.If by changing some properties of the traffic generator (for example changingcoding algorithm properties) traffic parameters can be changed to values η = 0.5,H = 0.5 and θ = 0.6, and the leaky bucket size is again b/r = 20, the linkutilisation would increase to ρ = 0.999 (sic!). Moreover, the simulation studyshows that the real drop probability will decrease as well. One can conclude thatit is better to use a coding algorithm which generates a higher mean value (ofcourse not too high) and better traffic parameters (described by η, H and θ) thana lower mean value and more correlated traffic.

(75, 4355) (78, 2749) (82, 1812)0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

(r, b)

p s

FMA(d,1), θ=-0.6

(75, 4355) (82, 1812) (90, 853)0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055FAR(1,d), φ = 0.6

(r, b)

p s

ǫ

Figure 6.4: The differences between the theoretical drop probability ǫ and thedrop probability ps obtained by a simulation study obtained for the FAR(1,d)and FMA(d,1) models. Traffic parameters: a = 60 kbit/s, η = 1.0 and H = 0.7.

Figure 6.4 illustrates the relation between ǫ and ps for a particular BC. Similarto the results obtained for a FGN model a theoretical drop probability value isover confidence intervals for almost all considered pairs (r, b). The accuracy valueis higher for a lower b.An overestimation obtained for η = 1.5 is in most cases not higher than 0.5,

i.e. the real drop probability is generally higher than ǫ/2. For η = 0.5, the

76 Burstiness curve

η φ H ρ b/r ǫ95% confidence interval

for psps/ǫ

0.5

0.6

0.500.950

3 0.203 (0.126, 0.148) 0.6755 0.144 (0.084, 0.101) 0.64610 0.067 (0.036, 0.044) 0.59920 0.017 (0.009, 0.012) 0.621

0.650.900

3 0.195 (0.120, 0.180) 0.7705 0.153 (0.086, 0.120) 0.67510 0.096 (0.046, 0.066) 0.58320 0.049 (0.019, 0.030) 0.505

0.850.750

3 0.118 (0.037, 0.11) 0.6555 0.100 (0.043, 0.085) 0.64110 0.078 (0.011, 0.039) 0.32420 0.057 (-0.000, 0.061) 0.533

-0.6

0.500.997

3 0.223 (0.036, 0.050) 0.1935 0.163 (0.013, 0.020) 0.10210 0.082 (0.005, 0.010) 0.09520 0.024 (0.000, 0.003) 0.077

0.650.975

3 0.157 (0.043, 0.059) 0.3255 0.115 (0.023, 0.036) 0.26110 0.063 (0.006, 0.012) 0.14920 0.025 (0.002, 0.006) 0.179

0.850.900

3 0.187 (0.019, 0.066) 0.2285 0.169 (0.040, 0.109) 0.44110 0.144 (0.025, 0.064) 0.31220 0.119 (0.012, 0.061) 0.308

Table 6.4: A comparison between the theoretical drop probability ǫ and the dropprobability ps obtained by a simulation study for the FAR(1,d) model; η = 0.5.

accuracy value depends on the Hurst parameters and an SRD structure. If theSRD structure is negative the overestimation is very high. The overestimation isaround 0.5 for a high Hurst parameter and a positive high SRD structure.

6.4 Overestimation analysis

The results presented in Tables 6.1-6.7 show that the overestimation is not stableand in some cases can take rather high values. This section offers an explanationwhy some high values of overestimation have been obtained and why they do not

6.4 Overestimation analysis 77

η φ H ρ b/r ǫ95% confidence interval

for psps/ǫ

1.5

0.6

0.500.800

3 0.246 (0.199, 0.216) 0.8425 0.193 (0.145, 0.160) 0.79010 0.115 (0.083, 0.090) 0.75620 0.046 (0.032, 0.038) 0.772

0.650.700

3 0.222 (0.175, 0.206) 0.8575 0.183 (0.135, 0.154) 0.78910 0.126 (0.074, 0.094) 0.66620 0.073 (0.039, .0510) 0.615

0.850.450

3 0.098 (0.041, 0.101) 0.7295 0.082 (0.023, 0.059) 0.50510 0.061 (0.021, 0.070) 0.74720 0.043 (0.009, 0.031) 0.472

-0.6

0.50 0.975

3 0.228 (0.124, 0.133) 0.5625 0.167 (0.064, 0.067) 0.39210 0.086 (0.020, 0.025) 0.26820 0.026 (0.004, 0.006) 0.208

0.650.900

3 0.187 (0.109, 0.127) 0.6325 0.143 (0.054, 0.071) 0.43810 0.087 (0.025, 0.034) 0.34420 0.041 (0.011, 0.015) 0.325

0.850.650

3 0.118 (0.045, 0.117) 0.6875 0.100 (0.030, 0.091) 0.60810 0.077 (0.012, 0.041) 0.34520 0.057 (0.011, 0.034) 0.402

Table 6.5: A comparison between the theoretical drop probability ǫ and the dropprobability ps obtained by a simulation study for the FAR(1,d) model; η = 1.5.

impact real deployments.The results presented in the dissertation are theoretical and they have been

derived with the following two main assumptions:

1. The queue process can be approximated by equation (5.7).

2. The overestimation obtained by taking Qmax instead of Qt (see equation(6.2)) is small enough from the application point of view.

Ad. 1 According to the conclusion given in Chapter 5 equation (5.7) Qt =At−ct is true only for a never empty queue. Assumption 1 does not influence the

78 Burstiness curve

η θ H ρ b/r ǫ95% confidence interval

for psps/ǫ

0.5

0.6

0.500.999

3 0.261 (0.024, 0.031) 0.1075 0.204 (0.008, 0.014) 0.05510 0.121 (0.002, 0.007) 0.04120 0.049 (0.000, 0.002) 0.020

0.650.990

3 0.213 (0.033, 0.046) 0.1895 0.171 (0.017, 0.032) 0.14610 0.112 (0.006, 0.014) 0.09420 0.061 (0.003, 0.010) 0.111

0.850.900

3 0.106 (0.018, 0.052) 0.3345 0.089 (0.009, 0.060) 0.39510 0.067 (0.009, 0.035) 0.33320 0.048 (0.003, 0.021) 0.257

-0.6

0.5 0.975

3 0.205 (0.101, 0.117) 0.5305 0.144 (0.057, 0.069) 0.43610 0.067 (0.023, 0.030) 0.40120 0.017 (0.005, 0.007) 0.392

0.650.900

3 0.126 (0.063, 0.090) 0.6085 0.085 (0.042, 0.066) 0.63010 0.040 (0.017, 0.023) 0.49520 0.013 (0.005, 0.011) 0.666

0.850.800

3 0.144 (0.066, 0.139) 0.7145 0.126 (0.005, 0.098) 0.41110 0.101 (0.016, 0.065) 0.40320 0.079 (0.022, 0.068) 0.575

Table 6.6: A comparison between the theoretical drop probability ǫ and the dropprobability ps obtained by a simulation study for the FMA(d,1) model; η = 0.5.

drop probability since a packet drop can occur only if the queue is not empty, i.e.during a busy period. However, equation (5.7) is true only if during the wholetime interval (0, t) the queue is never empty what can be denoted by

∀s ∈ (0, t) Qs = As − cs > 0 (6.5)

Since the FGN and FARIMA model values can be negative some differencesbetween equation (5.7) and the simulation queue length are inevitable. Therefore,differences between the theoretical drop probability ǫ and the simulation dropprobability ps are influenced by assumption 1.

6.4 Overestimation analysis 79

η θ H ρ b/r ǫ95% confidence interval

for psps/ǫ

1.5

0.6

0.50 0.993 0.250 (0.128, 0.136) 0.5265 0.191 (0.051, 0.056) 0.28110 0.108 (0.015, 0.019) 0.16020 0.039 (0.003, 0.006) 0.125

0.65 0.953 0.218 (0.128, 0.138) 0.6135 0.174 (0.055, 0.066) 0.34910 0.115 (0.019, 0.028) 0.20620 0.063 (0.009, 0.015) 0.192

0.85 0.753 0.139 (0.057, 0.111) 0.6055 0.120 (0.047, 0.103) 0.62810 0.096 (0.014, 0.040) 0.28220 0.074 (0.010, 0.028) 0.266

-0.6

0.50.90

3 0.275 (0.210, 0.225) 0.7905 0.221 (0.147, 0.154) 0.68110 0.139 (0.079, 0.086) 0.59720 0.063 (0.030, 0.035) 0.520

0.65 0.753 0.202 (0.157, 0.200) 0.8825 0.160 (0.119, 0.140) 0.81310 0.103 (0.065, 0.082) 0.72020 0.053 (0.025, 0.033) 0.555

0.85 0.503 0.108 (0.057, 0.120) 0.8165 0.091 (0.033, 0.100) 0.73010 0.069 (0.010, 0.045) 0.40020 0.050 (0.014, 0.041) 0.546

Table 6.7: A comparison between the theoretical drop probability ǫ and the dropprobability ps obtained by a simulation study for the FMA(d,1) model; η = 1.5.

Ad. 2 Note that Qmax = Q(t∗) as an envelope process has the maximum valuefor some time t denoted by t∗ (measured from the beginning of a busy period).For different network parameters the times t∗1 and t

∗2 can differ. Note that if t

∗1 is

long the accuracy would be lower since numerous busy periods are shorter thant∗1, and therefore, generate a lesser drop probability.

