Multi-fractional generalized Cauchy process and its application to teletraffic

Physica A xxx (xxxx) xxx

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Multi-fractional generalized Cauchy process and its application to teletraffic Ming Li

∗

Shanghai Key Laboratory of Multidimensional Information Processing, East China Normal University, 500 Dongchuan Rd., Shanghai 200241, PR China Ocean College, Zhejiang University, Zhejiang 310012, PR China

article

info

Article history: Received 11 June 2019 Available online xxxx MSC: 28A80 60G15 60G18 62M10 60K30 60G18 60E07 Keywords: Long-range dependence Hurst parameter Local self-similarity Fractal dimension Generalized Cauchy process Teletraffic

a b s t r a c t The contributions given in this paper are in two aspects. The first is to introduce a novel random function, which we call the multi-fractional generalized Cauchy (mGC) process. The second is to dissertate its application to network traffic for studying the multi-fractal behavior of traffic on a point-by-point basis. The introduced mGC process is with the time varying fractal dimension D(t) and the time varying Hurst parameter H(t). The representations of the autocorrelation function (ACF) and the power spectrum density (PSD) of the mGC process are proposed. Besides, the asymptotic expressions of the ACF and PSD of the mGC process are presented. The computation formula of D(t) is given. The mGC model may be a new tool to describe the multi-fractal behavior of traffic. Precisely, it may be used to reveal the local irregularity or local self-similarity (LSS), which is a small-time scale behavior of traffic, and global long-term persistence or long-range dependence (LRD), which is a large-time scale behavior of traffic, on a point-by-point basis. The cast study with real traffic traces exhibits that the variance of D(t) is much greater than that of H(t). Thus, the present mGC model may provide a novel way to explain the fact that traffic has highly local irregularity while its LRD is robust. © 2020 Elsevier B.V. All rights reserved.

1. Introduction Arrival teletraffic (traffic for short) modeling plays a role in communication systems (Gibson [1]). On circuit switched communication systems, such as traditional telephone networks, Erlang [2] gave the pioneering work on traffic modeling with the Poisson distribution, also see e.g., Brockmeyer et al. [3], Gall [4], Lin et al. [5], Manfield and Downs [6], Reiser [7], Akimaru and Kawashima [8], Bojkovic et al. [9]. The traffic model of the Poisson distribution type is simple and successfully applied to circuit switched communication networks (Gibson [1], Cooper [10]). However, things change in packet switched communications. On packet communication networks, such as the Internet, it was found that the traffic is non-Poisson in the late 1970s (Tobagi et al. [11, p. 1427]). In the 1980s, traffic property of non-Poisson was clearly noticed and the burstiness, which is a local property of traffic from a view of bounded modeling, was observed (Jain and Routhier [12]). ∗ Correspondence to: School of Information Science & Technology, East China Normal University, 500 Dongchuan Rd., Shanghai 200241, PR China. E-mail addresses: [email protected], [email protected]. URL: https://www.mendeley.com/profiles/ming-li173/. https://doi.org/10.1016/j.physa.2019.123982 0378-4371/© 2020 Elsevier B.V. All rights reserved.

Please cite this article as: M. Li, Multi-fractional generalized Cauchy process and its application to teletraffic, Physica A (2020) 123982, https://doi.org/10.1016/j.physa.2019.123982.

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In the 1990s, people began investigating traffic from the point of view of fractal time series. The fractal properties of traffic, namely, self-similarity (SS) and long-range dependence (LRD), were paid attention to (Michiel and Laevens [13], Li and Borgnat [14]). The focus of this research is on representing multi-fractal behavior of traffic. Let x(t) be traffic series. Denote by r(τ ) its autocorrelation function (ACF) given by r(τ ) = E[x(t)x(t + τ )],

(1.1)

where τ is time lag and E is the mean operator. Then, it was found the r(τ ) decays so slowly such that (Paxson and Floyd [15], Beran et al. [16])

∫

∞

r(τ )dτ = ∞.

(1.2)

−∞

In mathematics, x(t) is said to be of long-range dependence (LRD) or have long memory if its ACF satisfies (1.2) (Beran [17,18]). Eq. (1.2) can be equivalently expressed by r(τ ) ∼ c τ 2H −2

(τ → ∞),

(1.3)

where c > 0 is a constant and 0.5
1 f

(f → 0).

(1.4)

The above says that traffic is a type of 1/f noise, which reflects its LRD property in frequency domain. When a random function X (t) satisfies X (at) = aH X (t),

a > 0,

(1.5)

where the equality denotes the same probability distribution on both sides, we say that X (t) is an exact self-similar process with the SS measure of H. Traffic asymptotically follows (1.5) at large time scales (Paxson and Floyd [15], Leland et al. [20]). A stationary process that exactly follows (1.3)–(1.5) is the fractional Gaussian noise (fGn) in the Wely sense, which was introduced by Mandelbrot and van Ness [21]. Its ACF is given by CfGn (τ ) =

