A Multifractal Wavelet Model for the Generation of
Long-Range Dependency Traffic Traces with Adjustable Parameters
Miguel Tuberquia-David 1, Fernando Vela-Vargas 1, Hans López-Chávez 1, Cesar Hernández 1*
1 Faculty of Engineering and Technology, Universidad Distrital Francisco José de Caldas, Bogotá DC,
Colombia. Email: lmtuberquiad@correo.udistrital.edu.co, fjvelav@correo.udistrital.edu.co, hilopezc@udistrital.edu.co, cahernandezs@udistrital.edu.co
* Corresponding author
Abstract—The available multifractal traffic finite-length time series to implement performance test of the
management, control and admission algorithms, and level of service about M/M/1 models for WAN/LAN communication systems are very few and their recollection through current mechanisms is very slow due to the amount of data that must be obtained. Hence, it is necessary to develop a tool which synthesizes traces with multifractal features and allows the stochastic parameters configuration as its average, Hurst parameter and, multifractal spectrum width. This article describes the development of a proposed algorithm to generate multifractal traffic finite-length time series with a Hurst parameter and the multifractal spectrum width, sampling and adjustable, called MultiFractal Hurst Spectrum Width (MFHSW). The MFHSW algorithm is based on the MultiFractal Hurst model (MFH) and on the Multifractal Wavelet Model (MWM), to construct the time series through a binomial multiplicative cascade. The main contribution of the MFHSW algorithm is to allow adjusting both the Hurst parameter and the multifractal spectrum width, the aforementioned is achieved by appropriately modifying the beta distributions that conform the binomial cascade. Consequently, the impact developed by the algorithm to the trace generation with multifractal features will be the improvement in the simulation and data network modeling. The MFHW algorithm behaves as an expert system when inferring to distribution of the beta coefficients present in the scales that make part of the binomial cascade starting from the stochastic parameters configured by the user, and obtaining the corresponding time series through an inference engine. To validate the algorithm effectiveness, a trace with the Hurst parameter sampling and the multifractal spectrum width similar to the presented in a network traffic time series are synthetized. The MFHSW happens to be a promising tool for the modeling of time series applicable to diverse fields as the traffic engineering, finances, biomedical signals, among other real traces with multifractal features.
Keywords- multifractal spectrum, hurst parameter, multifractal wavelet model, binomial cascade, network traffic.
1. INTRODUCTION
The communication networks have been studied for more than two decades. It has been shown that the inter arrival times of the users’ demand arrivals and the demand itself on the network presents a correlation that persists through diverse time scales. It has been shown that such events can be properly categorized by using self-similar models (Alzate, 2001; Rudolf H Riedi & Vehel,
1997; Taqqu, Teverovsky, & Willinger, 1996). Outstanding studies
as the ones conducted in the Bellcore laboratories in the 80’s have set a standard for the modeling and study of the traffic in communication networks with the purpose of predicting, controlling and generating a better service (W.E. Leland &
Wilson, 1991) . It has been shown that the use of the discrete
wavelet transform as a synthesis and estimation tool for the fractal traffic analysis has been computationally effective (P. Abry,
Flandrin, Taqqu, & Veitch, 2000; Alzate Monroy, 2002).
Additionally, it proved that the time series of input packets of the users’ demand in a network presents a behavior of stochastic self-similarity, main characteristic of multifractal signals (Chen, Cai, &
Li, 1997; Contreras, Ospina, & Alzate, 2006; Will E. Leland, Taqqu,
Willinger, & Wilson, 1994). The multifractal nature of network
traffic was equally validated in the different time scales in the network (Rudolf H Riedi & Vehel, 1997; Wang & Qiu, 2005; Y. Yu,
Song, Fu, & Song, 2013) just like in other areas such as: financial
time series (Resta, 2004; Thompson & Wilson, 2014), climate modeling (Das & Ghosh, 2015), biomedical signals (Lopes &
Betrouni, 2009; L. Yu, Qi, & Introduction, 2012) among others.
However, such modeling must capture diverse characteristics of the multifractal behavior such as the long-range dependence (LRD), the variability in the scales and the fluctuations between small and large magnitudes (burstiness) (Ge, Fan, Zhu, Deng, &
Wang, 2014; Ihlen, 2012), just as the data generation strictly
positive. This last characteristic causes this task not to be modeled just with a process of Fractional Gaussian Noise (fGn)
(Fan & Li, 2015), because this process normally generates
in 1999, combines the multifractal features and the wavelet transform for the modeling of this type of traces. MWM involves the higher order statistic due to the multiplicative construction and it has given rise to the development of other models as the MultiFractal Hurst algorithm (MFH) (Contreras et al., 2006); The MFH algorithm is the basis of the current development, although it permits the configuration of the average and the Hurst parameter sampling, it does not adjust the multifractal spectrum width to a determined size by the user, which is the main difference in the current paper.
