IP-traffic theory and performance

Reading without meditation is sterile; meditation without reading is liable to error; prayer without meditation is lukewarm; meditation without prayer is unfruitful; prayer, when it is fervent, wins contemplation, but to obtain contemplation without prayer would be rare, even miraculous. Bernhard de Clairvaux (12th century) NobodycandenythatIP-basedtra?chasinvadedourdailylifeinmanyways and no one can escape from its di?erent forms of appearance. However, most people are not aware of this fact. From the usage of mobile phones - either as simple telephone or for data transmissions - over the new form of telephone service Voice over IP (VoIP), up to the widely used Internet at the users own PC, in all instances the transmission of the information, encoded in a digital form, relies on the Internet Protocol (IP). So, we should take a brief glimpse at this protocol and its constant companions such as TCP and UDP, which have revolutionized the communication system over the past 20 years. The communication network has experienced a fundamental change, which was dominated up to end of the eighties of the last century by voice appli- tion.Butfromthemiddleoftheninetieswehaveobservedadecisivemigration in the data transmission. If the devoted reader of this monograph reads the title 'IP tra?c theory and performance', she/he may ask, why do we have to be concerned with mod- ing IP tra?c, and why do we have to consider and get to know new concepts.

Author Notes

Professor Dr.- Ing. Christian Grimm has been working for more than ten years with measuring and modelling of data traffic in packet switched networks. In his PhD thesis he investigated complex methods for the traffic modelling in the World Wide Web. At present he is head of the division for research and development and new network services at the regional computing centre for lower saxony. Since 2003 he has an assistant professorship (Juniorprofessur) in computer networks at the faculty of electrical engineering and computer sciences at the university of Hannover.

Professor Dr. rer.nat. Georg Schlüchtermann finished his study in Mathematics in 1984. He habilitated 1994 at the Ludwig-Maximilians-Universität in functional analysis. Since 2001 he is apl. professor at the faculty for mathematics, computer sciences and statistics at the university of Munich. He is lecturing in the fields of traffic theory, mathematical modelling in mobile communication and finance mathematics.