σ 2 ε 2H −2

[(

2

] ⏐2H )2H ⏐ ⏐ τ ⏐2H ⏐ |τ | ⏐ |τ | ⏐ ⏐ , +1 + ⏐⏐ − 1⏐⏐ − 2 ⏐ ⏐ ε ε ε

(1.6)

where σ 2 = (H π )−1 Γ (1 − 2H) cos(H π ) stands for the intensity of fGn, ε > 0 is a parameter used for regularizing fractional Brownian motion (fBm) so that the regularized fBm is differentiable [21, p. 427–428]. FGn is of LRD if 0.5
(

)− 12−−HD

,

1 ≤ D<2, 0
(1.7)

The ACF C (τ ) is positive-definite for the above ranges of H and D. It is a completely monotone for 1 ≤ D<2, 0 ≤ H <1. When D = 1 and H = 0, C (τ ) reduces to the ACF of the usual Cauchy process. The condition for the GC process to be of LRD is 0.5
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An interesting and important property of traffic is its multi-fractal, see e.g., Abry et al. [24], Willinger et al. [40,41], Feldmann et al. [42], Veitch et al. [44], Stênico and Ling [45], Xu and Feng [46], Ostrowsky et al. [47], Vieira et al. [48], Rocha et al. [49], Budhiraja and Liu [50], Vieira and Lee [51], Masugi and Takuma [52], Fontugne et al. [53]. The challenging issue in this regard is to find a proper way to explain the highly local irregularity and robust LRD of traffic from the point of view of multi-fractals (Willinger et al. [40,41], Feldmann et al. [42], Fontugne et al. [53], Ribeiro et al. [54]). However, the reports of describing the multi-fractal behavior of traffic on a point-by-point basis are rarely seen. This paper aims at presenting the multi-fractional GC (mGC) process and applying it to describing the multi-fractal behavior of traffic on a point-by-point basis. By mGC process, we mean that the constants D and H in (1.7) are replaced by D(t) ∈ [1, 2] and H(t) ∈ (0.5, 1), respectively. In this way, D(t) may serve as a tool to describe the local irregularity or LSS of traffic on a point-by-point basis while H(t) may be used to characterize the global long-term persistence or LRD of traffic on a point-by-point basis. The rest of paper is organized as follows. In Section 2, we propose the ACF of the mGC process and its asymptotic expressions for τ → ∞ and τ → 0. Section 3 presents the PSD of the mGC process and its asymptotic expressions for ω→0 and ω → ∞. The computation formula of D(t) is given in Section 4. Case study with the demonstrations of D(t) and H(t) of real traffic is shown in Section 5. Discussions are in Section 6, which is followed by conclusions. 2. The mGc process 2.1. ACF of the mGC process Theorem 2.1. Let C (τ , t) = 1 + |τ |4−2D(t)

(

)− 12−−H(t) D(t)

,

(2.1)

where H : [0, ∞) → [0, 1), D : [0, ∞) → [1, 2). Then, C (τ , t) is an ACF. Proof. The function C (τ , t) is an even function since C (τ , t) = C (−τ , t).

(2.2)

It is positive-definite for 1 ≤ D(t)<2 and 0 ≤ H(t)<1. That is, C (τ , t) ≥ 0.

(2.3)

Besides, C (0, t) ≥ C (τ , t).

(2.4)

In addition, lim C (τ , t) = 0.

(2.5)

τ →∞

Thus, according to the theory of random processes (Nigam [55], Bendat and Piersol [56], Priestley [57]), C (τ , t) is an ACF. □ Definition 2.1. A random function X (t) is called the mGC process if its ACF is in the form C (τ , t) = ψ 2 1 + |τ |4−2D(t)

(

)− 12−−H(t) D(t)

,

(2.6)

where H : [0, ∞) → [0, 1), D : [0, ∞) → [1, 2), and ψ is the intensity of X (t). □ 2

X (t)

For the purpose of traffic modeling and the description of D(t) and H(t), we may use the random function ψ without the generality losing. In what follows, unless otherwise stated, we only consider the normalized ACF in the form expressed by (2.1). Note 2.1. C (τ , t) reduces to the ACF of the conventional generalized Cauchy (GC) (Li and Lim [36]) if both D(t) and H(t) are constants. □ For facilitating discussions, we denote

α (t) = 4 − 2D(t),

(2.7)

β (t) = 2 − 2H(t).

(2.8)

and

Using α (t) and β (t), C (τ , t) is written by C (τ , t) = 1 + |τ |α (t)

(

(t) )− βα(t)

,

(2.9)

Please cite this article as: M. Li, Multi-fractional generalized Cauchy process and its application to teletraffic, Physica A (2020) 123982, https://doi.org/10.1016/j.physa.2019.123982.