Accordingly, this paper describes the proposal of the MultiFractal Hurst Spectrum Width (MFHSW) model which allows adjusting the multifractal spectrum width (MSW) of a synthetic time series, also the Hurst parameter. The introduction of the multifractal spectrum to the MFH algorithm enables a more precise analysis of the time series, since it uses all the moments of q order. The proposed algorithm adjusts the Hurst parameter sampling and reallocates the MSW adjusting the beta distributions that conform the binomial cascade. Such binomial cascade is supported by MWM and at the moment of modifying its multiplicative coefficients it can adjust the MSW giving rise to the MFHSW algorithm. As input packets for the algorithm, the average of the trace is entered, the expected Hurst parameter, the MSW, the scales j1 and j2 that represent the octaves to be analyzed to calculate the Log-scale Diagram (LD), that allows determining the estimated H sampling and finally, the type of Wavelet to use.
Traditionally, the multifractal analysis is based on the trace parameterization. By contrast, the approach of this paper is the multifractal synthesis, this is, to generate multifractal traces that offer some expected characteristics by the users. In such a way, that is possible to specify the Hurst parameter, as the MSW, to obtain the corresponding trace.
The MFHSW algorithm is associated with an expert system
(Miller, 1986) as is described in Fig. 1. Initially, the user interface
captures the Hurst and the MSW parameters required by the user. The basis of rules is implemented through the LD and the multifractal spectrum (MS), because they are in charge of evaluating the Hurst and MSW parameters. The function of the inference engine is determining if the generated trace meets the predetermined Hurst and MSW parameter, and that their standard deviations are lower than 0.005 and 0.01 respectively. Finally, the working memory is associated with the generated trace without the verification of the basis of the rules and the inference engine.
Figure 1. MFHSW algorithm from the point of view of a system expert.
The rest of the article is structured as follows. In section II the most relevant works that approach the synthesize traces problem with multifractal characteristics are show. In section III the mathematical resources for the estimation of the Hurst parameter sampling and the MSW are described; the formalism of the discrete wavelet is presented to calculate the detail coefficients and thus calculate LD. In section VI is displayed how the scale coefficients proposed by MWM were modified with the purpose of establishing a relationship between the Hurst parameter and MSW. In section V the MFHSW algorithm is developed and validated through the comparison between a trace generated by the proposed algorithm and one as reference. Finally, in section VI the conclusions are drawn.
2. RELATED WORK
The first steps in generating traffic communications networks were presented at Intel Corporation by K. Kant (Kant, 1999). Under his direction, the generation of request times between consecutive arrivals in a Web server was investigated; this generation looked for asymptotically self-similar processes having multifractal characteristics in small time scales. To find a solution, a Markov process M/G/oo was used to allow the introduction of multifractal behaviors in small/medium scales of time without affecting asymptotic self-similarity in the trace. Using this information, Kant modified the M/G/oo process to scale in time and queue behavior in the aggregated traffic and studied irregularities at small scales of time due to the construction of the cascade (Kant, 1999). Although both the queue and the scale behavior traditionally depended on the second order of the cascade, Kant showed that the long timescales of mass distribution had greater influence compared to the number of stages in the cascade. Based on this, he established a cascade construction method to properly select the scale and queue properties using one or two stages of the mass redistribution derived from the real networks dynamics (Kant,
Later in 2004, at the IBM research laboratory in Tokyo, J. Shimizu
(Shimizu, 2004) proposed a method for generating a network
traffic pattern that could be used as a stress test on a web page. In the first step of the method, Shimizu generated a time series of positive root using a wavelet α-stable distribution to represent network traffic characteristics with both multifractibility and heavy queue. In the second step, the time series is restructured by Volterra filtering the high and low frequency components to finally re-synthesize the extracted components and obtain a designated traffic pattern by changing the components intensities. Shimizu simulated this method to generate traffic patterns containing long range dependence and an appropriate distribution of short-term (Shimizu, 2004).