Simulation studies have revealed traffic and leaky bucket parameters resultingin a higher overestimation. Two observations have been made; the first is thatthe buffer size increases overestimation what is logical since if the maximumqueue length is greater then the time of its occurrence t∗ is longer. The second

80 Burstiness curve

observation is that for less correlated traffic (especially with negative correlationfor small lags, i.e. negative SRD structure) an overestimation is higher than thisobtained for traffic being more correlated. The worst overestimation occurs forH ∼ 0.5, i.e. the weak LRD structure, a low traffic variation and strong negativecorrelation structure. Note that ps/ǫ for and positive correlation for short range,i.e. positive SRD structure, is higher or close to 0.5 and since the most importantin practise is positive SRD structure [61] overestimation from implementationpoint of view is not as high (theoretical results are closer to real ones) as it canbe concluded from some values of ps/ǫ presented in Tables 6.4-6.7. Moreover,resources needed to service a traffic with a weak LRD structure (H ∼ 0.5), lowvariation (η ∼ 0.5) and a negative SRD value can be approximated by the trafficmean value since such traffic does not have bursts (no LRD structure), has asmall variation and it behaves regularly (after sending a large amount of data itsends much less in regular periods negative SRD structure).It is important to note that the obtained results give the overestimation of

traffic resources (as it has been assumed). Therefore, during the design or mod-elling process the results can be used as the overestimation of needed resources.Future research combined with an implementation in a real environment can re-veal what is the discrepancy between resources that are needed by a real systemand these obtained by analysis presented in the dissertation.

6.5 Chapter summary

In this chapter a burstiness curve has been described. The first section presentsthe burstiness curve definition as well as the results presented in literature. Thedifficulties in computing a BC for a traffic trace has been described, too. Thesecond section presents a BC obtained by an envelope analysis for the FGNmodel. This solution has been proposed by the author. Then the properties ofthe obtained results have been analysed. At the end of this section the limitationsof using this model have been discussed along with computing an overestimationaccuracy analysed for different network parameters. The third section presentsthe results obtained for FARIMA models described in Chapter 3. Again, theaccuracy of overestimation and the influence of the SRD on the leaky bucketutilisation and the overestimation has been analysed. The last section providesa short explanation why the overestimation varies as well as justification of thelimitations of the presented models.

Chapter 7

Summary

The main result derived and analysed in this dissertation is the burstiness curvefor the FGN and FARIMA models. Moreover, a derivation of the BC for a givenenvelope process has been presented. The additional result of the dissertationis finding a distribution of accumulated FARIMA processes and, in consequence,the envelope processes of accumulated FARIMA processes.The goal of the dissertation has been defined as:

• finding an envelope process for a workload process modelled by one of threespecial cases of FARIMA processes,

• deriving a burstiness curve for a given envelope process,

• simulation study of an accuracy of a theoretical burstiness curve obtainedfor FGN and FARIMA models.

In the thesis the author concluded that it is possible to find an analyticalform of the burstiness curve for a workload process modelled by the accumulatedFARIMA and FGN processes. Since the burstiness curve provides a trade-off oftraffic shaping parameters at a fixed drop probability knowing its form can yieldin a flexible traffic shaping at the same drop probability.In order to obtain BCs for an accumulated FARIMA traffic an envelope pro-

cess of the accumulated FARIMA process has been found. Derivation of theenvelope process is equivalent to looking for accumulated FARIMA process dis-tributions since the envelope process is just a quantile line.An original derivation of the burstiness curve on the basis of an envelope

process has been proposed. According to this algorithm, burstiness curves forthe FARIMA and FGN models have been computed. The obtained burstinesscurves result in some overestimation of required resources. The overestimation

82 Summary

level has been validated by a simulation study. According to simulation results alimitation of the traffic parameters for which the obtained results are the upperbound have been considered with discussion about relationship between theselimitations and the real traffic parameters. This analysis shows that obtainedlimitations are much higher than the parameters received by analysis of the realtraffic.The main part of results presented in this dissertation have been published

in conference proceedings. In the paper [43] envelope processes of the FARIMAmodels have been presented, the burstiness curve of an FGN model has beendescribed in [42] and an influence of SRD and LRD structures on the drop prob-ability (using an envelope process) has been addressed in [44].The original results presented in the dissertation are:

• finding envelope processes for three types of accumulated FARIMA pro-cesses; these processes are:

– FARIMA(0,d,0),

– FARIMA(1,d,0) called in this dissertation FAR(1,d),

– FARIMA(0,d,1) called in this dissertation FMA(d,1),

• derivation of the burstiness curve given a traffic envelope process,

• simulation validation of burstiness curves obtained for the FGN and FARIMAmodels,

As a future research the author would like to compare the obtained resultsagainst real network measurements. On the basis of such a comparison a heuristiccorrection of the obtained equations could be proposed. The final equation (withthe heuristic correction) could be used as a tool for the design and managementof video servers or other traffic sources where a single user is limited by a leakybucket algorithm.Another promising research direction is to combine the envelope analysis with

such tools like the effective bandwidth [13, 50] and the network calculus [53]. Itis envisioned that these powerful techniques being integrated in a one tool willreduce greatly the observed overestimation.

Appendix A

In this appendix proof of equation (4.30) is presented.In order to prove equation (4.30) the simplification of the sum

∑n−1i=0 (n −

i) Γ(d+i+k)Γ(1−d+i+k) given by

Sn =

n−1∑

i=0

(n− i)Γ(d+ i+ k)

Γ(1 − d+ i+ k)=

=1

2d(1 + 2d)· Γ(1 + d+ k + n)

Γ(−d+ k + n)− n(1 + 2d) + (d+ k)

2d(1 + 2d)· Γ(d+ k)

Γ(−d+ k)

(A.1)

has to be proved.Note that

Sn+1 − Sn =

n∑

i=0

Γ(d+ i+ k)

Γ(1 − d+ i+ k)(A.2)

Therefore, before equation (4.30) is proved the simplification of sum (A.2)have to be found. We start from proving that

S′

n =

n∑

i=0

Γ(d+ i+ k)

Γ(1 − d+ i+ k)=

Γ(1 + d+ k + n)

2dΓ(1 − d+ k + n)− Γ(d+ k)

2dΓ(−d+ k)(A.3)

Equations (A.1) and (A.3) are proved by the mathematical induction. Westart from equation (A.3) and according to mathematical induction steps letprove that equation (A.3) is true for n = 0. We have to prove that

Γ(d+ k)

Γ(1 − d+ k)=

Γ(1 + d+ k)

2dΓ(1 − d+ k)− Γ(d+ k)

2dΓ(−d+ k)(A.4)

since [52]Γ(1 + x) = xΓ(x) (A.5)

84 Appendix A

we obtainΓ(d+ k)

(k − d)Γ(−d+ k)=

[(d+ k) − (k − d)]Γ(d + k)

2d(k − d)Γ(−d+ k)(A.6)

Therefore, the statement is true for n = 0.Assume that statement is true for n = m, i.e.

S′

m =

m∑

i=0

Γ(d+ i+ k)

Γ(1 − d+ i+ k)=

Γ(1 + d+ k +m)

2dΓ(1 − d+ k +m)− Γ(d+ k)

2dΓ(−d+ k)(A.7)

and we ask if

S′

m+1 =

m+1∑

i=0

Γ(d+ i+ k)

Γ(1 − d+ i+ k)

?=

Γ(1 + d+ k +m+ 1)

2dΓ(1 − d+ k +m+ 1)− Γ(d+ k)

2dΓ(−d+ k)(A.8)

since

S′

m+1 = S′

m +Γ(d+m+ 1 + k)

Γ(1 − d+ i+m+ 1)(A.9)

and we assume that equation (A.7) is true it is enough if we prove that

Γ(1 + d+ k +m+ 1)

2dΓ(1 − d+ k +m+ 1)− Γ(d+ k)

2dΓ(−d+ k)=

=Γ(1 + d+ k +m)

2dΓ(1 − d+ k +m)− Γ(d+ k)

2dΓ(−d+ k)+

Γ(d+m+ 1 + k)

Γ(1 − d+ k +m+ 1)

(A.10)

since terms Γ(d+k)2dΓ(−d+k) are the same on both sides of equation (A.10) so they can

be reduced. Next, using equation (A.5) we obtain

(1 + d+ k +m)Γ(1 + d+ k +m)

2d(1 − d+ k +m)Γ(1 − d+ k +m)=

=Γ(1 + d+ k +m)

2dΓ(1 − d+ k +m)+

Γ(1 + d+ k +m)

(1 − d+ k +m)Γ(1 − d+ k +m)

(A.11)

the obtained equation is divided by the common factor Γ(1+d+k+m)2d(1−d+k+m)Γ(1−d+k+m)

and, finally, we obtain

1 + d+ k +m = 1 − d+ k +m+ 2d (A.12)

what concludes the proof. �

Now we can move to equation (A.1). Since we are proving it by the mathe-matical induction we start from n = 1. We have to show that

Γ(d+ k)

Γ(1 − d+ k)=

=Γ(1 + d+ k + 1)

2d(1 + 2d)Γ(−d+ k + 1)− (1 + 2d)Γ(d+ k) + Γ(1 + d+ k)

2d(1 + 2d)Γ(−d+ k)

(A.13)

85

again using equation (A.5) we obtain

Γ(d+ k)

(−d+ k)Γ(−d+ k)=

=(d+ k + 1)(d+ k)Γ(d+ k)

2d(1 + 2d)(−d+ k)Γ(−d+ k)− [(1 + 2d) + (d+ k)]Γ(d+ k)

2d(1 + 2d)Γ(−d+ k)

(A.14)

next, we multiply both sides by 2d(1+2d)(−d+k)Γ(−d+k)Γ(d+k) and we obtain

2d(1 + 2d) = (d+ k + 1)(d+ k) − (−d+ k)(1 + 3d+ k) (A.15)

what is always true.So we can move to the second step and we assume that

Sn =

n−1∑

i=0

(n− i)Γ(d+ i+ k)