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Fig. 2.1. Plots of the ACF of mGC process. (a) Solid line: D(t) = 1.5. H(t) = 0.95. Dot line: D(t) = 1.5. H(t) = 0.75. Dash line: D(t) = 1.5. H(t) = 0.55. (b) Solid line: H(t) = 0.75. D(t) = 1.85. Dot line: H(t) = 0.75. D(t) = 1.65. Dash line: H(t) = 0.75. D(t) = 1.45. (c) D(t) = 1.9|cos(0.05t)|. H(t) = 0.5|cos(0.05t)|. (d) D(t) = 1.9|cos(0.05t)|. H(t) = 0.75. (e) D(t) = 1.75. H(t) = 0.5|cos(0.05t)|.

where 0 <α (t) ≤ 2, 0<β (t) ≤ 2. Because C (τ , t) is sufficient smooth on (0, ∞) and for τ → 0 C (0, t) − C (τ , t) ∼ c |τ |α (t) ,

(2.10)

according to Hall and Joy [58], Chan et al. [59], Kent and Wood [60], Adler [61], one has the time varying fractal dimension given by D(t) = 2 −

α (t) 2

.

(2.11)

Thus, α (t) is a time varying fractal index. Since lim C (τ , t) = |τ |−β (t) ,

τ →∞

(2.12)

the condition for the mGC process X (t) to be of LRD is 0 <β (t)< 1, which corresponds to 0.5
M. Li / Physica A xxx (xxxx) xxx

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2.2. Asymptotic expressions Asymptotically, lim C (τ , t) = |τ |−β (t) = |τ |2−2H(t) .

τ →∞

(2.13)

Thus, for large τ , we have the asymptotical expression of C (τ , t) in the form C (τ , t) ≈ |τ |−β (t) = |τ |2−2H(t)

for large τ .

(2.14)

For τ ≥ 10, C (τ , t) is well approximated by |τ |2−2H(t) . The above exhibits that, for large τ , C (τ , t) is solely associated with its LRD measure H(t) and the measure of local irregularity, D(t), may be neglected. On the other hand, if τ → 0, we have C (τ , t) = 1 − |τ |α (t) = 1 − |τ |4−2D(t) .

(2.15)

In fact, for small τ , C (τ , t) ≈ 1 − |τ |α (t) = 1 − |τ |4−2D(t) .

(2.16)

The above says that the LRD measure H(t) may be ignored and C (τ , t) simply relates to the LSS measure D(t) for small τ . 3. PSD of the mGC process 3.1. PSD Denote by S(ω, t) the PSD of the mGC process. Then (Nigam [55, Eq. (4.111)], Bendat and Piersol [56, Eq. (12.130)], Priestley [57], Priestley and Tong [62, Eq. (2.4)], we have S(ω, t) = F [C (τ , t)] =

∞

∫

C (τ , t)e−iωτ dτ ,

(3.1)

−∞

where F is the operator of Fourier transform. The above implies S(ω, t) =

∞

∫

1 + |τ |α (t)

(

(t) )− βα(t)

e−iωτ dτ .

(3.2)

−∞

For the mGC process with LRD, we have S(0, t) =

∞

∫

1 + |τ |α (t)

(

(t) )− βα(t)

dτ = ∞.

(3.3)

−∞

Thus, the computation of (3.2) should be done in the domain of generalized functions over the Schwartz space of test functions. Theorem 3.1. The PSD of the mGC process is expressed by

{[ ] } β (t) ∞ (−1)k Γ +k ∑ α (t) [ ] S(ω, t) = I1 (ω) ∗ Sa(ω) (t) Γ βα(t) Γ (1 + k) k=0 {[ ] } β (t) ∞ (−1)k Γ + k ∑ α (t) [ ] [π I2 (ω) − I2 (ω) ∗ Sa(ω)] , + β (t) Γ α(t) Γ (1 + k) k=0

(3.4)

where ∗ is the convolution operation, I1 (ω, t) = F |τ |α (t)k =

∫

∞

|τ |α(t)k e−iωτ dτ , ∫ ∞ [ −[β (t)+α(t)k] ] |τ |−[β (t)+α(t)k] e−iωτ dτ , I2 (ω, t) = F |τ | =

(

)

(3.5)

−∞

(3.6)

−∞

and Sa(ω) =

sin ω

ω

.

Please cite this article as: M. Li, Multi-fractional generalized Cauchy process and its application to teletraffic, Physica A (2020) 123982, https://doi.org/10.1016/j.physa.2019.123982.

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Proof. Let u(τ ) be the unit step function. Then, with the binomial series, C (τ , t) can be expanded to be

} {[ ] ⎧ ⎫ β (t) ∞ (−1)k Γ ⎨∑ ⎬ + k α (t) αk [ ] |τ | [u(τ + 1) − u(τ − 1)] C (τ , t) = (t) ⎩ ⎭ Γ βα(t) Γ (1 + k) k=0 {[ ] } ⎧ ⎫ β (t) ∞ (−1)k Γ ⎨∑ ⎬ + k α (t) [ ] |τ |−(β+αk) [u(τ − 1) + u(−τ − 1)]. + (t) ⎩ ⎭ Γ βα(t) Γ (1 + k) k=0

(3.7)

The first item on the right side of the above is the binomial expansion of C (τ , t) when |τ | <1 while the second is for |τ | > 1 (Li and Lim ( [63]). ) Note that F |τ |λ = −2 sin (λπ/2) Γ (λ + 1) |ω|−λ−1 for λ ̸ = −1, −3, · · · (Gelfand and Vilenkin [64]). Thus, for α k ̸= −1, −3, · · ·, we have ( ) F |τ |α (t)k = −2 sin [α (t)kπ/2] Γ [α (t)k + 1] |ω|−α (t)k−1 . (3.8) Similarly, for – (β + α k) ̸ = −1, −3, · · ·, we have F |τ |−[β (t)+α (t)k] = 2 sin {[β (t) + α (t)k] π/2} Γ {1 − [β (t) + α (t)k]} |ω|[β (t)+α (t)k]−1 .