An updated model was presented in 2010 by R. Zhao (Zhao &
Zhang, 2010) at China National Digital Switching System Center
department of computing and science, using the latency of the network frames, the Hurst exponent, and the types of network traffic. To predict their relationship, Zhao not only used the multifractal wavelet model based on multi–patterns, but also modified the Hurst parameter to generate multi-patterns of network traffic that include stochastic traffic and self-similar traffic at various distributions. All of this under a precise control of the frames latency to ensure the desired characteristics for the generated trace (Zhao & Zhang, 2010).
In 2012, H. Lopez (Chávez & Monroy, 2012) created an algorithm that generated traces of given length with configurable average sampling rate as well as Hurst parameter using the Multifractal Wavelet Model (MWM) proposed by Riedi in 1999 (R.H. Riedi et
al., 1999). His iterative algorithm built the fractals trace based on
a binomial conservative cascade to reduce the computational complexity validation tools such as diagram variance-time
(Chávez & Monroy, 2012).
Although the above models had multifractal characteristics and were based on wavelet synthesis, they do not consider a multifractal spectrum setting. This is a predominant factor when comparing queue times relative to the real time series as demonstrated in this research.
3. MATERIAL AND METHODS: MULTIFRACTAL
ANALYSIS
3.1 Discrete Wavelet Transform.
The discrete wavelet transform is a mathematically effective tool for the analysis and synthesis of fractal and multifractal data, since the wavelet basis obtaining is carried out from a mother wavelet expansion (P. Abry et al., 2000). That expansion works as a scaling operator; therefore, wavelet transform function family has an intrinsic feature of the invariability of the scale of the fractal phenomena (Alzate Monroy, 2002).
The signal construction from a mother wavelet 𝜓! is obtained
through the expansion or contraction of itself. This is
accomplished through a scaling parameter a, where ψa
represents the wavelet coefficient as is described by Eq. (1).
ψa τ = 1 aψ
τ
a (1)
Therefore, the continuous wavelet transform Tx(a,t) is defined as the coefficient set that are compared by means of inner products of a x signal. See Eq. (2).
TX a,t = x,ψa,t , a∈ R+, t∈ R (2)
For the discrete wavelet calculation, the scaling coefficients ϕj,k(t) are provided by the shifted and orthogonal scaling functions set as is illustrated by Eq. (3).
ϕj,kt = 2-j 2ϕo 2-jt-k , k∈ Z (3)
Thanks to the multi-resolution analysis theory, the presence of an obtained mother wavelet is shown from the scaling function ϕo(t), forming a Riesz basis function set as is described by the Eq. (4).
ψj,kt = 2-j 2ψo 2-jt-k , k∈ Z (4)
In this way, the discrete wavelet transform is determined from ϕo
and ψo as the inner product of the x(t) signal with the function
scaling coefficients and the wavelet transform as is shown in Eq. (6) (Alzate Monroy, 2002; Sheluhin, Smolskiy, & Osin, 2007)
aX J,k = !aX J,k ϕJ,k t ; dX j,k =TX 2j, 2jk
(6)
In such a way j is described as the logarithm on the base-2 of the scale, a=2j, called octave, these detail coefficients can be
obtained through a pyramidal algorithm based on high-pass filters and low-pass filters, obtaining the coefficients of those filters from ψo and ϕo (Abry et al., 2000;Flandrin, Gonçalves, &
Abry, 2009), see Fig. 2. The answer in octaves is achieved through
subsampling (↓ 2) removing one of every two aX-1[n] samples,
starting from the source signal. Using the high-pass filters and the subsampling, detailed coefficients are obtained, and with the low-pass filters and subsampling the next approximation (aX+1[n])
Figure 2. Algorithm for calculating the detail coefficients by filter bank
Once the detailed coefficients are obtained at the end of the pyramidal algorithm, the inherent redundancy of the continuous wavelet transform is reduced, since the contained information in a signal is expressed by the discrete transform in a two-dimension space (Sheluhin et al., 2007).
To have a full picture of the time series with multifractal behavior, Fig. 3a illustrates the inter arrival times as a function of the packet number in an Ethernet network, the reference trace of August 1989, taken in Bellcore Morris Research and Engineering Center in Morristown.