Γ(1 − d+ i+ k)=

=Γ(1 + d+ k + n)

2d(1 + 2d)Γ(−d+ k + n)− n(1 + 2d)Γ(d+ k) + Γ(1 + d+ k)

2d(1 + 2d)Γ(−d+ k)

(A.16)

and we show that

Sn+1 =

n∑

i=0

(n− i+ 1)Γ(d+ i+ k)

Γ(1 − d+ i+ k)=

Γ(1 + d+ k + n+ 1)

2d(1 + 2d)Γ(−d+ k + n+ 1)−

− (n+ 1)(1 + 2d)Γ(d+ k) + Γ(1 + d+ k)

2d(1 + 2d)Γ(−d+ k)

(A.17)

Note that

Sn+1 = Sn + S′

n (A.18)

Since Sn+1 − Sn = S′

n we have to prove that

Γ(1 + d+ k + n+ 1)

2d(1 + 2d)Γ(−d+ k + n+ 1)− (n+ 1)(1 + 2d)Γ(d+ k) + Γ(1 + d+ k)

2d(1 + 2d)Γ(−d+ k)=

=Γ(1 + d+ k + n)

2d(1 + 2d)Γ(−d+ k + n)− n(1 + 2d)Γ(d+ k) + Γ(1 + d+ k)

2d(1 + 2d)Γ(−d+ k)+

+Γ(1 + d+ k + n)

2dΓ(1 − d+ k + n)− Γ(d+ k)

2dΓ(−d+ k)

(A.19)

we divide both sides by 2d and move all fractions with similar denominators on

86 Appendix A

one side

Γ(1 + d+ k + n+ 1)

(1 + 2d)Γ(−d+ k + n+ 1)− Γ(1 + d+ k + n)

(1 + 2d)Γ(−d+ k + n)−

− Γ(1 + d+ k + n)

Γ(1 − d+ k + n)=

(n+ 1)(1 + 2d)Γ(d+ k) + Γ(1 + d+ k)

(1 + 2d)Γ(−d+ k)−

− n(1 + 2d)Γ(d+ k) + Γ(1 + d+ k)

(1 + 2d)Γ(−d+ k)− Γ(d+ k)

Γ(−d+ k)

(A.20)

Now we focus only on the left side of equation (A.20) and by using equation (A.5)we have

(1 + d+ k + n)Γ(1 + d+ k + n)

(1 + 2d)(−d+ k + n)Γ(−d+ k + n)−

− Γ(1 + d+ k + n)

(1 + 2d)Γ(−d+ k + n)− Γ(1 + d+ k + n)

(−d+ k + n)Γ(−d+ k + n)=

=(1 + d+ k + n) − (−d+ k + n) − (1 + 2d)

(−d+ k + n)(1 + 2d)

Γ(1 + d+ k + n)

Γ(−d+ k + n)= 0

(A.21)

The right side of equation (A.20) is

[(n+ 1)(1 + 2d) + (d+ k)]Γ(d+ k)

(1 + 2d)Γ(−d+ k)−

− [n(1 + 2d) + (d+ k)]Γ(d+ k)

(1 + 2d)Γ(−d+ k)− Γ(d+ k)

Γ(−d+ k)=

=(n+ 1)(1 + 2d) − n(1 + 2d) − (1 + 2d)

(1 + 2d)

Γ(d+ k)

Γ(−d+ k)= 0

(A.22)

Since the left and right side of equation (A.20) equal 0 the proof has been com-pleted. �

Note that to prove equation (4.30) it is enough to show that the variance ofsum of variables with the FARIMA(0,d,0) distribution is

D2Zn = 2Γ(−1 − 2d)

(1

Γ2(−d) − (−1)n

Γ(−d− n)Γ(−d+ n)

)

(A.23)

On the base of equation (2.12) we can write

D2Zn = D2z

(

2

n∑

i=0

(n− i)ρi − n

)

(A.24)

Variance of the FARIMA(0,d,0) process has the following form [38]

D2z =(−2d)!

[(−d)!]2 (A.25)

87

and the autocorrelation function has the form [38]

ρi =(−d)!(i+ d− 1)!

(i− d)!(d − 1)!(A.26)

From equations (A.24), (A.25) and (A.26) we obtain

(−2d)!

[(−d)!]2

(

2

n∑

i=0

(n− i)(−d)!(i+ d− 1)!

(i− d)!(d− 1)!− n

)

(A.27)

Since [52]x! = Γ(x+ 1) (A.28)

for x 6= −1,−2, · · · we have

Γ(−2d+ 1)

Γ2(−d+ 1)

(

2Γ(−d+ 1)

Γ(d)

n∑

i=0

(n− i)Γ(i+ d)

Γ(i− d+ 1)− n

)

(A.29)

Note that∑n

i=0(n− i)Γ(i+d)

Γ(i−d+1) is a special case of equation (A.1) where k = 0

therefore, we can write

Γ(−2d+ 1)

Γ2(−d+ 1)

(

2Γ(−d+ 1)

Γ(d− 1 + 1)·

·(

Γ(1 + d+ n)

2d(1 + 2d)Γ(−d+ n)− n(1 + 2d)Γ(d) + Γ(1 + d)

2d(1 + 2d)Γ(−d)

)

− n

) (A.30)

by a simple algebraic transformation and by using equation (A.5) we obtain

Γ(−2d+ 1)

Γ(1 − d)·

·(

Γ(1 + d+ n)

d(1 + 2d)Γ(−d+ n)Γ(d)− n

dΓ(−d) − Γ(1 + d)

d(1 + 2d)Γ(−d)Γ(d)+

n

dΓ(−d)

) (A.31)

As [52]

Γ(x)Γ(1 − x) =π

sin(πx)(A.32)

we have

Γ(−2d+1)

(π

sin(π(−d−n))

d(1 + 2d)Γ(−d+ n)Γ(−d− n) πsin(πd)

+1

d(1 + 2d)Γ2(−d)

)

(A.33)

88 Appendix A

Note that sin is an odd function and sin(a+nπ) = (−1)n sin(a) for integer n,therefore, we have

Γ(−2d+ 1)

d(2d+ 1)

( −(−1)n

Γ(−d+ n)Γ(−d− n)+

1

Γ2(−d)

)

(A.34)

As Γ(−2d+1) = −2dΓ(−2d) = 2d(2d+1)Γ(−2d−1) equation (A.23) is identicalto equation (A.34) what concludes deriving equation (4.30).

Appendix B

This appendix presents the derivation of equation (4.31). Again, it is needed toderive a similar sum as in Appendix A, denoted by

Sn =

n−1∑

i=0

(n− i)Γ(d− i+ k)

Γ(1 − d− i+ k)=

=1

2d(1 + 2d)· Γ(1 + d+ k − n)

Γ(−d+ k − n)+n(1 + 2d) − (k − d)

2d(1 + 2d)· Γ(1 + d+ k)

Γ(1 − d+ k)

(B.1)

As the proof of equation (B.1) is similar to the proof given in Appendix A itis not presented.Note that it is enough to prove that the sum of random variables with the

FAR(1,d) distribution has the following variance

D2Zn = c1·

·(

c2 + (−1)n 1 −2 F1(1, 1 + d− n;−d− n;φ) −2 F1(1, 1 + d+ n;−d+ n;φ)

Γ(−d− n)Γ(−d+ n)

)(B.2)

where

c1 =2Γ(−1 − 2d)

1 − φ2(B.3)

and

c2 =22F1(1, 1 + d;−d;φ) − 1

Γ2(−d) (B.4)

On the base of equation (A.24) we can write

D2Zn = 2

n∑

i=0

(n− i)γi − nγ0 (B.5)

90 Appendix B

On the base of [38] γi for the FAR(1,d) process has the following form

γi =(−1)i(−2d)!

(i− d)!(−i− d)!·

· 2F1(1, d+ i; 1 − d+ i;φ) +2 F1(1, d− i; 1 − d− i;φ) − 1

1 − φ2

(B.6)

As equation (B.6) uses hypergeometric functions it is important to recall itsdefinition. The hypergeometric function can be described as [87]

2F1(a, b; c; z) =

∞∑

k

(a)k(b)k

(c)k

zk

k!(B.7)

where (a)k is Pochhammer Symbol denoted by [88]

(a)k =Γ(a+ k)

Γ(a)(B.8)

Therefore the variance can be written as

D2Zn =

2

n∑

i=0

(n− i)(−1)i(−2d)!

(i− d)!(−i− d)!

(2F1(1, d+ i; 1 − d+ i;φ)

1 − φ2+

+2F1(1, d− i; 1 − d− i;φ) − 1

1 − φ2

)

−

− n(−2d)!

(−d)!(−d)!22F1(1, d; 1 − d;φ) − 1

1 − φ2

(B.9)

First note that c1 given by equation (4.32) can be extracted from bracketsfrom equation (B.9) as (−2d)! = 2d(1 + 2d)Γ(−1 − 2d). Next we will focus onparticular terms of equation (B.9). Let start from

2

n∑

i=0

(n− i)(−1)id(1 + 2d)

(i− d)!(−i− d)!2F1(1, d+ i; 1 − d+ i;φ) (B.10)

Using equations (B.7) and (B.8) we get

2n∑

i=0

(n− i)(−1)id(1 + 2d)

(i− d)!(−i− d)!