[

]

Since F [u(τ + 1) − u(τ − 1)] = 2Sa(ω), the Fourier transform of C (τ , t) is given by (3.4). The proof completes.

(3.9) □

3.2. Asymptotic expressions of PSD For ω → 0, we have S(ω, t) =

∫

∞

|τ |−β (t) e−iωτ dτ = 2 sin [β (t)π/2] Γ [1 − β (t)] |ω|β (t)−1

−∞

= 2 sin {[H(t) − 1] π } Γ [3 − 2H(t)] |ω|

2H(t)−3

,

(3.10)

ω → 0.

Note 3.1. If X (t) is of LRD, 0.5
∫

∞

1 − |τ |α (t) e−iωτ dτ = 2πδ (ω) −

[

]

∞

=−

∞

|τ |α(t) e−iωτ dτ

−∞

−∞

∫

∫

|τ |α(t) e−iωτ dτ = 2 sin [α (t)π/2] Γ [α (t) + 1] |ω|−α(t)−1

(3.11)

−∞

= 2 sin {[2 − D(t)] π} Γ [5 − 2D(t)] |ω|2D(t)−5 ,

ω → ∞.

Note that we consider S(ω, t) for ω → ∞. Thus, δ (ω) = 0 in the above. Since 1 ≤ D(t)<2, the PSD of the mGC process for ω → ∞ is also a kind of 1/f noise but it merely relates to D(t) or α (t). 4. Computations of D(t) and H (t) 4.1. Computing D(t) of the mGC process Theorem 4.1. Denote by ε positive infinitesimal, i.e., ε → 0+ . Then, D(t) =

4 − {ln [X (t + ε ) − X (rt)] − ln [X (t + ε ) − X (t)]}

,

2 ln r where the equality is in the sense of the both sides have the same probability distribution.

(4.1)

Proof. The mGC process X (t) is said to be locally self-similar of order v for r > 0 if (Lim and Li [37, p. 2937], Adler [61]) X (s) − X (rt) = r v [X (s) − X (t)] ,

|t − s| → 0,

(4.2)

where the equality means that the both sides have the same probability distribution. When v = α (t), the above becomes X (s) − X (rt) = r α (t) [X (s) − X (t)] ,

| t − s| → 0 .

(4.3)

Let s = t + ε . Then, we have X (t + ε ) − X (rt) = r α (t) [X (t + ε ) − X (t)] = r 4−2D(t) [X (t + ε) − X (t)] .

(4.4)

Please cite this article as: M. Li, Multi-fractional generalized Cauchy process and its application to teletraffic, Physica A (2020) 123982, https://doi.org/10.1016/j.physa.2019.123982.

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Table 5.1 Four real traffic traces on the Ethernet measured by BC. Trace name

Starting time of measurement

Duration

Number of packets

pAug.TL pOct.TL OctExt.TL OctExt4.TL

11:25AM, 29Aug89 11:00AM, 05Oct89 11:46PM, 03Oct89 2:37PM, 10Oct89

52 min 29 min 34.111 h 21.095 h

1 1 1 1

million million million million

Table 5.2 Four wide-area TCP traces recorded by DEC. Trace name

Record date

Duration

Number of packets

DEC-pkt-1.TCP DEC-pkt-2.TCP DEC-pkt-3.TCP DEC-pkt-4.TCP

08Mar95 09Mar95 09Mar95 09Mar95

10PM–11PM 2AM–3AM 10AM–11AM 2PM–3PM

3.3 3.9 4.3 5.7

million million million million

Performing the logarithm operations on the both sides of (4.4) yields (4.1). This finishes the proof. □ Note 4.1. When ε is a small positive number, we have D(t) ≈

4 − {ln [X (t + ε ) − X (rt)] − ln [X (t + ε ) − X (t)]} 2 ln r

.

(4.5)

4.2. Computing H(t) of the mGC process Because C (τ , t) ≈ |τ |2−2H(t) for large τ , the ACF of the mGC process is equivalent to that of the modified multi-fractional Gaussian noise (Li and Zhao [65]). Therefore, the computation of H(t) of the mGC process may adopt that utilized in the multi-fractional Brownian motion (mfBm) introduced by Peltier and Levy-Vehel [66,67], also see Guevel and LevyVehel [68], Ayache et al. [69], Falconer and Levy-Vehel [70], Muniandy and Lim [71], Muniandy et al. [72]. To be precise,

√

H(t) = −

log( π /2Sn (j)) log(N − 1)

,

(4.6)

where Sn (j) =

m N −1

j+n ∑

|X (i + 1) − X (i)| ,

1 < n < N,

(4.7)

j=0

where m is the largest integer not exceeding N/n and t = j/(N − 1).