3.2 Log-scale Diagram (LD).
When there is little correlation between the detailed coefficient, it guarantees that the temporary estimators have a minimum variance, in this way, it is estimated that the dx(j,⋅) detailed
coefficient variance, is given by Eq. (7) (Sheluhin et al., 2007;Flandrin et al., 2009)
μj=n1
j dX j,k
2
nj
k=1 (7)
Where nj is the detailed coefficient number in octave j and µj is
the E[|dx(j,⋅)|2] estimator. As soon as the detailed coefficients are
gathered and the second statistic moment obeys the power law with a 2H-1 exponent. Identifying H as the Hurst parameter, which represents the exponent for the self-similarity of the second stochastic moment, which allows estimating the Hurst parameter as is described by Eq. (8).
log2μj= 2H-1 j +log2C (8)
When log2µj is graphed as a function of octave j, LD is obtained,
see Fig. 3b. Once a linear regression is done over the calculated estimators for each octave and calculating the slope, H
estimation is obtained with confidence intervals determined by the estimated H standard deviation.
Figure 3. a) Trace BC-pAug89 of Bellcore labs. b) Log-Scale Diagram estimates (0.70324 < H = 0.71181<0.72038)
for the reference trace.
The structure of invariability to the scale of the BC-pAug89 trace can be explained as a structure with self-correlative (Flandrin et
al., 2009) and self-similar behavior (Vega & Alzate, 2002). When H
estimation is already calculated using LD, it is possible to obtain a computationally effective tool for the data analysis with self-similar features (Will E. Leland et al., 1994) as a consequence, fractals [1].
3.3Linear Multiscale Diagram
Extending the Eq. (7) estimator not only for the second statistic moment q=2, but also for moments that belong to the real number set, positive as well as negative, and denoting a q as the order of the estimator, Eq. (9) is obtained.
μjq =1
nj dX j,k
q nj
k=1 (9)
In self-similar processes where 0.5<H<1, thus μjqis the estimator
of E dX j,k q (i.e. 𝜇!≈E d
X j,k q ), the estimator obeys the
power law, and it can be extended to Eq. (10) (P. Abry et al.,
2000).
E dX j,k q = Cq2jζq+ q
2 (10)
Where Cq is a q function and ζ(q) is a function that allows
distinguishing between monofractal and multifractal processes, hence, the order scaling exponent q (P. Abry et al., 2000)is estimated as well as in Eq. (8).
𝑞ℎ and are monofractal with ℎ=𝐻(Sheluhin et al., 2007). The multiscale diagram is obtained when using the exponent
𝛼 𝑞 =ζ q +𝑞 2, but the objective on this investigation is to show how the value of H decreases as long the order q increases. Therefore the linear multiscale diagram (LMD) estimating the scaling exponent orders q by 𝐻 𝑞 =𝛼 𝑞 /𝑞-1 2 plotted against
q(P. Abry et al., 2000).
In order to have a behavior representation of those exponents as a function of order q, the linear multiscale diagram (MD) is used (see Fig. 4a) where H is projected in function of order q for q
values between -15 and 15.
It can be seen in the LMD that for monofractal time series, the variation of H(q) for positive q values is taken a lineal behavior opposite situation with multifractal time series. It is worth mentioning that in the intersection of H(q) for both cases is close to q=2, thus reaffirming the requirement to project q not only to the order 2. In this research the comparison is done from the lowest q value q=-15, until the highest value of order q=15, thus it is possible to differentiate the behaviors regarding q for monofractal and multifractal signals. These q values between [-15,15] were taken because if this interval is increased the H(q) values are bounded by an asymptote and, in computational terms the calculation of the estimated values is increased. Inasmuch as the algorithm is an approximation it was limited to this interval, because the statistical analysis of the trace is not modified.
Figure 4. Comparison Time Series between Monofractal and Multifractal a) Linear Multiscale Diagram. b) Legendre transform step one, calculating τ(q) from H(q) c) Legendre transform step two, calculating D(q) from τ(q)
d) Multifractal Spectrum H(q) against D(q).
3.4 Multifractal Spectrum
A multifractal time series can be analyzed with different methods, in this research the Legendre Transform is implemented. This procedure measures the singularity dimension of order q
denoted D(q) as a function of the resolution H(q) (Meakin, 1998).
D(q) is a linear transformation from the scales to statistic moments, because the function that maps the sampling scales
against the respective statistic moments is not linear (R.H. Riedi,
Crouse, Ribeiro, & Baraniuk, 1999). Even though the Legendre
method has a serious disadvantage of covering with a convex structure the singularities D(q) constructs a curve between the pair of point (qi,D(qi)) and(qi+1,D(qi+1)) despite the fact that any other point exists between qi and qi+1. This curve is present to construct the parameter order of mass exponent τ(q) (Meakin,
1998). However from a computational point of view is taken the
Legendre Transform since it is more efficient than other methods
(Rudolf H Riedi & Vehel, 1997).
