An efficient, three-dimensional, anisotropic, fractional Brownian motion and truncated fractional Levy motion simulation algorithm based on successive random additions

2003 ◽  
Vol 29 (1) ◽  
pp. 15-25 ◽  
Author(s):  
Silong Lu ◽  
Fred J. Molz ◽  
Hui Hai Liu
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Three Dimensional ◽  
Simulation Algorithm ◽  
Motion Simulation ◽  
Lévy Motion ◽  
Fractional Lévy Motion

scholarly journals Is Network Traffic Appriximated by Stable Lévy Motion or Fractional Brownian Motion?

2002 ◽  
Vol 12 (1) ◽  
pp. 23-68 ◽  
Author(s):  
Thomas Mikosch ◽  
Sidney Resnick ◽  
Holger Rootzén ◽  
Alwin Stegeman
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Network Traffic ◽  
Lévy Motion

MULTISCALING TRANSPORT EQUATION ON FRACTALS

Fractals ◽  
1999 ◽  
Vol 07 (02) ◽  
pp. 105-111
Author(s):  
QIUHUA ZENG ◽  
HOUQIANG LI
Keyword(s):  
Brownian Motion ◽  
Diffusion Equation ◽  
Transport Equation ◽  
Fractional Brownian Motion ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Conservation Of Mass ◽  
Fractal Media ◽  
General Transport ◽  
Fractional Transport

The general transport equation of multiscaling disordered fractal media in three-dimensional case is derived from conservation of mass. Multiscaling fractional transport equation is obtained on the basis of discussing Brownian motion, fractional Brownian motion and standard diffusion equation of fractals, which is consistent with the result of literature.

Generate Breath Flow Having Fractal Signal Feature Using Weierstrass Function Combination

2011 ◽  
Vol 393-395 ◽  
pp. 796-799
Author(s):  
Meng Chao Li ◽  
Zhong Hai He
Keyword(s):  
Mechanical Ventilation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Medical Patient ◽  
Motion Simulation ◽  
Weierstrass Function ◽  
Patient Simulator ◽  
Function Combination ◽  
Fractal Signal ◽  
Self Similar

Fractal signal feature in breath flow is verified by many articles. So the generate fractal feature have two meanings, one to decrease damage to lung in mechanical ventilation because of natural similar, two to increase similarity in breath simulation used in medical patient simulator. The main feature of fractal signal is self-similar. Some algorithms have been proposed using fractional Brownian motion simulation. In this paper we use Weierstrass function combination to generate fractal signal. The method includes all fractal features and easy to realize in algorithm compared with fractional Brownian motion.

Extremal behavior of heavy-tailed ON-periods in a superposition of ON/OFF processes

2002 ◽  
Vol 34 (01) ◽  
pp. 179-204 ◽  
Author(s):  
Alwin Stegeman
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Network Traffic ◽  
High Speed ◽  
Empirical Studies ◽  
High Threshold ◽  
Data Traffic ◽  
Lévy Motion ◽  
Extremal Behavior ◽  
Many Sources

Empirical studies of data traffic in high-speed networks suggest that network traffic exhibits self-similarity and long-range dependence. Cumulative network traffic has been modeled using the so-called ON/OFF model. It was shown that cumulative network traffic can be approximated by either fractional Brownian motion or stable Lévy motion, depending on how many sources are active in the model. In this paper we consider exceedances of a high threshold by the sequence of lengths of ON-periods. If the cumulative network traffic converges to stable Lévy motion, the number of exceedances converges to a Poisson limit. The same holds in the fractional Brownian motion case, provided a very high threshold is used. Finally, we show that the number of exceedances obeys the central limit theorem.

scholarly journals Simulation of a strictly φ-sub-Gaussian generalized fractional Brownian motion

2021 ◽  
pp. 11-19
Author(s):  
O. I. Vasylyk ◽  
I. I. Lovytska
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Series Representation ◽  
Random Processes ◽  
Specific Class ◽  
Programming Environment ◽  
Motion Simulation ◽  
Actuarial Mathematics ◽  
R Programming ◽  
Simulation Parameters

In the paper, we consider the problem of simulation of a strictly φ-sub-Gaussian generalized fractional Brownian motion. Simulation of random processes and fields is used in many areas of natural and social sciences. A special place is occupied by methods of simulation of the Wiener process and fractional Brownian motion, as these processes are widely used in financial and actuarial mathematics, queueing theory etc. We study some specific class of processes of generalized fractional Brownian motion and derive conditions, under which the model based on a series representation approximates a strictly φ-sub-Gaussian generalized fractional Brownian motion with given reliability and accuracy in the space C([0; 1]) in the case, when φ(x) = (|x|^p)/p, |x| ≥ 1, p > 1. In order to obtain these results, we use some results from the theory of φ-sub-Gaussian random processes. Necessary simulation parameters are calculated and models of sample pathes of corresponding processes are constructed for various values of the Hurst parameter H and for given reliability and accuracy using the R programming environment.

Extremal behavior of heavy-tailed ON-periods in a superposition of ON/OFF processes

2002 ◽  
Vol 34 (1) ◽  
pp. 179-204 ◽  
Author(s):  
Alwin Stegeman
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Network Traffic ◽  
High Speed ◽  
Empirical Studies ◽  
High Threshold ◽  
Data Traffic ◽  
Lévy Motion ◽  
Extremal Behavior ◽  
Many Sources

Empirical studies of data traffic in high-speed networks suggest that network traffic exhibits self-similarity and long-range dependence. Cumulative network traffic has been modeled using the so-called ON/OFF model. It was shown that cumulative network traffic can be approximated by either fractional Brownian motion or stable Lévy motion, depending on how many sources are active in the model. In this paper we consider exceedances of a high threshold by the sequence of lengths of ON-periods. If the cumulative network traffic converges to stable Lévy motion, the number of exceedances converges to a Poisson limit. The same holds in the fractional Brownian motion case, provided a very high threshold is used. Finally, we show that the number of exceedances obeys the central limit theorem.

LABYRINTH CHAOS

2007 ◽  
Vol 17 (06) ◽  
pp. 2097-2108 ◽  
Author(s):  
J. C. SPROTT ◽  
KONSTANTINOS E. CHLOUVERAKIS
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Three Dimensional ◽  
Bifurcation Parameter ◽  
Cyclic Symmetry ◽  
Three Dimensional Flow ◽  
Attractor Dimension ◽  
Dimensional Flow ◽  
Hamiltonian Chaos ◽  
Limiting Value

A particularly simple and mathematically elegant example of chaos in a three-dimensional flow is examined in detail. It has the property of cyclic symmetry with respect to interchange of the three orthogonal axes, a single bifurcation parameter that governs the damping and the attractor dimension over most of the range 2 to 3 (as well as 0 and 1) and whose limiting value b = 0 gives Hamiltonian chaos, three-dimensional deterministic fractional Brownian motion, and an interesting symbolic dynamic.

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