(4.8)

5. Case study The real traffic data provided by Bellcore (BC) in 1989 and Digital Equipment Corporation (DEC) in March 1995 were used for the work on traffic modeling and analysis from the point of view of fractals, see e.g., Paxson and Floyd [15], Leland et al. [20], Abry et al. [24], Li and Lim [36], Li [38], Willinger et al. [41], Feldmann et al. [42], Veitch et al. [44], Roughan et al. [73], Abry and Veitch [74], Li [75], Li et al. [76,77]. The research by Fontugne et al. [53] exhibits that stochastic properties of traffic remain the same from the early data by BC in 1989 to the recent data by the MAWI (Measurement and Analysis on the WIDE Internet) Working Group Traffic Archive (Japan) in 2015, respectively. In this case study, we shall reveal that, by using the mGC model, the property of Var[D(t)] ≫ Var[H(t)] holds for the traffic data by BC in 1989, DEC in 1995 to those by MAWI in 2019, providing a new evidence to support the point of view stated by Fontugne et al. [53] as well as Borgnat et al. [78]. 5.1. Data The file names of the real traffic traces used in this work are BC-pAug89.TL, BC-pOct89.TL, BC-Oct89Ext.TL, BCOct89Ext4.TL, which are Ethernet traffic collected at BC, DEC-PKT-1.TCP, DEC-PKT-2.TCP, DEC-PKT-3.TCP, DEC-PKT-4.TCP, which are wide-area TCP traffic recorded at DEC, MAWI-pkt-1.TCP MAWI-pkt-2.TCP MAWI-pkt-3.TCP, MAWI-pkt-4.TCP, which are wide-area TCP traffic measured by MAWI, see Tables 5.1–5.3. Please cite this article as: M. Li, Multi-fractional generalized Cauchy process and its application to teletraffic, Physica A (2020) 123982, https://doi.org/10.1016/j.physa.2019.123982.

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M. Li / Physica A xxx (xxxx) xxx Table 5.3 Four wide-area TCP traces recorded by MAWI. Trace name

Starting record time

Duration

Number of packets

MAWI-pkt-1.TCP MAWI-pkt-2.TCP MAWI-pkt-3.TCP MAWI-pkt-4.TCP

2:00PM, 2:00PM, 2:00PM, 2:00PM,

6.77208 min 6.65965 min 12.55740 min 12.66541 min

741 404 742 638 482 564 576 495

18Apr2019 19Apr2019 20Apr2019 21Apr2019

Table 5.4 Variances of D(i) and H(i) Trace name

Var[D(i)]

Var[H(i)]

pAug.TL pOct.TL OctExt.TL OctExt4.TL DEC-pkt-1.TCP DEC-pkt-2.TCP DEC-pkt-3.TCP DEC-pkt-4.TCP MAWI-pkt-1.TCP MAWI-pkt-2.TCP MAWI-pkt-3.TCP MAWI-pkt-4.TCP

0.093 0.125 0.072 0.116 0.104 0.112 0.130 0.119 0.118 0.128 0.173 0.173

1.143 9.152 3.416 7.953 1.352 1.550 2.203 1.768 2.208 1.619 6.076 9.232

× × × × × × × × × × × ×

Var[D(i)] ≫ Var[H(i)] 10−3 10−4 10−4 10−3 10−5 10−4 10−5 10−4 10−5 10−5 10−6 10−6

Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

5.2. Demonstrations Let x[t(i)] be the number of bytes in a packet at the time t(i) (i = 0, 1, 2, . . . ), where t(i) is the timestamp of the ith packet. In the case study, we consider traffic series x(i) that represents the packet size or packet length of the ith packet. Then, D(i) and H(i) are denoted as the fractal dimension and the Hurst parameter of the ith packet, respectively. Fig. 5.1 Shows the first 1024 points of 12 traffic traces, Fig. 5.2 indicates their D(i)s to characterize their local irregularity on a point-by-point basis, and Fig. 5.3 illustrates their H(i)s to demonstrate their global long-term persistence on a point-by-point basis. We summarize the values of Var[D(i)] and Var[H(i)] of 12 traces in the captions of Figs. 5.2–5.3 and also in Table 5.4. The results exhibit that Var[D(i)] is much greater than Var[H(i)], implying that the fluctuation of the local irregularity of traffic is considerably larger than that of its globally long-term persistence. Therefore, the present mGC model provides a new way to study the multi-fractal behavior of traffic and novel tool to investigate the local irregularity and LRD of traffic. 6. Discussions Li and Lim [36] proposed the results in traffic modeling with constant D and H, as well as varying D and H on an interval-by-interval basis based on the GC process. The case of D and H being constants corresponds to the mono-fractal GC process. On the other side, that D and H vary on an interval-by-interval basis corresponds to the case of multi-scale GC process. The present mGC process is a multi-fractal case, which implies that D and H vary on a point-by-point basis. Note that a challenging issue in traffic modeling and analysis is to explain its multi-fractal property in that traffic has highly local irregularity and robust LRD (Paxson and Floyd [15], Abry et al. [24], Willinger et al. [40,41], Feldmann et al. [42], Veitch et al. [44], Willinger and paxson [79]). The present mGC model has the time varying fractal dimension D(t) and the Hurst parameter H(t). It may separately characterize the local irregularity and LRD of traffic on a point-by-point basis. Therefore, it may yet be a new tool to describe the multi-fractal behavior of traffic. Note that it needs ε → 0 for the computation of D(t), see Theorem 4.1. However, in practice, a traffic trace x(i) (i = 0, 1, 2, . . .) is discrete. Thus, the minimum ε is 1 in the numerical computation of D(i) when (4.5) is used. That may cause computation errors in a way with respect to D(i). Our future work will be on a numeric computation how to improve the computation accuracy of D(i) of traffic in discrete time series. Although this research gives a multi-fractal model of traffic, it may be applied to fractal time series in other areas, e.g., those in [80–102]. Applying the present mGC model to other fractal time series is another future work. 7. Conclusions We have presented a traffic model of multi-fractional generalized Cauchy (mGC) process by introducing a time varying autocorrelation function (ACF) C (τ , t) in Theorem 2.1. Its asymptotic expressions for τ → ∞ and τ → 0 are given in (2.13) and (2.15), respectively. In addition, we have given the power spectrum density (PSD) of the mGC process in Theorem 3.1. Its asymptotic expressions for ω → 0 and ω → ∞ are given in (3.10) and (3.11), respectively. The mGC model contains Please cite this article as: M. Li, Multi-fractional generalized Cauchy process and its application to teletraffic, Physica A (2020) 123982, https://doi.org/10.1016/j.physa.2019.123982.

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Fig. 5.1. First 1024 data of traffic traces. (a) x(i) of pAug.TL. (b) x(i) of pOct.TL. (c) x(i) of OctExt.TL. (d) x(i) of OctExt4.TL. (e) x(i) of DEC-pkt-1.TCP. (f) x(i) of DEC-pkt-2.TCP. (g) x(i) of DEC-pkt-3.TCP. (h) x(i) of DEC-pkt-4.TCP. (i) x(i) of MAWI-pkt-1.TCP. (j) x(i) of MAWI-pkt-2.TCP. (k) x(i) of MAWI-pkt-3.TCP. (l) x(i) of MAWI-pkt-4.TCP.

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Fig. 5.2. First 1024 data of D(i) of traffic traces. (a) D(i) of pAug.TL with Var[D(i)] = 0.093. (b) D(i) of pOct.TL with Var[D(i)] = 0.125. (c) D(i) of OctExt.TL with Var[D(i)] = 0.072. (d) D(i) of OctExt4.TL with Var[D(i)] = 0.116. (e) D(i) of DEC-pkt-1.TCP with Var[D(i)] = 0.104. (f) D(i) of DEC-pkt-2.TCP with Var[D(i)] = 0.112. (g) D(i) of DEC-pkt-3.TCP with Var[D(i)] = 0.130. (h) D(i) of DEC-pkt-4.TCP with Var[D(i)] = 0.119. (i) D(i) of MAWI-pkt-1.TCP with Var[D(i)] = 0.118. (j) D(i) of MAWI-pkt-2.TCP with Var[D(i)] = 0.128. (k) D(i) of MAWI-pkt-3.TCP with Var[D(i)] = 0.173. (l) D(i) of MAWI-pkt-4.TCP with Var[D(i)] = 0.173.

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Fig. 5.3. First 1024 data of H(i) of traffic traces. (a) H(i) of pAug.TL with Var[H(i)] = 1.143 × 10−3 . (b) H(i) of pOct.TL with Var[H(i)] = 9.152 × 10−4 . (c) H(i) of OctExt.TL with Var[H(i)] = 3.416 × 10−4 . (d) H(i) of OctExt4.TL with Var[H(i)] = 7.953 × 10−3 . (e) H(i) of DEC-pkt-1.TCP with Var[H(i)] = 1.352 × 10−5 . (f) H(i) of DEC-pkt-2.TCP with Var[H(i)] = 1.550 × 10−4 . (g) H(i) of DEC-pkt-3.TCP with Var[H(i)] = 2.203 × 10−5 . (h) H(i) of DEC-pkt-4.TCP with Var[H(i)] = 1.768 × 10−4 . (i) H(i) of MAWI-pkt-1.TCP with Var[H(i)] = 2.082 × 10−5 . (j) H(i) of MAWI-pkt-2.TCP with Var[H(i)] = 1.619 × 10−5 . (k) H(i) of MAWI-pkt-3.TCP with Var[H(i)] = 6.076 × 10−6 . (l) H(i) of MAWI-pkt-4.TCP with Var[H(i)] = 9.232 × 10−6 .

