Investigation of the Properties of the HEAF Estimator Using Simulation Experiments and MPEG-encoded Video Traces Karim Mohammed Rezaul Algirdas Pakstas Robert Gilchrist Department of Computing, Department of CCTM Department of CCTM Communications Technology and London Metropolitan University, UK London Metropolitan University, UK Mathematics (CCTM) a.pakstas(@,londonmet. ac. uk r.gilchrist(@,londonmet. ac. uk London Metropolitan University, UK morekba 786(@,yahoo. co. uk Abstract- More than a decade ago it was discovered that some it with other estimators, two different types of simulation LAN traffic exhibits self-similar rather than Poisson behaviour studies were performed. The first one is using fractional and there is ongoing research towards finding and improving Gaussian noise (fGn) sequences generated by Dietrich- suitable estimators which may help to characterize various Newsam algorithm [8, 9], which generates exact self-similar types of network traffic. Such characterization can be poten- sequences. The second one is using a fractional tially applied for control purposes such as traffic shaping, load auegressive m ong one 1S usmg a II]. balancing, etc. The Hurst exponent is used to measure the in- autoregressive moving average (FARIMA) process [10, 11]. tensity of long-range dependence (LRD) in the network traffic. After that the same set of estimators is tested on the trace Despite having several existing estimators, LRD analysis is still files for MPEG-encoded musical videos and movies. impeded by the difficulty of actual identification of its intensity. The rest of the paper is organised as follows. Section II This paper continues work on estimating the Hurst exponent shows the relationship between the autocorrelation function from the autocorrelation function, a so-called HEAF estimator (ACF) and LRD. Section III introduces the HEAF estimator. introduced earlier by the authors. It also compares HEAF with a few existing estimators such as Wavelet, Higuchi, aggregated The results of simulation experiments are presented in sec- variance time (V/T) and Rescaled-range (R/S). The simulation tion IV and experiments on MPEG-encoded movie traces in studies show that HEAF can be used to capture the LRD in the Section V. The properties of the HEAF estimator are shown network traffic if true process is fGn and FARIMA. in section VI. II. RELATIONSHIP BETWEEN LRD AND ACF I. INTRODUCTION The stationary process X is said to be a long-range depend- Self-similar and long-range dependent (LRD) characteristics ent process if its ACF is non-summable [12] meaning that of internet traffic have attracted the attention of researchers 0 since 1994 when it was discovered that some aspects of ,Pk = LAN traffic exhibits self-similar rather than Poisson behav- k=-x iour [1]. Research towards finding and improving suitable The details of how ACF decays with k are of interest be- estimators which may help to characterise various types of cause the behaviour of the tail of ACF completely deter- network traffic has continued ever since. It is especially im- mines its summability. According to [1], Xis said to exhibit portant to understand the link between self-similarity and long-range dependence if long-range dependence of network traffic and performance (22H) of the networks because such characterization can be poten- Pk L(t)k , as (2.1) tially applied for control purposes such as traffic shaping, 1 load balancing, etc. where - < H < 1 and L(.) slowly varies at infinity, i.e., The Hurst exponent (or Hurst parameter, H) which more 2 than a half century ago was proposed for analysis of long- lim L(xt) = 1 for all x > 0 term storage capacity of reservoirs [2] is nowadays used to t-o L(t) measure the intensity of LRD in the network traffic. A num- ber of methods have been proposed to estimate the Hurst Equation (2.1) implies that the LRD is characterized by an parameter (H). Some of the most popular include the aggre- autocorrelation function that decays hyperbolically rather gated variance time (V/T) [3], Rescaled-range (R/S) [1, 2], than exponentially fast. Higuchi method [4], wavelet-based method [5, 6] and so on. In all these methods, H is calculated by taking the slope . HA:A'USEXO NTBATCRELIN from a log-log plot. FNTO'ETMTR This paper continues work on the new estimator introduced W eeitoueanwetmtrb xedn h p earlier which is named Hurst Exponent by Autocorrelation W ee1toueanwetmtrb xedn h p Functio (HAF [7] as wel as coprn it inasmlto proach of Kettani and Gubner [13]. As in [13], for a given experiment with the known methodIs (WSavelet, Higuchi, observed data X,:(i.e. Xl1,......... , Xv), the sample auto- aggregated variance time (V/T) and Rescaled-range (R/S)). correlation function can be calculated by the following HEAF estimates H by a process which is simple, quick and method: reliable. In order to investigate HEAF features and compare I1-4244-9708-8/06/$20.OO ©2006 IEEE. 276