For the calculation of the singularity dimension τ(q) is used as an intermediate step and it is calculated as Eq. (11).
τq = qH(q)-1 (11)
Fig. 4b shows how the transformation from H(q) to τ(q) is, in order to obtain the singularity dimension of order q, see Eq. (12)
(Kantelhardt et al., 2002).
Dq ≡τqq-1=qH(q)q-1-1 (12)
In Fig. 4c the transformation from τ(q) to D(q) is shown . Finally, the multifractal spectrum (MS) is obtained when graph D(q) against H(q) and it is shown in multifractal Fig. 4d. The resulting multifractal spectrum is a large arc where the difference between the maximum and minimum H(q) is called the multifractal spectrum width (Ihlen, 2012). MS form can be approximated to a second order polynomial function and measure it is width through the zero cross of the function when D(q)=0. The arc produced by monofractal time series (green) compared with the arc of multifractal time series (orange) is less indicating on multifractal time series greater stochastic structure.
The signal analysis with multifractal and monofractal characteristics is developed by using the statistic moments q or in the scale (Stênico, Lee, & Vieira, 2013) as shown in Eq. (13).
Z ct = dcHZ(t) (13)
Eq. (13) describes a monofractal process where H is in 0.5<H<1; =d
interval; (read, equally distributed) Eq. (13) is generalized with the purpose of analyzing the bigger multiscale relationships, given by
Z(ct) =dM(C)Z(t), where Z(t) and M(C) are independent stochastic processes (Stênico et al., 2013).
The scaling factor M(C) is a random variable whose distribution does not depend on t, for monofractal processes there is
M(C)=CH. For multifractal processes H(c) is a function which
depends on c and not a constant as for the monofractal processes, indicating more variability in the scale, see Eq. (14).
Hence, for traces with monofractal behavior is enough with obtaining the H parameter to describe the power law.
When analyzing the scale variability, the order q moments and the changes in the time series, it can be seen that when extremely large magnitudes are present in the time series, analyzed through higher scales, for order q>0 moments, the MS right side formation is determined; otherwise, when there are extremely small magnitudes in the time series, analyzed through lower scales, for order q<0 moments, the MS left side formation is determined. Consequently, when a truncation appears on MS right side the time series present more fluctuations with extremely small magnitudes, the contrary happens when a truncation appears on MS left side indicating more fluctuations with extremely larger magnitudes (Thompson & Wilson, 2014).
3.5Multiplicative Cascades and MFH algorithm
The MWM consists of Haar Wavelet transform and the structure seen in Eq. (15).
𝑊!,!= 𝐴!,!𝑈!,! (15)
where 𝐴!,! is a random variable which takes values in the interval
from -1 until 1, assuring the fulfillment of |Wj,k|≤Uj,k ∀ j,k so, the
scale coefficients are positive and symmetric regarding zero (R.H.
Riedi et al., 1999). The multipliers Aj,k=2Bj,k-1, with Bj,k coefficients
equal to P, are random variables identically distributed that take value in the interval from zero to one and they are symmetric regarding 0.5.
The relationship between MWM and the multiplicative cascades takes place when the Haar transform is used as the multiplicative cascade coefficients (R.H. Riedi et al., 1999); establishing the multiplicative coefficients as random variables, with an average of 0.5, that can take values between zero and one. With these parameters the positive data generation with multifractal characteristics is guaranteed.
The implemented algorithm starts with U0,0 value of iteration, this
quantity is distributed for two intervals according with the P value parameter and its complement as is shown in Fig. 5; then, in the third scale these values are distributed in 2 and each couple has a different P. This process continues until N scale, where it will have 2n intervals with U0,0 initial fractions, resulting in a Conservative
Binomial Cascade (CBC) (Contreras et al., 2006).
The MFH algorithm generates traces with positive data and with LRD, obeying a power law (hence its fractality), adjusting the H
parameter and its average through the proposed relationship in
(R.H. Riedi et al., 1999), in which the Wavelet coefficient
multipliers are given by Eq. (16).
k = 22!-1-1
2-22!-1 (16)
Thus, adjusting the H parameter is achieved by configuring the binomial cascade symmetry in the beta distribution by means of Eq. (16), this is how the MFH generates 2n length traces, and
through LD and the confidence intervals, it is verified that the estimated H is in the closest range to the H entered by the user.