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separate representations of the time varying fractal dimension D(t) and the time varying Hurst parameter H(t) on a point-by-point basis. We have proposed the computation formula of D(t) in Theorem 4.1. The case study has shown that Var[D(t)] ≫ Var[H(t)], providing a new view of multi-fractal behavior of traffic. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under the project grant numbers, 61672238 and 61272402. The author highly appreciates the ACM SIGCOMM for the real traffic data. Specifically, thanks go to Will Leland ([email protected]) and Dan Wilson ([email protected]) for the traffic traces measured by Bellcore in Table 5.1, Jeff Mogul ([email protected]) of Digital’s Western Research Lab (WRL) [20,103,104], Digital Equipment Corporation (DEC) for the traffic traces DEC-pkt-TCP-n (n = 1, . . ., 4) in Table 5.2 [15,105], the MAWI Working Group traffic archive of the WIDE Project (Japan) for the traces in Table 5.3 [106,107]. I am grateful to Miss Ying Dong for her collecting data by MAWI. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of NSFC or the Chinese government. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

J.D. Gibson, The Communications Handbook, IEEE Press, 1997. A.K. Erlang, Telefon-ventetider. et stykke sandsynlighedsregning, Mat. Tidsskr. B (1920) 25–42. E. Brockmeyer, H.L. Halstrom, A. Jensen, The life of A. K. Erlang, Trans. Danish Acad. Tech. Sci. (2) (1948) 23–100. F. Le Gall, One moment model for telephone traffic, Appl. Math. Model. 6 (6) (1982) 415–423. P. Lin, B. Leon, C. Stewart, Analysis of circuit-switched networks employing originating-office control with spill-forward, IEEE Trans. Commun. 26 (6) (1978) 754–766. D. Manfield, T. Downs, On the one-moment analysis of telephone traffic networks, IEEE Trans. Commun. 27 (8) (1979) 1169–1174. M. Reiser, Performance evaluation of data communication systems, Proc. IEEE 70 (2) (1982) 171–196. H. Akimaru, K. Kawashima, Teletraffic: Theory and Applications, Springer-Verlag, 1993. Z. Bojkovic, M. Bakmaz, B. Bakmaz, Originator of teletraffic theory, Proc. IEEE 98 (1) (2010) 123–127. R.B. Cooper, Introduction to Queueing Theory, Elsevier, 1981. F.A. Tobagi, M. Gerla, R.W. Peebles, E.G. Manning, Modeling and measurement techniques in packet communication networks, Proc. IEEE 66 (11) (1978) 1423–1447. R. Jain, S. Routhier, Packet trains-measurements and a new model for computer network traffic, IEEE J. Sel. Areas Commun. 4 (6) (1986) 986–995. H. Michiel, K. Laevens, Teletraffic engineering in a broad-band era, Proc. IEEE 85 (12) (1997) 2007–2033. M. Li, P. Borgnat, Foreword to the special issue on traffic modeling, its computations and applications, Telecommun. Syst. 43 (3–4) (2010) 145–146. V. Paxson, S. Floyd, Wide area traffic: The failure of Poison modeling, IEEE/ACM Trans. Netw. 3 (3) (1995) 226–244. J. Beran, R. Shernan, M.S. Taqqu, W. Willinger, Long-range dependence in variable bit-rate video traffic, IEEE Trans. Commun. 43 (2-3-4) (1995) 1566–1579. J. Beran, Statistics for Long-Memory Processes, Chapman & Hall, 1994. J. Beran, Statistical methods for data with long-range dependence, Statist. Sci. 7 (4) (1992) 404–416. I. Csabai, 1/f noise in computer network traffic, J. Phys. A: Math. Gen. 27 (12) (1994) L417–L421. W.E. Leland, M.S. Taqqu, W. Willinger, D.V. Wilson, On the self-similar nature of ethernet traffic (extended version), IEEE/ACM Trans. Netw. 2 (1) (1994) 1–15. B.B. Mandelbrot, J.W. van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (4) (1968) 422–437. B. Tsybakov, N.D. Georganas, Self-similar processes in communications networks, IEEE Trans. Inform. Theory 44 (5) (1998) 1713–1725. A. Adas, Traffic models in broadband networks, IEEE Commun. Mag. 35 (7) (1997) 82–89. P. Abry, R. Baraniuk, P. Flandrin, R. Riedi, D. Veitch, Multiscale nature of network traffic, IEEE Signal Process. Mag. 19 (3) (2002) 28–46. T. Karagiannis, M. Molle, M. Faloutsos, Long-range dependence: Ten years of Internet traffic modeling, IEEE Internet Comput. 8 (5) (2004) 57–64. W.-B. Gong, Y. Liu, V. Misra, D. Towsley, Self-similarity and long range dependence on the Internet: A second look at the evidence, origins and implications, Comput. Netw. 48 (3) (2005) 377–399. Ian W.C. Lee, A.O. Fapojuwo, Stochastic processes for computer network traffic modeling, Comput. Commun. 29 (1) (2005) 1–23. S. Stoev, M.S. Taqqu, C. Park, J.S. Marron, On the wavelet spectrum diagnostic for Hurst parameter estimation in the analysis of Internet traffic, Comput. Netw. 48 (3) (2005) 423–445. S. Bregni, L. Jmoda, Accurate estimation of the Hurst parameter of long-range dependent traffic using modified Allan and Hadamard variances, IEEE Trans. Commun. 56 (11) (2008) 1900–1906. I. Lokshina, Study on estimating probabilities of buffer overflow in high-speed communication networks, Telecommun. Syst. 62 (2) (2016) 269–302. J.-S.R. Lee, S.-K. Ye, H.-D.J. Jeong, ATMSim: An anomaly teletraffic detection measurement analysis simulator, Simul. Model. Pract. Theory 49 (2014) 98–109. M. Pinchas, Cooperative multi PTP slaves for timing improvement in an fGn environment, IEEE Commun. Lett. 22 (7) (2018) 1366–1369. B.B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, New York, 1982. M. Li, Fractal time series — a tutorial review, Math. Probl. Eng. 2010 (2010) 157264, 26 pages. T. Gneiting, M. Schlather, Stochastic models that separate fractal dimension and Hurst effect, SIAM Rev. 46 (2) (2004) 269–282. M. Li, S.C. Lim, Modeling network traffic using generalized Cauchy process, Physica A 387 (11) (2008) 2584–2594. S.C. Lim, M. Li, Generalized Cauchy process and its application to relaxation phenomena, J. Phys. A: Math. Gen. 39 (12) (2006) 2935–2951. M. Li, Record length requirement of long-range dependent teletraffic, Physica A 472 (2017) 164–187. M. Li, J.-Y. Li, Generalized Cauchy model of sea level fluctuations with long-range dependence, Physica A 484 (2017) 309–335. W. Willinger, R. Govindan, S. Jamin, V. Paxson, S. Shenker, Scaling phenomena in the Internet critically, Proc. Natl. Acad. Sci. USA 99 (Suppl. 1) (2002) 2573–2580.