∞∑

k

Γ(k+1)Γ(1)

Γ(d+i+k)Γ(d+i)

Γ(1−d+i+k)Γ(1−d+i)

φk

Γ(k + 1)(B.11)

91

As the second sum is absolutely summable series for |φ| < 1 we can changethe order of summation and write

2∞∑

k

n∑

i=0

(n− i)(−1)id(1 + 2d)

Γ(1 + i− d)Γ(1 − i− d)

Γ(d+ i+ k)Γ(1 − d+ i)

Γ(d+ i)Γ(1 − d+ i+ k)φk (B.12)

From equation (A.32) we have Γ(1 − i− d)Γ(d + i) = πsin(π(i+d)) and we can

write

2

∞∑

k

φkn∑

i=0

(n− i)(−1)i sin(π(d+ i))d(1 + 2d)

π

Γ(d+ i+ k)

Γ(1 − d+ i+ k)(B.13)

as sin(a+ iπ) = (−1)i sin(a) for integer i we have

2

∞∑

k

sin(dπ)d(1 + 2d)

πφk

n∑

i=0

(n− i)Γ(d+ i+ k)

Γ(1 − d+ i+ k)(B.14)

As the sum in equation (B.14) is the same as the sum in equation (A.1) weobtain

sin(dπ)

π

∞∑

k

(Γ(1 + d+ k + n)

Γ(−d+ k + n)− [n(1 + 2d) + (d+ k)]

Γ(d+ k)

Γ(−d+ k)

)

φk (B.15)

Let process the next term of equation (B.9)

2n∑

i=0

(n− i)(−1)id(1 + 2d)

(i− d)!(−i− d)!2F1(1, d− i; 1 − d− i;φ) (B.16)

By a similar transformation we obtain

2

∞∑

k

sin(dπ)d(1 + 2d)

πφk

n∑

i=0

(n− i)Γ(d− i+ k)

Γ(1 − d− i+ k)(B.17)

and using equation (B.1) we have

sin(dπ)

π

∞∑

k

(Γ(1 + d+ k − n)

Γ(−d+ k − n)+ [n(1 + 2d) − (k − d)]

Γ(1 + d+ k)

Γ(1 − d+ k)

)

φk (B.18)

The third term in equation (B.9) has the following form

−2nd(1 + 2d)

Γ2(1 − d)2F1(1, d; 1 − d;φ) (B.19)

92 Appendix B

Again, by using the hypergeometric function definition (B.7) together withequations (A.5) and (A.32) we obtain

−2nd(1 + 2d) sin(dπ)

π

∞∑

k=0

Γ(d+ k)

Γ(1 − d+ k)φk (B.20)

The last term of equation (B.9) with (−2d)! has the following form

−2

n∑

i=0

(n− i)(−1)i(−2d)!

(i− d)!(−i− d)!+ n

(−2d)!

(−d)!(−d)! (B.21)

As equation (B.21) equals −D2Zn where Zn has the FARIMA(0,d,0) distri-bution we get

−2Γ(−1− 2d)

(1

Γ2(−d) − (−1)n

Γ(−d− n)Γ(−d+ n)

)

(B.22)

Finally, we can rewrite equation (B.9) linking equations (B.15), (B.18), (B.20),and (B.22) in the following form

D2Zn =

= c1

(sin(dπ)

π

∞∑

k

(Γ(1 + d+ k + n)

Γ(−d+ k + n)−

− [n(1 + 2d) + (d+ k)]Γ(d+ k)

Γ(−d+ k)+

Γ(1 + d+ k − n)

Γ(−d+ k − n)+

+ [n(1 + 2d) − (k − d)]Γ(1 + d+ k)

Γ(1 − d+ k)−

− 2nd(1 + 2d)Γ(d+ k)

Γ(1 − d+ k)

)

φk − 1

Γ2(−d)+

+(−1)n

Γ(−d− n)Γ(−d+ n)

)

(B.23)

Let focus on the term

sin(dπ)

π

∞∑

k

Γ(1 + d+ k + n)

Γ(−d+ k + n)φk (B.24)

and rewrite it in the form

sin(dπ)Γ(1 + d+ n)

πΓ(−d+ n)

∞∑

k

Γ(1 + d+ k + n)Γ(−d+ n)

Γ(−d+ k + n)Γ(1 + d+ n)φk (B.25)

93

Note that the infinite sum in equation (B.25) is a hypergeometric function(see equations (B.7) and (B.8)), therefore, we obtain

(−1)n sin(dπ + nπ)Γ(1 + d+ n)

πΓ(−d+ n)2F1(1, 1 + d+ n;−d+ n;φ) (B.26)

Equation (A.32) yields

(−1)n

Γ(−d− n)Γ(−d+ n)2F1(1, 1 + d+ n;−d+ n;φ) (B.27)

The same procedure can be used to the Γ(1+d+k−n)Γ(−d+k−n) term and we obtain

(−1)n

Γ(−d− n)Γ(−d+ n)2F1(1, 1 + d− n;−d− n;φ) (B.28)

Now let us consider the remaining terms of the infinite sum in equation (B.23)

sin(dπ)

π

∞∑

k

(

− [n(1 + 2d) + (d+ k)]+

+ [n(1 + 2d) − (k − d)]d+ k

k − d+

−2nd(1 + 2d)

−d+ k

) Γ(d+ k)

Γ(−d+ k)φk

(B.29)

By a simple algebraic operation and equation (A.5) equation (B.29) gets theform

−2 sin(dπ)

π

Γ(1 + d)

Γ(−d)

∞∑

k

Γ(1 + d+ k)Γ(−d)Γ(1 + d)Γ(−d+ k)

φk (B.30)

Again, the sum is a hypergeometric function and since Γ(−d)Γ(1 + d) =π

− sin(dπ) (see equation (A.32)) we obtain for equation (B.30)

22F1(1, 1 + d;−d;φ)

Γ2(−d) (B.31)

Finally, substituting terms of equation (B.23) by formulae (B.27), (B.28) and(B.31) we obtain relationship (??).

94 Appendix B

Appendix C

This appendix presents the derivation of equation (4.34).

Note that it is enough to prove that the sum of random variables with theFAR(d,1) distribution has the following variance

D2Zn = 2Γ(−1 − 2d)

(2θ + d(1 + θ)2

dΓ2(−d)+

(−1)n((n2 − d2)(θ − 1)2 − 2θd(1 + 2d))

Γ(1 − d − n)Γ(1 − d + n)

)

(C.1)

On the base of [38] γi for the FMA(d,1) process has the form

γi =(−1)i(−2d)!

(i− d)!(−i− d)!

(1 − θ)2i2 − (1 − d)[(1 − d)(1 + θ2) − 2θd]

i2 − (1 − d)2(C.2)

Let rewrite γi in a handier way

γi =(−1)i(−2d)!(1 + θ2)

Γ(i− d+ 1)Γ(−i− d+ 1)− (−1)i(−2d)!2θ(d(1 − d) − i2)

Γ(i− d+ 2)Γ(−i− d+ 2)(C.3)

According to equation (B.5) we have to prove that

D2Zn =

= 2n∑

i=0

(n− i)( (−1)i(−2d)!(1 + θ2)

Γ(i− d+ 1)Γ(−i− d+ 1)− (−1)i(−2d)!2θ(d(1 − d) − i2)

Γ(i− d+ 2)Γ(−i− d+ 2)

)

−

−n((−2d)!(1 + θ2)

Γ2(1 − d)− (−2d)!2θd(1 − d)

Γ2(2 − d)

)

(C.4)

96 Appendix C

Note that

D2Zn+1 −D2Zn =

= 2n∑

i=0

( (−1)i(−2d)!(1 + θ2)

Γ(i− d+ 1)Γ(−i− d+ 1)− (−1)i(−2d)!2θ(d(1 − d) − i2)

Γ(i− d+ 2)Γ(−i− d+ 2)

)

︸︷︷︸

Sn

−

−( ((−2d)!(1 + θ2)

Γ(−d+ 1)Γ(−d+ 1)− (−2d)!2θd(1 − d)

Γ(−d+ 2)Γ(−d+ 2)

)

(C.5)

First we prove by mathematics induction that the sum appearing in equation(C.5) equals

Sn =(−2d)!

2

( 1 + θ2

Γ2(1 − d)− (−1)n(1 + θ2)

dΓ(−d− n)Γ(1 − d+ n)+

+2θ

Γ(2 − d)Γ(−d) − (−1)n2θ(n2 + n+ d2)

dΓ(1 − d− n)Γ(2 − d+ n)

) (C.6)

We start form n = 0, therefore, we have to prove that

(−2d)!(1 + θ2)

Γ2(−d+ 1)− (−2d)!2θd(1 − d)

Γ2(−d+ 2)=

(−2d)!

2

( 1 + θ2

Γ2(1 − d)−

− 1 + θ2

dΓ(−d)Γ(1 − d)+

2θ

Γ(2 − d)Γ(−d) − 2θd

Γ(1 − d)Γ(2 − d)

) (C.7)

Let divide both sides by (−2d)! and move all terms with the 1 + θ2 factorto the left side, all terms with 2θ to the right side and reduce to a commondenominator factors where Γ(·) functions have similar arguments. Therefore, weobtain

1 + θ2

Γ2(−d+ 1)− 1 + θ2

2Γ2(1 − d)− 1 + θ2

2Γ2(1 − d)=

=2θd(1 − d)

Γ2(−d+ 2)− 2θd(1 − d)

2Γ2(2 − d)− 2θd(1 − d)

2Γ2(2 − d)

(C.8)

what completes the proof for n = 0Next we assume that Sn is given by equation (C.6) and based on this assump-

tion we prove the form of Sn+1. Since

Sn+1 − Sn =(−1)n+1(−2d)!(1 + θ2)

Γ(n− d+ 2)Γ(−n− d)−

− (−1)n+1(−2d)!2θ(d(1 − d) − (n+ 1)2)

Γ(n− d+ 3)Γ(−n− d+ 1)

(C.9)

97

our task is to prove that

(−2d)!