Figure 5. Synthesis of MWM by CBC
4. CALCULATION: HURST PARAMETER AND
MULTIFRACTAL SPECTRUM WIDTH
The classic structure of the multiplicative cascade is modified in this research work when two random numbers with beta distribution are associated on Bj,k coefficients. As it was aforementioned, beta parameter is controlled by k (Eq. (16)) and k is determined by H, thus, every scale has a different random number but all of them use the same beta distribution, in other words, all of them use the same k.
Without changing the range of values for the coefficients in the interval between cero and one, the H parameter of beta distribution is going to be modified, where H parameter depends on H proposed by the user.
This is carried out with the aim of changing the H of the k coefficients and modifying the beta distributions to keep the Hurst parameter sampling of the obtained trace and observing how this affects on MSW.
The group of the scales was divided into two, in one of those a beta parameter was assigned with a H1 equal to the Hurst
parameter sampling in the trace; in the second group another beta parameter with a H2, in such a way that the average between
H1 and H2 is equal to 0.75, that value represents half of the
allowed interval for Hurst parameter (0.5<H<1). Therefore, H1 and
H2 obey Eq (17).
In consequence, the scales are sought without modifying the Hurst parameter sampling but the MSW. H1 and H2 used values were 0.71 and 0.79 respectively.
According to Fig. 6d, MSW has a gradual increase from the j=8 to octave j=16, and it remains invariant in the interval scale (2< Js
<8) y (16< Js <20).
To carry out the experiment, 1000 time series were synthesized for each assignation in the different Js levels. Two histograms were graphed for each level of scaling and, they illustrate the 1000 traces average of the estimated Hurst parameter and the average of MSW. In Fig. 6a and 6b, the histograms exemplify the calculation of the average of the estimated Hurst Parameter and the average of MSW respectively, for the octave js=12 level.
Figure 6. Behavior of the algorithm around the scales by distributing two values of 𝑘 at differents scales of 𝑗! a) Histogram of H with Average of 0.79586 Octave j =12, b) Histogram of MSW with average of 4.9068, Octave j=12, c)
Standard deviation of Hurst parameter d) Standard Deviation for wide Spectrum
In Fig. 6, it is shown that the octaves higher than j=14 determine the value of Hurst parameter with an average of 0.75. It is also evident that MSW is modified when the octaves are higher than j>10.
From the aforesaid experiment, it is stated that, if an H sampling is wanted for a trace and a particular MSW, the following step is to provide Hurst parameter for all the trace on the higher scales than Js =12, that is to say, H2=Hexpected, and on the lower octaves
(H1) it is necessary to assign another H parameter sampling that
permits adjusting the MSW to the expected value. In order to have greater control over the trace, a Hurst parameter equal to H2 is assigned over the octaves higher than twelve, so as to the
estimated Hurst parameter sampling from the lower octaves than twelve has a close value to H2 . To manipulate the MSW, a Hurst
parameter equal to H1 is assigned to lower octaves than twelve
and depending on the assigned value the MS trace will increase or decrease. Thus, H1 value is related to MSW value as is shown in Fig. 6d. When an octave higher than twelve is taken, the range
of values of MSW increases from one scale to another. By inferring MSW behavior (Fig. 6c) and the H parameter sampling (Fig. 6d), this paper suggests that the octave which will divide the assignation of H1 and H2 it will be J=16. This was due to the fact
that this octave is in the midpoint between the highest value of possible scales (Jmax=20) and the point where the H1 influence
over the spectrum width (J=12) is proved.
In order to know the behavior between the relationship H1 with
MSW, some values between 0.55 and 0.95, distributed each 0.05, were assigned. With the purpose of comparing the obtained results, the value of 0.71181 is assigned to H2, which represents
the estimated H parameter sampling for the reference trace BC-pAug89.
When conducting a test of 1000 iterations for each assigned H2 in each scale and doing for each test two histograms, one that estimates the average Hurst parameter and the other one the MSW, it can be observed the behavior in the time series when an ordered assignation of H2 is performed and how this affects MSW. Fig. 7 collects the obtained data for this experiment, modifying the H2 values between [0.55, 0.95] and distributing them in octaves j>4, j>10, j>14, j>16 and j>18, which describe the average behavior of the histograms (H parameter sampling Fig. 7a and MSW Fig. 7b).
By observing this behavior, it is possible to perceive that for close scales to the highest octave j=20, the MSWs values differ between 1.5 and 12.5, they vary more in contrast with the close scales to the lowest octave j=2, where MSW differs between 4.9 and 6.3. Thereby, obtaining a very interesting result where it is possible to assign on the close scales to the highest octave a greater MSW range of values.