1 Introduction to IP Traffic	p. 1
1.1 TCP/IP Architecture Model	p. 1
1.1.1 Physical Layer	p. 3
1.1.2 Data Link Layer	p. 4
1.1.3 Network Layer	p. 5
1.1.4 Transport Layer	p. 5
1.1.5 Application Layer	p. 8
1.2 Aspects of IP Modeling	p. 9
1.2.1 Levels of Modeling	p. 10
1.2.2 Traffic Relations	p. 13
1.2.3 Asymmetry in IP Traffic	p. 17
1.2.4 Temporal Behavior	p. 17
1.2.5 Network Topology	p. 18
1.3 Quality of Service	p. 19
1.3.1 Best Effort Traffic	p. 19
1.3.2 Time Sensitive Data Traffic	p. 20
1.3.3 Overprovisioning	p. 20
1.3.4 Prioritization	p. 21
1.4 Why Traditional Models Fail	p. 22
2 Classical Traffic Theory	p. 29
2.1 Introduction to Traffic Theory	p. 29
2.1.1 Basic Examples	p. 29
2.1.2 Basic Processes and Kendall Notation	p. 32
2.1.3 Basic Properties of Exponential Distributions	p. 33
2.2 Kolmogorov Equation	p. 34
2.2.1 State Probability	p. 36
2.2.2 Stationary State Equation	p. 37
2.3 Transition Processes	p. 38
2.4 Pure Markov Systems M/M/n	p. 40
2.4.1 Loss Systems M/M/n	p. 40
2.4.2 Queueing Systems M/M/n	p. 44
2.4.3 Application to Teletraffic	p. 52
2.5 Special Traffic Models	p. 58
2.5.1 Loss Systems M/M/[infinity]	p. 58
2.5.2 Queueing Systems of Engset	p. 58
2.5.3 Queueing Loss Systems	p. 59
2.6 Renewal Processes	p. 60
2.6.1 Definitions and Concepts	p. 60
2.6.2 Bounds for the Renewal Function	p. 65
2.6.3 Recurrence Time	p. 67
2.6.4 Asymptotic Behavior	p. 68
2.6.5 Stationary Renewal Processes	p. 71
2.6.6 Random Sum Processes	p. 71
2.7 General Poisson Arrival and Serving Systems M/G/n	p. 73
2.7.1 Markov Chains and Embedded Systems	p. 73
2.7.2 General Loss Systems M/G/n	p. 74
2.7.3 Queueing Systems M/G/n	p. 74
2.7.4 Heavy-Tail Serving Time Distribution	p. 81
2.7.5 Application of M/G/1 Models to IP Traffic	p. 95
2.7.6 Markov Serving Times Models GI/M/1	p. 102
2.8 General Serving Systems GI/G/n	p. 107
2.8.1 Loss Systems	p. 107
2.8.2 The Time-Discrete Queueing System GI/G/1	p. 109
2.8.3 GI/G/1 Time Discrete Queueing System with Limitation	p. 117
2.9 Network Models	p. 122
2.9.1 Jackson's Network	p. 122
2.9.2 Systems with Priorities	p. 130
2.9.3 Systems with Impatient Demands	p. 131
2.9.4 Conservation Laws Model	p. 133
2.9.5 Packet Loss and Velocity Functions on Transmission Lines	p. 134
2.9.6 Riemann Solvers	p. 141
2.9.7 Stochastic Velocities and Density Functions	p. 146
2.10 Matrix-Analytical Methods	p. 148
2.10.1 Phase Distribution	p. 148
2.10.2 Examples for Different Phase Distributions	p. 153
2.10.3 Markovian Arrival Processes	p. 156
2.10.4 Queueing Systems MAP/G/1	p. 161
2.10.5 Application to IP Traffic	p. 173
3 Mathematical Modeling of IP-based Traffic	p. 181
3.1 Scalefree Traffic Observation	p. 181
3.1.1 Motivation and Concept	p. 181
3.1.2 Self-Similarity	p. 183
3.2 Self-Similar Processes	p. 184
3.2.1 Definition and Basic Properties	p. 184
3.2.2 Fractional Brownian Motion	p. 190
3.2.3 [alpha]-stable Processes	p. 194
3.3 Long-Range Dependence	p. 202
3.3.1 Definition and Concepts	p. 203
3.3.2 Fractional Brownian Motion and Fractional Brownian Noise	p. 207
3.3.3 Farima Time Series	p. 211
3.3.4 Fractional Brownian Motion and IP Traffic - the Norros Approach	p. 218
3.4 Influence of Heavy-Tail Distributions on Long-Range Dependence	p. 226
3.4.1 General Central Limit Theorem	p. 226
3.4.2 Heavy-Tail Distributions in M/G/[infinity] Models	p. 233
3.4.3 Heavy-Tail Distributions in On-Off Models	p. 235
3.4.4 Aggregated Traffic	p. 240
3.5 Models for Time Sensitive Traffic	p. 245
3.5.1 Multiscale Fractional Brownian Motion	p. 245
3.5.2 Norros Models for Differentiating Traffic	p. 249
3.6 Fractional Levy Motion in IP-based Network Traffic	p. 259
3.6.1 Description of the Model	p. 259
3.6.2 Calibration of a Fractional Levy Motion Model	p. 260
3.7 Fractional Ornstein-Uhlenbeck Processes and Telecom Processes	p. 261
3.7.1 Description of the Model	p. 261
3.7.2 Fractional Ornstein-Uhlenbeck Gaussian Processes	p. 262
3.7.3 Telecom Processes	p. 263
3.7.4 Representations of Telecom Processes	p. 263
3.7.5 Application of Telecom Processes	p. 265
3.8 Multifractal Models and the Influence of Small Scales	p. 267
3.8.1 Multifractal Brownian Motion	p. 267
3.8.2 Wavelet-Based Multifractal Models	p. 270
3.8.3 Characteristics of Multifractal Models	p. 280
3.8.4 Multifractal Formalism	p. 292
3.8.5 Construction of Cascades	p. 296
3.8.6 Multifractals, Self-Similarity and Long-Range Dependence	p. 308
3.9 Summary of Models for IP Traffic	p. 316
4 Statistical Estimators	p. 321
4.1 Parameter Estimation	p. 321
4.1.1 Unbiased Estimators	p. 322
4.1.2 Linear Regression	p. 329
4.1.3 Estimation of the Heavy-Tail Exponent [alpha]	p. 335
4.1.4 Maximum Likelihood Method	p. 344
4.2 Estimators of Hurst Exponent in IP Traffic	p. 349
4.2.1 Absolute Value Method (AVM)	p. 349
4.2.2 Variance Method	p. 352
4.2.3 Variance of Residuals	p. 354
4.2.4 R/S Method	p. 356
4.2.5 Log Periodogram - Local and Global	p. 359
4.2.6 Maximum Likelihood and Whittle Estimator	p. 363
4.2.7 Wavelet Analysis	p. 368
4.2.8 Quadratic Variation	p. 379
4.2.9 Remarks on Estimators	p. 380
5 Performance of IP: Waiting Queues and Optimization	p. 383
5.1 Queueing of IP Traffic for Perturbation with Long-Range Dependence Processes	p. 383
5.1.1 Waiting Queues for Models with Fractional Brownian Motion	p. 384
5.1.2 Queueing in Multiscaling FBM	p. 392
5.1.3 Fractional Levy Motion and Queueing in IP Traffic Modeling	p. 395
5.1.4 Queueing Theory and Performance for Multifractal Brownian Motion	p. 405
5.2 Queueing in Multifractal Traffic	p. 411
5.2.1 Queueing in Multifractal Tree Models	p. 411
5.2.2 Queueing Formula	p. 417
5.3 Traffic Optimization	p. 423
5.3.1 Mixed Traffic	p. 423
5.3.2 Optimization of Network Flows	p. 424
5.3.3 Rate Control: Shadow Prices and Proportional Fairness	p. 436
5.3.4 Optimization for Stochastic Perturbation	p. 442
5.3.5 Optimization of Network Flows Using an Utility Approach	p. 449
References	p. 465
Index	p. 479

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