2

( 1 + θ2

Γ2(1 − d)− (−1)n(1 + θ2)

dΓ(−d− n)Γ(1 − d+ n)+

2θ

Γ(2 − d)Γ(−d)−

− (−1)n2θ(n2 + n+ d2)

dΓ(1 − d− n)Γ(2 − d+ n)

)

+(−1)n+1(−2d)!(1 + θ2)

Γ(n− d+ 2)Γ(−n− d)−

− (−1)n+1(−2d)!2θ(d(1 − d) − (n+ 1)2)

Γ(n− d+ 3)Γ(−n− d+ 1)=

(−2d)!

2

( 1 + θ2

Γ2(1 − d)−

− (−1)n+1(1 + θ2)

dΓ(−d− n− 1)Γ(2 − d+ n)+

2θ

Γ(2 − d)Γ(−d)−

− (−1)n+12θ((n+ 1)2 + n+ 1 + d2)

dΓ(−d− n)Γ(3 − d+ n)

)

(C.10)

Again, we move all terms with the 1 + θ2 factor to the left side and termswith the 2θ factor to the right side. We also divide the both sides by the (−2d)!factor and reduce the same fractions. We have

− (−1)n(1 + θ2)

2dΓ(−d− n)Γ(1 − d+ n)+

(−1)n+1(1 + θ2)

Γ(n− d+ 2)Γ(−n− d)+

+(−1)n+1(1 + θ2)

2dΓ(−d− n− 1)Γ(2 − d+ n)=

(−1)n2θ(n2 + n+ d2)

2dΓ(1 − d− n)Γ(2 − d+ n)+

+(−1)n+12θ(d(1 − d) − (n+ 1)2)

Γ(n− d+ 3)Γ(−n− d+ 1)−

− (−1)n+12θ((n+ 1)2 + n+ 1 + d2)

2dΓ(−d− n)Γ(3 − d+ n)

(C.11)

Note that to prove equality (C.11) we have to prove that the both sides ofequation (C.11) equal 0. Let us consider the left and the right sides separatelyand begin with the left one. Without loosing generality we can assume that n isan even number and, therefore, remove the (−1)n terms. Therefore, we have

(1 + θ2)(

− 1 − d+ n

2dΓ(−d− n)Γ(2 − d+ n)− 2d

2dΓ(n− d+ 2)Γ(−n− d)+

− −d− n− 1

2dΓ(−d− n)Γ(2 − d+ n)

)

=(1 + θ2)

2dΓ(n− d+ 2)Γ(−n− d)(

− 1 + d− n− 2d+ d+ n+ 1)

= 0

(C.12)

98 Appendix C

and the right side has the following form

2θ( (n2 + n+ d2)

2dΓ(1 − d− n)Γ(2 − d+ n)− (d(1 − d) − (n+ 1)2)

Γ(n− d+ 3)Γ(−n− d+ 1)+

+((n+ 1)2 + n+ 1 + d2)

2dΓ(−d− n)Γ(3 − d+ n)

)

= 2θ( (n2 + n+ d2)(2 − d+ n)

2dΓ(1 − d− n)Γ(3 − d+ n)−

− (d(1 − d) − (n+ 1)2)2d

2dΓ(n− d+ 3)Γ(−n− d+ 1)+

(−d− n)((n+ 1)2 + n+ 1 + d2)

2dΓ(1 − d− n)Γ(3 − d+ n)

)

=

=2θ

2dΓ(1 − d− n)Γ(3 − d+ n)

(

(n2 + n+ d2)(2 − d+ n)−

− (d(1 − d) − (n+ 1)2)2d+ (−d− n)((n+ 1)2 + n+ 1 + d2))

= 0

(C.13)

that completes the proof. �

Equation (C.1) is proved by the mathematical induction as well but beforethe proof is presented equation (C.1) is rewritten in a more convenient form

D2Zn =(−2d)!

d(2d+ 1)

( 1 + θ2

Γ2(−d) +2θ(d+ 1)

dΓ2(−d) −

− (−1)n(1 + θ2)

Γ(−d− n)Γ(−d+ n)− (−1)n2θ(n2 + d+ d2)

Γ(1 − d− n)Γ(1 − d+ n)

) (C.14)

We start from showing that equation (C.14) is true for n = 1 so we have toprove that

(−2d)!

d(2d+ 1)

( 1 + θ2

Γ2(−d) +2θ(d+ 1)

dΓ2(−d) +1 + θ2

Γ(−d− 1)Γ(−d+ 1)

+2θ(1 + d+ d2)

Γ(−d)Γ(2 − d)

)

=(−2d)!(1 + θ2)

Γ2(1 − d)− (−2d)!2θd(1 − d)

Γ2(2 − d)

(C.15)

In the same manner as for previous proofs we carry over terms with the 1+θ2

factor to the left side and terms with 2θ factor to the right one.

1 + θ2

d(2d+ 1)Γ2(−d) +1 + θ2

(2d+ 1)dΓ(−d− 1)Γ(−d+ 1)− 1 + θ2

Γ2(−d+ 1)=

= −2θd(1 − d)

Γ2(2 − d)− 2θ(d+ 1)

d2(2d+ 1)Γ2(−d) − 2θ(1 + d+ d2)

(2d+ 1)dΓ(−d)Γ(2 − d)

(C.16)

The left side equals

1 + θ2

(2d+ 1)Γ2(1 − d)

(

d+ (−1)(−d− 1) − (2d+ 1))

= 0 (C.17)

99

The right side equals

2θ(1 − d)

(2d+ 1)Γ2(2 − d)

(

(1 + d+ d2) − (d+ 1)(1 − d) − d(2d+ 1))

= 0 (C.18)

Now let us assume that equation (C.14) is valid for some n and we prove thatit is valid for n+ 1. Since

Zn+1 = Zn + 2Sn − γ0 (C.19)

where Sn is described by equation (C.6), we have to show that

(−2d)!

d(2d+ 1)

( 1 + θ2

Γ2(−d) +2θ(d+ 1)

dΓ2(−d) − (−1)n(1 + θ2)

Γ(−d− n)Γ(−d+ n)−

− (−1)n2θ(n2 + d+ d2)

Γ(1 − d− n)Γ(1 − d+ n)

)

+ (−2d)!( 1 + θ2

Γ2(1 − d)−

− (−1)n(1 + θ2)

dΓ(−d− n)Γ(1 − d+ n)+

2θ

Γ(2 − d)Γ(−d)−

− (−1)n2θ(n2 + n+ d2)

dΓ(1 − d− n)Γ(2 − d+ n)

)

−( (−2d)!(1 + θ2)

Γ2(1 − d)− (−2d)!2θd(1 − d)

Γ2(2 − d)

)

=

=(−2d)!

d(2d+ 1)

( 1 + θ2

Γ2(−d) +2θ(d+ 1)

dΓ2(−d) −

− (−1)n+1(1 + θ2)

Γ(−d− n− 1)Γ(−d+ n+ 1)− (−1)n+12θ((n+ 1)2 + d+ d2)

Γ(−d− n)Γ(2 − d+ n)

)

(C.20)

Let us divide both sides by (−2d)!, reduce the same terms and carry over allterms with the 1 + θ2 factor to the left side and the terms with 2θ to the rightone. Firstly we consider the left side only. Again, without loosing generality wecan assume that n is an even number.

− 1 + θ2

d(2d+ 1)Γ(−d− n)Γ(−d+ n)+

1 + θ2

Γ2(1 − d)

− 1 + θ2

dΓ(−d− n)Γ(1 − d+ n)− 1 + θ2

Γ2(1 − d)−

− 1 + θ2

d(2d+ 1)Γ(−d− n− 1)Γ(−d+ n+ 1)=

=(1 + θ2) (−(−d+ n) − (2d+ 1) − (−d− n− 1))

d(2d+ 1)Γ(−d− n)Γ(1 − d+ n)= 0

(C.21)

100 Appendix C

The right side is

− 2θ(n2 + d+ d2)

d(2d+ 1)Γ(1 − d− n)Γ(1 − d+ n)+

2θ

Γ(2 − d)Γ(−d)−

− 2θ(n2 + n+ d2)

dΓ(1 − d− n)Γ(2 − d+ n)+

2θd(1 − d)

Γ2(2 − d)−

− 2θ((n+ 1)2 + d+ d2)

d(2d+ 1)Γ(−d− n)Γ(2 − d+ n)=

=2θ[−(n2 + d+ d2)(1 − d+ n) − (n2 + n+ d2)(2d+ 1)]

d(2d+ 1)Γ(1 − d− n)Γ(2 − d+ n)+

+2θ[(d+ n)((n+ 1)2 + d+ d2)]

d(2d+ 1)Γ(1 − d− n)Γ(2 − d+ n)=

=2θ[−d− n− 2dn− n2(2 + d+ n) + (d+ n)(n+ 1)2]

d(2d+ 1)Γ(1 − d− n)Γ(2 − d+ n)= 0

(C.22)

what completes the proof. �

Appendix D

The following appendix concerns the unit computation for SS processes. Thefirst equation worth analysing is the SS definition given by

{Zat} d= aH{Zt} (D.1)

If Zt models a real process its values have some units. Let [u] denotes thisunit. Since both sides of equation (D.1) have to be described in the same unitsboth a and H parameters have to be unitless.Note that in equation (D.1) we have two units, the first is the process unit

Z·[u] and the second one is the time unit t[T]. Therefore, equation (D.1) withboth units shown explicitly is given by

