## Fast, approximate synthesis of fractional gaussian noise for generating self-similar network traffic (1997)

Venue: | Computer Communication Review |

Citations: | 62 - 2 self |

### BibTeX

@ARTICLE{Paxson97fast,approximate,

author = {Vern Paxson},

title = {Fast, approximate synthesis of fractional gaussian noise for generating self-similar network traffic},

journal = {Computer Communication Review},

year = {1997},

volume = {27},

pages = {5--18}

}

### Years of Citing Articles

### OpenURL

### Abstract

Recent network traffic studies argue that network arrival processes are much more faithfully modeled using statistically self-similar processes instead of traditional Poisson processes [LTWW94, PF95]. One difficulty in dealing with selfsimilar models is how to efficiently synthesize traces (sample paths) corresponding to self-similar traffic. We present a fast Fourier transform method for synthesizing approximate self-similar sample paths for one type of self-similar process, Fractional Gaussian Noise, and assess its performance and validity. We find that the method is as fast or faster than existing methods and appears to generate close approximations to true self-similar sample paths. We also discuss issues in using such synthesized sample paths for simulating network traffic, and how an approximation used by our method can dramatically speed up evaluation of Whittle's estimator for H, the Hurst parameter giving the strength of long-range dependence present in a self-similar time series.

### Citations

1840 | On the self-similar nature of {Ethernet} traffic," presented at - Leland, Taqq, et al. - 1993 |

1513 | An Introduction to Probability Theory and its - FELLER - 1966 |

1476 | Wide area traffic: the failure of Poisson Modelling
- Paxson, Floyd
(Show Context)
Citation Context ...tically self-similar processes [LTWW94], which have much different theoretical properties than Poisson processes. A subsequent investigation suggests that the same holds for wide-area network traffic =-=[PF95]-=-. The strength of self-similar models is that they are able to incorporate long-range dependence, which informally means significant correlations across arbitrarily large time scales. For many network... |

735 | End-to-end internet packet dynamics
- Paxson
- 1999
(Show Context)
Citation Context ...cially large transfers, as they contribute the most to long-range dependence) is primarily shaped by congestion avoidance, then it is vital to include TCP effects when simulating networks. We show in =-=[P97]-=- that packet loss is not a rare event in the Internet, so we would expect that often TCP transfers are indeed operating in the congestion avoidance regime. Related to the previous point, if the goal o... |

623 | Self-similarity through high-variability: statistical analysis of Ethernets LAN traffic at the source level
- Willinger, Taqqu, et al.
- 1997
(Show Context)
Citation Context ...ThenSnis asymptotically (asnapproaches1) a self-similar process [LTWW94]. ofSn This method is particularly attractive because it matches empirical evidence of the behavior of Ethernet traffic sources =-=[WTSW97]-=-. The principle difficulty with using a simulation for synthesizing a self-similar process is that one must trade off speed of computation (lown) against the degree of agreement with a true self-simil... |

475 | Analysis, Modeling and Generation of Self-similar VBR Video Traffic
- Garrett, Willinger
- 1994
(Show Context)
Citation Context ...thors evaluated sample paths of only 800 points, and used a heuristic for assessing the quality of the generated FBM [SLN94]. A fifth method, due to Hoskings, is discussed by Garrett and Willinger in =-=[GW94]-=-. This algorithm generates sample paths from a fractional ARIMA process, which are asymptotically self-similar. Hoskings algorithm has the very attractive property of being exact, but its running time... |

305 | Experimental Queueing Analysis with Long Range Dependence Packet Traffic
- Erramilli, Narayan, et al.
- 1996
(Show Context)
Citation Context ... Computational Sciences Division of the United States Department of Energy under Contract No. DE-AC03-76SF00098. presence of long-range dependence can completely alter the tail of queue waiting times =-=[ENW96]-=-. The theory of self-similar stochastic processes is not nearly as well-developed as that for Poisson processes. But given the strong empirical evidence that self-similar models are much better than P... |

231 | Wavelet analysis of long-range dependent traffic
- Abry, Veitch
- 1998
(Show Context)
Citation Context ..., given the assumption that the power spectrum of the underlying process does indeed correspond tof(;H). 2Alternative estimation techniques based on wavelet decompositions are discussed in [KK93] and =-=[AV97]-=-. 34 The Fourier Transform method In this section we present a method, based on the Discrete Time Fourier Transform (DTFT), for synthesizing fractional Gaussian noise. The strategy behind our method ... |