Figure 7. a) Variations of H b) Variations of MSW to distribute a 0.55< Hs<0.95 for different scales
between 2< js<20.
mathematical expression for MSW. It must be taking into consideration that the family of curves was only proposed for H1=0.71181, therefore it can only be possible to get time series with H parameter sampling equal to H1, but with MSW between
1.5 and 12.5.
5. RESULTS: MFHSW ALGORITHM
By doing a linear regression for the octave 16 the Eq. (18) is obtained, which links WMS with H2 for a fixed H1 and equal to
0.71181.
HW W = -23.71881 W + 0.9761 (18)
The result obtained with Eq. (18) is linked to MFH algorithm to finally achieve the proposed algorithm, which not only adjusts the Hurst parameter value, but it also allows calculating the MS and adjusting its respective MSW, this algorithm was denominated MFHSW, which is described by Fig. 8.
The MFHSW algorithm has as input parameters: the number of n decomposition, the average of m trace, the expected Hurst parameter h, and the expected MSW, j1 and j2 scales (that represent the octaves to be analyzed in order to calculate LD) and the type of wavelet to use Wvt.
Figure 8. Example of the algorithm for H = 0.71181
The MFHSW algorithm behavior (Fig. 8) is as follows: initially k is divided into two k1 and k2. k1 for the conservative cascade, its
represents the value that controls the trace and determines the H value =0.71181. k2 is calculated through the octave 16 linear regression (equation 17), accordingly, a binomial cascade is formed. Once the trace is formed, the next step is to calculate the detail coefficients through the estimator and LD is generated, which allows the calculation of H sampling with a maximum error of 0.005. As soon as this condition is completed, it is appropriate to continue with the calculation of hq and Dq to generate the Ms. MSW is calculated by means of the linear regression and it is compared with the expected MSW, verifying the error to be less than 0.01.
By completing this process, the trace is obtained with the expected features, if one of the users’ requirements was not fulfilled, the process is restarted in order to generate a new trace and the process is performed once again until the expected features are achieved.
So as to evaluate the MFHSW algorithm, h=0.71181 and a MSW (that complies with 1.5≤WMS≤5 which is increasing) are entered
and, MSW is modified maintaining a fixed H, as is illustrated in Fig. 9.
Figure 9. a) Multifractal traces generated by MFHSW, b) verification by LD of H same for all the traces and c)
different MSW for each trace verified by MS
In Fig. 9, it is observed that LD slopes are equal but their cut-off point with y axis are higher inasmuch as MSW increases gradually (Fig. 9c) as well as the time series features in which the changes for large magnitudes (Fig. 9a) are contemplated at the same time MSW is increasing. The previous statement reinforces the MFHSW algorithm functionality, demonstrating that it has the capacity of maintaining a fixed H established by the user, while MSW is also adjusted and established by the user. Different MSWs delimit different features in the time series structure.
Finally, in order to validate MFHSW algorithm, the calculated parameters are entered into it for the (BC-pAug89) reference trace, e.g. an average of 0.0031, a H=0.71181 parameter and a WMS=1.3958, with the aim of generating a similar trace (Fig. 10).
Figure 10. Comparison between reference trace and synthetic trace generated by MFHSW Algorithm. a) Time Series BC-pAug89 b) Synthetic Trace c) Log-Scale Diagram
d) Multifractal Spectrum
6. DISCUSSION
According to the obtained results, the MFHSW algorithm meets the objectives of adjusting the MSW and obtaining a sampling H similar to the one in the reference trace, consequently it becomes in a tool of great relevance and utility in the generation of multifractal traffic finite-length time series, since currently there are no papers that show similar contribution in this area. However, it is also important to mention two limitations that the proposed algorithm presents. First, the synthesized trace by the algorithm MFHSW shows a maximum inter-arrival time of 2.6s which is too high in comparison to the 0.34s of the reference trace. Second, although the synthesized and the reference trace show a similar stochastic structure and their mean value is the same, their magnitudes differ because the MFHSW only has an influence on big magnitudes, shaping only to the right side of the MS. The aforementioned shows a disadvantage when a time series is needed where the fluctuations with extremely small magnitudes predominate.
The proposed teletraffic simulation tool in this paper is based on stochastic models that attack the problem of data generation from the scale, in comparison with cases like the one proposed
by (Wang, Zhang, & Zhang, 2015) that attacks the problem from a
spatiotemporal perspective of the base stations.