{Zat[T]}[u] d= aH{Zt[T]}[u] (D.2)

Now if time expanded from 1[T] to t[T] then a = t[T]/1[T] and equation (D.1)gets a form

{Zt[T]} d=(t[T]1[T]

)H

{Z1[T]} (D.3)

with omitted process units.According to equation (D.3), it is possible to compute the Zt value showed in

Figure D.1 for two different time units. However, for time measured in seconds

the Zt[s] value is given by Z30[s] =(

30[s]1[s]

)H

Z1[s], for time measured in minutes

the Zt[min] value is given by Z0.5[min] =(

0.5[min]1[min]

)H

Z1[min]. Since Zt distributiondoes not depend on the time unit we get

(30[s]1[s]

)H

Z1[s]d=

(0.5[min]1[min]

)H

Z1[min] (D.4)

102 Appendix D

0 10 20 30 40 50 60 70 80−100

−50

0

50

100

150

200

time [s]

valu

es [u

]

Z1 for t in [s]Z1 for t in [min]

Zt = tHZ1

Figure D.1: The influence of t unit on Z1 distribution.

and the relation between Z1[s] and Z1[min]

Z1[s]d=

(1

60

)H

Z1[min] (D.5)

In a general case equation (D.5) for two time units [T1] and [T2] has thefollowing form

Z1[T1]d= TH

a Z1[T2] (D.6)

where Ta yields how many [T2] units is included in a single [T1] unit. For example,if [T1] unit is second and [T2] unit is minutes than Ta is 1/60 since 1/60 minutesgive 1 second.Equation (D.6) links the Zt distribution obtained for different time

units.Let us apply the above remarks to the Norros model of the accumulated traffic

presented in Chapter 5.An accumulated workload process for a time unit [T1] is given by

At[T1] = a[T1]t[T1] + σ[T1]

(t[T1]

1[T1]

)H

Z1 (D.7)

103

where a[T1] and σ[T1] denotes a and σ obtained for a particular time unit [T1] andZ1 is a normalised process. For a new time unit [T2] we have correspondingly

At′[T2] = a[T2]t′[T2] + σ[T2]

(t′[T2]

1[T2]

)H

Z1 (D.8)

where t and t′ describe the same time moment but given in different time units,i.e. tTa = t′, therefore, we have

At[T1] = At′[T2] (D.9)

From equation (D.9) we have two equations linking a and σ parameters ob-tained for different time units

a[T1]t[T1] = a[T2]t′[T2] ⇒ a[T1] = a[T2]Ta (D.10)

σ[T1]tH = σ[T2]t

′H ⇒ σ[T1] = σ[T2](Ta)H (D.11)

For example, if we know that a process has mean a = 10 kbit/s and standarddeviation σ = 20 kbit/s with the Hurst parameterH = 0.8 and we want to changethe time unit to [ms] and the process unit to [bit] the new mean and standarddeviation values are a = 10 bit/ms and σ ≈ 79.6 bit/ms.The FARIMA model is more difficult to analyse. Firstly, we have to note that

the FARIMA process includes an SRD that is dependent on a particular unit ofdiscrete time n. This problem is showed in Figure D.2.

ρ = a

ρ =?

n = 1 n = 2 n = 3 n = 4

m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8

Figure D.2: An example of two different divisions of the time scale for the discretetime process.

In the first case, the workload process has been measured with time precision∆t (Figure D.2). For this process the correlation structure has been computedand it is known that the correlation for a single time step ∆t (the correlationbetween a sample n = 1 and n = 2) is ρ = a, as denoted in Figure D.2. If anew time precision is used, for example ∆t/2, a new process is obtained. For

104 Appendix D

this new process the time unit is described in Figure D.2 by m. Note that thenew process is obtained not by data analysis but by a simple change of the timeunit, therefore, it is impossible to decide of the value of the correlation between asample m = 1 and m = 2. Therefore, for the FARIMA process we cannot simplychange the time scale without a new investigation of the considered data.However, for a changing time scale for longer periods where n and a new time

unit m values are large (more than 100) the approximation that the standarddeviation of the new process obtained for a new time scale is given by equation(D.11) is valid since the obtained results are similar. The results are similar sincethe autocorrelation function of the FARIMA process for a large n value behaveslike an autocorrelation function of the increments of the H − ss, si process [38].According to this approximation the results for large H values in Chapters 5 and6 have been obtained.

Bibliography

[1] Abry P., Flandrin P., Taqqu M., Veitch D.: Wavelets for the Analysis, Es-timation and Synthesis of Scaling Data, in Self-Similar Network Traffic andPerformance Evaluation, pp. 39-88, Wiley, New York 2000.

[2] Abry P., Veitch D.: Wavelet Analysis of Long-Range Dependence Traffic,IEEE Transactions on Information Theory, 44(1), pp. 2-15, January 1998.

[3] Aczel A. D.: Statystyka w Zarządzaniu, Wydawnictwo Naukowe PWN,Warszawa 2000.

[4] Adas A.: Traffic Models in Broadband Networks, IEEE CommunicationsMagazine, 35(7), pp. 82-89, July 1997.

[5] Addie R. G., Zukerman M., Neame T. D.: Broadband Traffic Modeling:Simple Solutions to Hard Problems, IEEE Communications Magazine, 36(8),pp. 88-95, August 1998.

[6] Ansari N., Liu H., Shi Y. Q., Zhao H.: On Modeling MPEG Video Traffics,IEEE Transaction on Broadcasting, 48(4), pp. 337-347, December 2002.

[7] Barford, P., Bestavro, A., Bradley, A., and Crovella, M. E.: Changes in WebClient Access Patterns: Characteristics and Caching Implications, WorldWide Web, Special Issue on Characterization and Performance Evaluation2, pp. 15-28, January 1999.

[8] Beran J.: Statistic for Long-Memory Processes, Chapman&Hall, New York1994.

[9] Box G.E.P, Jenkins G.: Time Series Analysis: Forecasting and Control,Holden-Day, San Francisco 1976.

[10] Bylina B.: The WZ Factorization and the Use of the BLAS Library in theNumerical Solution of Markov Chains, Proceedings of EuroNGI Workshop:

106 BIBLIOGRAPHY

New Trends in Modeling, Quantitive Methods and Measurements, JacekSkalmierski Computer Studio, pp. 135–144, Gliwice 2004.

[11] Cappe O., Moulines E., Pesquet J.-C., Petropulu A., Yang X.: Long-RangeDependence and Heavy-Tail Modeling for Teletrac Data, IEEE Signal Pro-cessing Magazine, 19(3), pp. 14-27, May 2002.

[12] Crovella M. E., Bestavros A.: Self-Similarity in World Wide Web Traf-fic: Evidence and Possible Causes, IEEE/ACMTransactions on Networking,5(6), pp. 835-846, December 1997.

[13] Chang C.-S., Thomas J.A.: Effective Bandwidth in High Speed Digital Net-works, IEEE Journal on Selected Areas in Communications, 13(6), pp. 1091-1100, August 1995.

[14] Chong S., Li S.-Q.: Probabilistic Burstiness Curve Based Connection Con-trol for Real-Time Multimedia Services in ATM Networks, IEEE Journal onSelected Areas in Communications, vol. 15(6), pp. 1072-1086, August 1997.

[15] Czachórski T.: Modele Kolejkowe w Ocenie Efektywności Sieci i SystemówKomputerowych, Wydawnictwo Pracowni Komputerowej Jacka Skalmier-skiego, Gliwice 1999.

[16] Czachórski T.: Forming Optical Packets, a Diffusion Approximation Model,12th Polish Teletraffic Symposium, pp. 17-26, Poznań, Poland 2005.

[17] Dąbrowski M., Burakowski W.: Assessment of Token Bucket Parametersby On-Line Traffic Measurements, 10th Polish Teletraffic Symposium, pp.321-334, Cracow, Poland 2003.

[18] Dang T.D., Molnar S.: On the Effects of non-Stationarity in Long-RangeDependent Tests, Periodica Polytechnica, Electrical Engineering, 43(4), pp.227-250, 1999.

[19] Dinda P., O’Hallaron D.: Host Load Prediction using Linear Models, ClusterComputing, 3(4), pp. 265-280, June 2000.

[20] Duffield N., O’Connell N.: Large Deviations and Overflow Probabilities forthe General Single-Server Queue, with Applications, Mathematical Proceed-ings of the Cambridge Philosophical Society, 118, pp. 363-374, 1995.

[21] Embrechts P., Maejima M.: Selfsimilar Processes, Princeton UniversityPress, New Jersey 2002.

BIBLIOGRAPHY 107

[22] Erlang A.K.: Solution of Some Problems in the Theory of Probabilities ofSignificance in Automatic Telephone Exchanges, The Post Office ElectricalEngineer’s Journal 10, pp. 189–197. 1917-18.

[23] Erramilli A., Narayan O., Willinger W.: Experimental Queueing Analysiswith Long-Range Dependent Packet Traffic, IEEE/ACM Transactions onNetworking, 4(2), pp. 209-223, April 1996.

[24] Erramilli A., Narayan O., Willinger W.: Fractal Queueing Models, in: Fron-tiers in Queuing, J. Dshalalow (Ed.), CRC Press, pp.245-269, Boca Raton1997.

[25] Erramilli A., Roughan M., Veitch D., Willinger W.: Self-Similar Traffic andNetwork Dynamics, Proceedings of the IEEE, 90(5), pp. 800-819, May 2002.

[26] Feller W.: An Introduction to Probability Theory and its Applications, Vol-ume II, Sons Inc., New York 1966.