193 | Empirically-Derived Analytic Models of Wide-Area TCP - Paxson - 1994 |

187 |
Wavelet analysis and synthesis of fractional Brownian motion
- FLANDRIN
- 1992
(Show Context)
Citation Context ...A fourth method involves computing wavelet coefficients corresponding to a wavelet transform of FBM. The coefficients are then used with an inverse wavelet transformation to yield sample paths of FBM =-=[F92]-=-. The method is only approximate because the wavelet coefficients are not independent, but it is difficult to capture their interdependence. The author of [F92] points out that the RMD method is essen... |

160 |
Traffic Modeling for Telecommunications Networks
- Frost, Melamed
- 1994
(Show Context)
Citation Context ...nt in a self-similar time series. 1 Introduction When modeling network traffic, packet arrivals are often assumed to be Poisson processes because such processes have attractive theoretical properties =-=[FM94]-=-. Recent work, however, argues convincingly that local-area network traffic is much better modeled using statistically self-similar processes [LTWW94], which have much different theoretical properties... |

110 |
Self-Similarity in HighSpeed Packet Traffic: Analysis and Modeling of Ethernet Traffic Measurements
- Willinger, Taqqu, et al.
- 1995
(Show Context)
Citation Context ...1For a slowly-varying functionL,limt!1L(tx)=L(x)=1for all x>0. Constants and logarithms are examples of slowly-varying functions. 2 for synthesizing sample paths for self-similar processes—see ofR(t) =-=[WTLW95]-=- for a more complete discussion and citations— but they have drawbacks: Consider an alternating renewal processR(t)in which the on and off periods have durations from a “heavytailed” (e.g., Pareto) di... |

104 | Estimators for long-range dependence: an empirical study - Taqqu, Teverovsky, et al. - 1995 |

100 |
Statistical methods for data with long-range dependence
- BERAN
- 1992
(Show Context)
Citation Context ...lient characteristics of these processes. In this paper we present a fast algorithm for generating approximate sample paths for a type of self-similar process known as fractional Gaussian noise (FGN) =-=[B92b]-=-. The algorithm is based on synthesizing sample paths that have the same power spectrum as FGN. These sample paths can then be used in simulations as traces of self-similar network traffic. The key to... |

73 |
Long-range dependence: a review. Statistics: An Appraisal
- Cox
- 1984
(Show Context)
Citation Context ...implementing our method. 12 Self-similar processes We begin with two definitions. A stationary process is longrange dependent (LRD) if its autocorrelation functionr(k) is nonsummable (i.e.,Pkr(k)=1) =-=[C84]-=-. Thus, the definition of long-range dependence applies only to infinite time series. The simplest models with long-range dependence are self-similar processes, which are characterized by hyperbolical... |

64 |
Contributions toward real-time services on packet switched networks
- Garrett
- 1993
(Show Context)
Citation Context ...tion, however, suggests an alternate way for converting the FGN sample path to a representation consistent with a physical process, namely the transformationyi=2xi. This transformation both preservesH=-=[G93]-=- (so the transformed process remains long-range dependent) and results in a physical arrival process that, as shown above, in some cases more closely matches measured (wide-area) network traffic.fyigd... |

58 |
Self-similar traffic generation: The random midpoint displacement algorithm and its properties
- Lau, Erramilli, et al.
- 1995
(Show Context)
Citation Context ...self-similar, so again one must trade off length of computation for degree of self-similarity. A third method of synthesizing a self-similar process is the “Random Midpoint Displacement” (RMD) method =-=[LEWW95]-=-, which works by progressively subdividing an interval over which to generate the sample path. At each division, a Gaussian displacement is used to determine the value of the sample path at the midpoi... |

42 | Long-Range Dependence: A Review," in Statistics: An Appraisal - Cox - 1984 |

37 |
Variable-bit-rate Video Traffic and Long Range Dependence
- Beran, Sherman, et al.
- 1995
(Show Context)
Citation Context ...rocessfXtgt=0;1;2;:::is asymptotically r(k)=1=2(k+1)2H�2k2H+(k�1)2H: self-similar if (1) for Hurst parameterHsatisfying1=2<H<1andL a slowly-varying function;1and the process is exactly selfsimilar if =-=[BSTW95]-=- [C84, p.59]: For any processfXtgt=0;1;2;:::we can consider an “aggregated” versionfX(m) ofX(m) tgconstructed by partitioningfXtg into non-overlapping blocks ofmsequential elements and constructing a ... |