Having an algorithm to synthetize traffic traces with a given sample mean, Hurst parameter and Multifractal Spectrum Width, we can study several issues of network performance. For example, we have developed several queue simulation scenarios, fed with synthetic traffic traces obtained from the proposed algorithm, which allowed us to observe that the average waiting time, in an LRD/M/1 queue, is closer to the reference when the
Multifractal Spectrum Width is considered. Fig. 11 shows the comparative results between traces obtained with the MFH and MFHSW algorithm. For the algorithm MFH, 1000 traces were synthesized with sample H and mean equal to the sample reference parameters. For the algorithm MFHSW, 1000 traces were synthesized with sample H, mean, and Multifractal Spectrum Width equal to the sample reference parameters. The length of all traces was 1000000.
Figure. 11. Average Waiting Times comparison in a MF/M/1/∞/∞ queue, with exponentially distributed service times (memory less: M) and Multifractal arrivals
(synthetic traces with LRD: MF).
In addition, in (Chávez & Monroy, 2012) we present a simple example that has very important implications. Several network parameters and traffic conditions can be inferred from packet dispersion measurements. In an active probing scheme, a sender transmits packets of given length at given instants of time and a receiver collects them, taking note of their arrival times. Under heavy load conditions, these measurements can be used to infer the cross-traffic over the tight-link in a path. In this application, it is widely believed that the probing traffic should drive the path close to congestion in order to obtain a high correlation between the dispersion measurements and the cross-traffic. Here we show through simulation that this is the case for SRD traffic but, if the cross traffic exhibits LRD, a very low probing traffic can suffice to estimate the cross traffic. For small H, a high correlation coefficient requires a very small measurement timescale, T, and high utilization, ro. As we increase the Hurst parameter, the correlation coefficient becomes less dependent on T and ro. This suggests the existence of very efficient probing techniques for LRD cross-traffic estimation.
traffic (Kirichek, Golubeva, Kulik, & Koucheryavy, 2016). In
(Kirichek, Golubeva, Kulik, & Koucheryavy, 2016) the data
recollection is implemented through a sensor network and the traffic parameters are obtained with WireShark. As above, the MFHSW algorithm can provide a database for the next simulations, avoiding repeating the capture process.
In the next generation technologies and networks is needed to count on simulation tools that allow doing an appropriate planning to assure a determined level of quality of service and at the same time minimize costs and time. In this regard, the proposed algorithm provides the required multifractal traffic traces in the aforementioned simulations like in (Ivanova et al.,
2015) where the teletraffic model simulation focused on
management problems of resources for 5G networks is done.
The application of the MFHSW algorithm spreads to different knowledge areas, for example, in the case of theoretical neuroscience, (Zheng, Gao, Sanchez, Principe, & Okun, 2005)
seeks to characterize through a multifractal multiplicative cascade recordings of neural firing. In this case, the cascade model singularity dimension of order q as function of q is graphed.
7. CONCLUSIONS
The developed MFHSW algorithm showed satisfactory results to modify the multifractal spectrum width without changing the selected Hurst parameter. Even though this paper only presents one experiment for a fixed H sampling, the methodology that was carried out, may be expanded to obtain a family of curves that allows obtaining traces with any Hurst parameter among the allowed values and adjusting the multifractal spectrum width to any target value.
MFHSW could be considered functional for the network traffic modeling, with the objective of guaranteeing the service quality, however, it could be expanded to all the research areas that require the use of traces with multifractal characteristics.
Although the multifractal spectrum width could be modified through the proposed MFHSW algorithm, it can be observed that the changes in the multifractal spectrum are in the right side, indicating fluctuations of large magnitudes. With the purpose of generating a greater control on the spectrum width it is necessary to modify the relationship between the multifractal spectrum width and the Hurst parameter, so that the left side of the multifractal spectrum can also be modified. The aforementioned will allow obtaining a better adjustment in the time series structure.
Although the found method achieved satisfactorily the objective of modifying the multifractal spectrum width without affecting the Hurst parameter value, it is suggested for further work to use a neural network and train it in order to avoid the MFHSW algorithm generates a new trace when this one does not fulfill with the expected requirements by the user; but by means of the synthetic trace, the time series structure can be modified, thus the spectrum shape changes and adjusts to the parameters set by the user.
Further work is the synthetic trace evaluation with the aim of estimating the service times between the server and the user, using the queuing theory, in the context where the user requires a demand to the system and the server is the support station for the user’s request, comparing these service times with the ones of the reference traces.
ACKNOWLEDGMENTS
The authors wish to thank Center for Research and Scientific Development of Universidad Distrital Francisco José de Caldas, for the supporting and funding during the course of this research project.
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