[27] Fitzek F., Reisslein M.: MPEG-4 and H.263 Video Traces for Network Per-formance Evaluation IEEE Network, 15(6), pp. 40-54, November/December2001.

[28] Floyd S., Jacobson V.: Random Early Detection Gateways for CongestionAvoidance, IEEE/ACM Transactions on Networking, 1(4), pp. 397-413, Au-gust 1993.

[29] Fonseca N., Mayor G., Neto C.: On the Equivalent Bandwidth of Self-Similar Sources, ACM Transactions on Modeling and Computer Simulation(TOMACS), 10(2), pp. 104-124, April 2000.

[30] Fonseca N., Mayor G., Neto C.: Policing and Statistical Multiplexing ofSelf-Similar Sources, Institute of Computing, Brazil 1999.

[31] Gallardo J. R., Makrakis D. Orozco-Barbosa L.: Use of alpha-Stable Self-Similar Stochastic Processes for Modeling Traffic in Broadband Networks,Performance Evaluation, 40(1-3), pp. 71-98, March 2000.

[32] Garrett M., Willinger W.: Analysis, Modeling and Generation of Self SimilarVBR Video Traffic, in Proceedings SIGCOMM’94, pp. 269-280, August 1994.

[33] Guerrero M.L., Barbosa L.O., Makrakis D.: On the Use of ProbabilisticEnvelope Processes in Resource Allocation, IEEE Canadian Conference onElectrical & Computer Engineering (CCECE), pp. 993- 996 2(4-7), Montreal,Quebec May 2003.

108 BIBLIOGRAPHY

[34] Harmantzis F., Hatzinakos d.: Heavy Network Traffic Modeling and Simu-lation using Stable FARIMA Processes, in Proceedings of The 19th Interna-tional Teletraffic Congress (ITC19), Beijing, China, September 2005.

[35] Harmantzis F. C., Hatzinakos D., Lambadaris I.: Effective Bandwidths andTail Probabilities for Gaussian and Stable Self-Similar Traffic, in Proceed-ings of IEEE International Conference on Communications (ICC 2003),26(1), pp. 1515-1520, May 2003.

[36] Hlavacs H., Kotsis G., Steinkellner C.: Traffic Source Modeling, TechnicalReport No. TR-99101, Institute of Applied Computer Science and Informa-tion Systems, University of Vienna, Austria 1999.

[37] Hong S. H., Park R.-H., Lee C. B.: Hurst Parameter Estimation of LongRange Dependent VBR MPEG Video Traffic in ATM Network, Journal ofVisual Communication and Image Representation, 12(1), pp. 44-65, March2001.

[38] Hosking J. R. M.: Fractional Differencing, Biometrika, 68, pp. 165-176, 1981.

[39] Jagerman, D., Melamed B., Willinger W.: Stochastic Modelling of TrafficProcesses, in: Frontiers in Queuing, J. Dshalalow (Ed.), CRC Press, pp.271-320, Boca Raton 1997.

[40] Jakubowski J., Sztencel R.: Wstęp do Teorii Prawdopodobieństwa, SCRIPT,Warszawa 2000.

[41] Janowski L.: Analiza Możliwości Modelowania Ruchu Samopodobnego zaPomocą Łańcuchów Markowa, Praca Dyplomowa Magisterska (promotor Z.Papir) Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie,Kraków 2002.

[42] Janowski L., Papir Z.: Analysis of a Burstiness Curve for FBM Traffic andToken Bucket Shaping Algorithm, 8th International Conference on Telecom-munications (ConTEL), pp. 439-442, Zagreb, Croatia July 2005.

[43] Janowski L., Papir Z.: An Envelope Process for FARIMA Traffic Model, XIIPolskie Sympozjum Ruchu Telekomunikacyjnego (PSRT), pp. 27-35, Poznań,Polska September 2005.

[44] Janowski L., Papir Z.: The Influence of an SRD Structure of LRD Traffic ona Drop Probability, The 4th Polish-German Teletraffic Symposium (PGTS),pp. 141-150, Wrocław, Poland September 2006.

BIBLIOGRAPHY 109

[45] Jędruś S.: Modelowanie Natężenia Ruchu Pakietów w Sieciach Komput-erowych z Wykorzystaniem Miar Multifraktalnych, Rozprawa Doktorska,promotor T. Czachórski, Instytut Informatyki Teoretycznej i Stosowanej Pol-skiej Akademii Nauk, Gliwice 1999.

[46] Kalden R., Ibrahim S.: Searching for Self-Similarity in GPRS, The 5thAnnual Passive & Active Measurement Workshop, PAM 2004, France, April2004.

[47] Karasaridis A.: Broadband Network Traffic Modeling, Management and FastSimulation Based on α-stable Self Similar Process, PhD thesis, Electrical andComputer Engineering, University of Toronto, Canada 1999.

[48] Karagiannis T.: The SELFIS Tool, http://www.cs.ucr.edu/ tkarag/Selfis/Selfis.html, 2002.

[49] Karagiannis T., Faloutsos M.: SELFIS: A Tool For Self-Similarity and Long-Range Dependence Analysis, 1st Workshop on Fractals and Self-Similarityin Data Mining: Issues and Approaches (in KDD) Edmonton, Canada, July2002.

[50] Kelly F. P.: Notes on Effective Bandwidths, Stochastic Networks: Theoryand Applications, 4, pp. 141-168, 1996.

[51] King P. J. B., Larijani A. H.: Queueing Consequence of Self-similar Traffic,Fourteenth UK Computer and Telecommunications Performance Engineer-ing Workshop, pp. 182-186, University of Edinburgh 1998.

[52] Korpal E.: Funkcje Specjalne, UWND AGH, 2001.

[53] Kumar A., Manjunath D., Kuri J.: Communication Networking an Analyt-ical Approach, Morgan Kaufman Publishers, 2004.

[54] Leland W. E., Taqqu M. S., Willinger W., Wilson D. V.: On the Self-Similar Nature of Ethernet Traffic (Extended Version), IEEE Transactionson Networking, 2, pp. 1-15, February 1994.

[55] Lelarge M., Liu Z. and Xia H.: Asymptotic Tail Distribution of End-to-EndDelay in Networks of Queues with Self-Similar Cross Traffic, 12th INFORMSApplied Proabability Conference (INFORMS), pp. 2352- 2363, June 2004.

[56] López-Ardao J. C., Argibay-Losada P., Rodriguez-Rubio R. F.: On ModelingMPEG Video at the Frame Level Using Self-similar Processes, MIPS 2004,pp. 37-48, November 2004.

110 BIBLIOGRAPHY

[57] Low S. and Varaiya E.: A Simple Theory of Traffic and Resource Alloca-tion in ATM, in Proceedings of IEEE GLOBECOM’91, 3, pp. 1633-1637,December 1991.

[58] Marquardt T.: Fractionally Integrated ARMA Processes in Discrete andContinuous Time, M.Sc. thesis, Munich University of Technology (TUM),Mathematic Center, January 2004.

[59] Massoulie L., Simonian A.: Large Buffer Asymptotics for the Queue withFBM Input, Journal of Applied Probability, 36(3), pp. 894-906, 1999.

[60] Mayor G., Silvester J.: Time Scale Analysis of an ATM Queueing Systemwith Long-Range Dependent Traffic, Proceedings of IEEE INFOCOM, pp.205-212, 1997.

[61] Melamed B.: An Overview of TES Processes and Modeling Methodology, In:Donatiello L., Nelson R. (eds.): Performance Evaluation of Computer andCommunications Systems. Springer (LNCS), pp. 359-393, 1993.

[62] Molnar S., Dang T. D.: Pitfalls in Long Range Dependence Testing and Esti-mation, in Proceedings of IEEE GLOBECOM 2000, pp. 662-666, November2000.

[63] Molnar S., Dang T. D., Maricza I.: On the Queue Tail Asymptotics for Gen-eral Multifractal Traffic, Proceedings of the Second International IFIP-TC6Networking Conference on Networking Technologies, Services, and Proto-cols, pp. 105-116, May 2002.

[64] Muscariello L., Mellia M., Meo M., Ajmone Marsan M., Cigno R. Lo.:A Simple Markovian Approach to Model Internet Traffic at Edge Routers,COST279 Technical Report, talk in the COST279 meeting in Karlskrona(Sweden), May 2003.

[65] Narayan O.: Exact Asymptotic Queue Length Distribution for FractionalBrownian Traffic, Advances in Performance Analysis, 1(1), pp. 39-63, 1998.

[66] NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/, 2002.

[67] Nolan J. Information on Stable Distributions, http://academic2.american.edu/∼jpnolan/stable/stable.html, 2005.

[68] Norros I.: A Storage Model with Self-Similar Input, Queueing Systems, 16,pp. 387-396, 1994.

BIBLIOGRAPHY 111

[69] Papir Z.: Ruch Telekomunikacyjny i Przeciążenia Sieci Pakietowych, WKŁWarszawa, Warszawa 2001.

[70] Riedi R. H., Crouse M. S., Ribeiro V., Baraniuk R. G.: A MultifractalWavelet Model With Application to Network Traffic, IEEE Transactions onInformation Theory, 45(3), pp. 992-1018, April 1999.

[71] Robert S., Le Boudec J.-Y.: A Markov Modulated Process for Self-SimilarTraffic, in Internationales Begegnungs und Forschungszentrum fuer Infor-matki, Schloss Dagsthul, Saarbr ucken, Germany, pp. 1-14, September 1995.

[72] Robert S., Le Boudec J.-Y.: On a Markov Modulated Chain Exhibiting Self-Similarities over Finite Timescale, Performance Evaluation, Volume 27-28,pp. 159-173, October 1996.