33 |
Fractal estimation from noisy data via discrete fractional Gaussian noise (DFGN) and the Haar basis
- Kaplan, Kuo
- 1993
(Show Context)
Citation Context ...timator ofH, given the assumption that the power spectrum of the underlying process does indeed correspond tof(;H). 2Alternative estimation techniques based on wavelet decompositions are discussed in =-=[KK93]-=- and [AV97]. 34 The Fourier Transform method In this section we present a method, based on the Discrete Time Fourier Transform (DTFT), for synthesizing fractional Gaussian noise. The strategy behind ... |

22 |
A Goodness-of-Fit Test for Time Series with Long Range Dependence
- Beran
- 1992
(Show Context)
Citation Context ...ptably close to theHwe intended. An important point regarding Whittle's estimator bears repeating: it is not a test for whether a sample of a time series is consistent with long-range dependence (see =-=[B92a]-=- for such a test). Rather, it is an estimator ofH, given the assumption that the power spectrum of the underlying process does indeed correspond tof(;H). 2Alternative estimation techniques based on wa... |

19 |
Extending self-similarity for fractional Brownian motion
- Kaplan, Kuo
- 1994
(Show Context)
Citation Context ...be a method of synthesizing self-similar sample paths that consistently integrates the presence of LRD and SRD. One such method, based on the Haar wavelet transform, is discussed by Kaplan and Kuo in =-=[KK94]-=-. At the moment their method is somewhat limited due to difficulties in parameter estimation, but still appears promising. In general, we believe wavelet methods hold great promise for characterizing ... |

13 |
Accurate synthesis of fractional Brownian motion using wavelets
- Stoksik, Lane, et al.
- 1994
(Show Context)
Citation Context ...d. A later study claims a high degree of accuracy when using the method, but the authors evaluated sample paths of only 800 points, and used a heuristic for assessing the quality of the generated FBM =-=[SLN94]-=-. A fifth method, due to Hoskings, is discussed by Garrett and Willinger in [GW94]. This algorithm generates sample paths from a fractional ARIMA process, which are asymptotically self-similar. Hoskin... |

12 | Estimation, Testing and Prediction for Self-Similar and related - Beran - 1986 |

9 |
Estimators for Long-range Dependence: An
- Taqqu, Teverovsky
- 1995
(Show Context)
Citation Context ...ationship [C84], so this anomaly may simply be due to stronger long-range dependence (i.e., 6higher values ofH) requiring longer sample paths to exhibit their true dependence; and (2) the authors of =-=[TTW95]-=- found that estimatingHfrom a variance-time plot introduces a bias towards underestimation ofH, which may account for the discrepency. Because of this latter consideration, we marked the underestimate... |

7 |
Resolving time-series structure with a controlled wavelet transform
- Schiff
- 1992
(Show Context)
Citation Context ...trum without compromising the selfsimilarity property. a sequence of complex values such thatjzij=q^fiand the phase ofziis uniformly distributed between 0 and2. The random phase technique, taken from =-=[S92]-=-, preserves the power spectrum (and thus autocorrelation) corresponding to f^fig, but ensures that different sample paths generated using the method will be independent. It also makes the marginal dis... |

3 |
An accurate fractional Brownian motion generator, Physica A Statistical Mechanics and its Applications 208
- Rambaldi, Pinazza
- 1994
(Show Context)
Citation Context ...ery attractive property of being exact, but its running time isO(n2)for generatingnpoints, quite slow; the authors report that generating 171,000 points required 10 CPU hours. Finally, the authors of =-=[RP94]-=- discuss generating FBM using an approximation to the definition of FBM. Instead of computing for each new sample point the correlational contribution of all the previous sample points (which is one w... |

2 |
Estimation, Testing, and Prediction for Self-Similar and
- Beran
- 1986
(Show Context)
Citation Context ...3) are inA(;H)=2sin(H)�(2H+1)(1�cos) f(;H)=A(;H)jj�2H�1+B(;H) tricately tied to the estimated power spectrum of the process. For B(;H)=1Xj=1(2j+)�2H�1+(2j�)�2H�1 an FGN process, the power spectrum is =-=[B86]-=-: (4) , where: The main B(;H) difficulty with using Eqn. 4 to compute the power spectrum is the vexing infinite summation in the ex+ad0 ad1+bd1+ad2+bd2+ad3+bd3 3+bd0 3+ad0 pression forB(;H), for which... |