[73] Rolls D.A.: Limit Theorems and Estimation for Structural and AggregateTeletraffic Models, Ph.D. thesis, Queen’s University, Kingston, Canada 2003.

[74] Roughan M., Veitch D.: Measuring Long-Range Dependence Under Chang-ing Traffic Conditions, in Proceedings of IEEE INFOCOM’99, pp. 1513-1521, 1999.

[75] Sadek N., Khotanzad A., Chen T.: ATM Dynamic Bandwidth AllocationUsing F-ARIMA Prediction Model, 12th International Conference on Com-puter Communications and Networks (ICCCN 2003), October 2003.

[76] Stoev S., Taqqu M., Park C., Michailidis G., Marron J.S.: LASS: a Toolfor the Local Analysis of Self-Similarity Computational Statistics and DataAnalysis, 50, pp. 2447-2471, 2006.

[77] Stallings W.: High-Speed Networks: TCP/IP and ATM Design Principles,Prentice Hall, New Jersey 1998.

[78] Taqqu M.S., Willinger W., Sherman R.: Proof of a Fundamental Result inSelf-Similar Traffic Modeling, Computer Communication Review, 27, pp.5-23, 1997.

[79] Tryfonas C., Varma A.: Efficient Algorithms for Computation of the Bursti-ness Curve of Video Sources, IEEE Transactions on Multimedia, 6(6), pp.862-875, December 2004.

[80] Uhlig S., Bonaventure O., Rapier C.: 3D-LD : a Graphical Wavelet-basedMethod for Analyzing Scaling Processes, In Proceedings of the 15th ITCSpecialist Seminar, July 2002.

112 BIBLIOGRAPHY

[81] Uhlig S.: Non-Stationarity and High-Order Scaling in TCP Flow Arrivals: aMethodological Analysis, Computer Communication Review, 34(2), pp. 9-24,2004.

[82] Valaee S., Gregoire J.-C.: An Estimator of Regulator Parameters in aStochastic Setting, IEEE/ACM Transactions on Networking, 13, pp. 1376-1389, December 2005.

[83] Valaee S., Gregoire J.: Resource Allocation for Video Streaming in WirelessEnvironment, in Proceedings of IEEE International Symposium on WirelessPersonal Multimedia Communications, pp. 1103-1107, 2002.

[84] Veitch D., Abry P.: A Statistical Test for the Time Constancy of ScalingExponents, IEEE Transactions on Signal Processing, 49(10), pp. 2325-2334,October 2001.

[85] Veitch D.: Code for the Estimation of Scaling Exponentshttp://www.cubinlab. ee.mu.oz.au/∼darryl/secondorder code.html, Novem-ber 2002.

[86] Weisstein E.: Gamma Function, From MathWorld–A Wolfram Web Re-source, http://mathworld.wolfram.com/GammaFunction.html

[87] Weisstein E.: Hypergeometric Function, From MathWorld–A Wolfram WebResource, http://mathworld.wolfram.com/HypergeometricFunction.html

[88] Weisstein E.: Pochhammer Symbol, From MathWorld–A Wolfram Web Re-source, http://mathworld.wolfram.com/PochhammerSymbol.html

[89] Xue F., Liu J., Shu Y., Zhang L.: Traffic Modeling Based on FARIMA Mod-els in Proceedings of IEEE Canadian Conference on Electrical and ComputerEngineering (CCECE99), pp. 162-167, May 1999.

[90] Zieliński T. P. Reprezentacje Sygnałów Niestacjonarnych Typu Czas-Częstotliwość i Czas-Skala, Wydawnictwo AGH, Kraków 1994.

[91] Zhou Y., Sethu H.: On the Effectiveness of Buffer Sharing in MultimediaServer Network Switches with Self-Similar Traffic, in Proceedings of IEEEInternational Conference on Communications (ICC 2002), 25(1), pp. 2533-2536, April 2002.

Acronyms and symbols

index

ARIMA Integrated AutoregRession and Moving Average processAR(p) AutoRegression process of order pBC Burstiness CurveCLT Central Limit TheoremDT Drop TailEBC Effective Burstiness CurveFAR(p,d) Fractional AutoRegression process of order pFARIMA Fractional ARIMAFBM Fractional Brownian MotionFGN Fractional Gaussian NoiseFLB Fractional Leaky BucketFMA(d,q) Fractional Moving Average process of order qH − ss H-self-similar process or self-similar process with Hurst

parameter HH − ss, si H-self-similar process with stationary incrementsiid independent identically distributedLB Leaky BucketLFSM Linear Fractional Stable MotionLog-FSM Log-Fractional Stable MotionLRD Long Range DependenceMA(q) Moving Average process of order qMMPP Markov Modulated Poisson ProcessPBC Probability Burstiness CurveQoS Quality of ServiceSRD Short Range DependenceSS Self-SimilarVoD Video on Demand

114 Acronyms and symbols index

In the dissertation, generally, an accumulated process is described by a capitalletter, for example Zn, and the corresponding increment process by a lowercaseletter, zi is the increment process of the Zn process.

d= equality in distributionα α-stable distribution parameterβ autocorrelation function decay rateΓ(·) Gamma function∆t time interval∇ (1 −B)ǫ exceed probabilityη index of dispersionθ moving average parameterΘ(B) moving average polynomialπi weights of invertible representation of FARIMA and

ARMA processρ(k) autocorrelation function computed for lag kρ utilisation (ration between workload process and service

rate)σ scale parameter of variability of A(t) workload processτb busy period lengthφ autoregressive parameterΦ(B) autoregressive polynomialψi weights of stationary representation of FARIMA and

ARMA processan Gaussian Noise process or an increment of At processA(t) envelope process of a continuous time process At

At workload processa mean of a workload processb bucket sizeB backshift operator or a buffer sizeBH(t) Fractional Brownian Motion processc link capacitycov(X,Y ) X and Y covarianced difference parameterD2X X varianceDmax maximum delaydH busy period lengthEt set of values that can be obtained by a stochastic process

for time tEX X excepted value

115

F (·) cumulative distribution function of a normalised Gaus-sian distribution

FX(·) cumulative distribution function of X distribution2F1(a, b; c; z) hypergeometric functionH Hurst parameter (measure of a self-similarity)k F−1(1− ǫ) where F (·) is a cumulative distribution of the

normalised Gaussian distributionP (A) probability of event Apb busy period probabilitypd drop probabilityps drop probability obtained by simulation studyQn queue length processQmax maximum of an envelope process of a queue processr token accumulate rateS set of time valuest∗ time for which Qmax is obtainedzi random process of increments of Zn computed from

equation zi = Zn+i − Zn

z(m)i accumulated process z in time scale m.Zn discrete time stochastic processZ(n) envelope process of a discrete time process Zn

Zn ∼ Exp( 1n ) Zn variable with exponential distribution with 1

n param-eter

Zn ∼ U(0, n) Zn variable with uniform distribution on an interval (0,n)Zt continuous time stochastic process

Index

B, see Backshift operatorG/G/1, 21H , 15H − ss, siincrement process, 10process, 10

α-stable distribution, 43H − ss, si process, 45probability computation, 46sum, 44symmetric, 44

β, 15∇d, 26generalisation, 29

Accumulated FARIMA processes, see FARIMAmodels

Accumulated FGN process, see FGNmodel

AR process, 26ARIMA process, 26autocorrelation, 27definition, 27invertible, 27stationary, 27

Autocorrelation function, 10, 11asymptotic behaviour, 11

Autocovariance function, 12

Backshift operator, 26Binomial series, 29Bucket size, 64Burstiness curve, 67

application, 68definition, 68FARIMA models, 73FGN model, 69overestimation analysis, 76

Busy period, 54, 58α-stable model, 58FARIMA model, 59FGN model, 58

Central Limit Theorem, 12Class of service, 21

Diffusion model, 21Drip Tail, 60

α-stable model, 62FARIMA model, 62FGN model, 61

Envelope processα-stable model, 45application, 41FAR(1,d) model, 48FARIMA(0,d,0) model, 47FBM process, 42FGN model, 42FMA(d,1) model, 48for a bounded stochastic process,38

for deterministic function, 37for stochastic processes, 39notation, 40

Exceed probability, 39

INDEX 117

FAR(1,d)variance, 51

FARIMA model, 55FARIMA process, 29

σa, 34an distribution, 34algorithm, 34applications, 34autocorrelation, 30definition, 29distribution transformation, 34FAR(1,d), 31FARIMA(0,d,0), 30FARIMA(0,d,1), see FMA(d,1)FARIMA(1,d,0), see FAR(1,d)FMA(d,1), 32generalisation, 35properties, 30

FARIMA(0,d,0)variance, 50

FBM, 15, 23definition, 23normalised, 24

FGN, 15, 23definition, 24

FGN model, 55FMA(d,1)variance, 51

Fractals, 7Fractional Brownian Motion, see FBMFractional Gaussian Noise, see FGN

Graphic test, 14

Hurst parameter, 7, 14interval, 11

Index of dispersion, 70

LB utilisation, 70Leaky Bucket, 54, 64LFSM, 45

Limit properties, 13Log-FSM, 45LRD, 11, 30, 46definition, 12

MA process, 26Markovchains, 21process, 21

MMPP, 21

Norros model, 42

ON/OFF, 21

QoS parameters, 53Queue, 17network, 21process, 53assumption, 56equation, 56

Self-similarity, 7definition, 7increment process, 9

SRD, 12, 30, 46Sumvariance, 11

Time scalechangecontinues, 7discrete time, 8example, 8engineering, 21

token accumulation rate, 64

Unit computation, 58, 101

Variancedecreasing, 14sum, 11

Video traffic, 34, 71

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