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M. Li / Physica A xxx (xxxx) xxx

13

[41] W. Willinger, M.S. Taqqu, R. Sherman, D.V. Wilson, Self-similarity through high-variability: Statistical analysis of Ethernet LAN traffic at the source level, IEEE/ACM Trans. Netw. 5 (1) (1997) 71–86. [42] A. Feldmann, A.C. Gilbert, W. Willinger, T.G. Kurtz, The changing nature of network traffic: Scaling phenomena, ACM SIGCOMM Comput. Commun. Rev. 28 (2) (1998) 5–29. [43] M. Li, W. Zhao, Representation of a stochastic traffic bound, IEEE Trans. Parallel Distrib. Syst. 21 (9) (2010) 1368–1372. [44] D. Veitch, N. Hohn, P. Abry, Multifractality in TCP/IP traffic: The case against, Comput. Netw. 48 (3) (2005) 293–313. [45] J.W. de G. Stênico, L.L. Ling, General solution to the losses estimation for multifractal traffic, J. Franklin Inst. B 351 (10) (2014) 4904–4922. [46] Y. Xu, H. Feng, Revisiting multifractality of TCP traffic using multifractal detrended fluctuation analysis, J. Stat. Mech. Theory Exp. 2014 (2014) P02007. [47] L.O. Ostrowsky, N.L.S. da Fonseca, C.A.V. Melo, A multiscaling traffic model for UDP streams, Simul. Model. Pract. Theory 26 (2012) 32–48. [48] F.H.T. Vieira, F.G.C. Rocha, J.A. dos Santos, Loss probability estimation and control for OFDM/TDMA wireless systems considering multifractal traffic characteristics, Comput. Commun. 35 (2) (2012) 263–271. [49] E. Rocha, P. Salvador, A. Nogueira, Can multiscale traffic analysis be used to differentiate Internet applications? Telecommun. Syst. 48 (1–2) (2011) 19–30. [50] A. Budhiraja, X. Liu, Multiscale diffusion approximations for stochastic networks in heavy traffic, Stochastic Process. Appl. 121 (3) (2011) 630–656. [51] F.H.T. Vieira, L.L. Lee, Adaptive wavelet-based multifractal model applied to the effective bandwidth estimation of network traffic flows, IET Commun. 3 (6) (2009) 906–919. [52] M. Masugi, T. Takuma, Multi-fractal analysis of IP-network traffic for assessing time variations in scaling properties, Physica D 225 (2) (2007) 119–126. [53] R. Fontugne, P. Abry, K. Fukuda, D. Veitch, K. Cho, P. Borgnat, H. Wendt, Scaling in Internet traffic: A 14 year and 3 day longitudinal study, with multiscale analyses and random projections, IEEE/ACM Trans. Netw. 25 (4) (2017) 2152–2165. [54] V.J. Ribeiro, Z.-L. Zhang, S. Moon, C. Diot, Small-time scaling behavior of Internet backbone traffic, Comput. Netw. 48 (3) (2005) 315–334. [55] N.C. Nigam, Introduction to Random Vibrations, The MIT Press, 1983. [56] J.S. Bendat, A.G. Piersol, Random Data: Analysis and Measurement Procedure, fourth ed., John Wiley & Sons, 2010. [57] M.B. Priestley, Spectral Analysis and Time Series, Academic Press, 1981. [58] P. Hall, R. Roy, On the relationship between fractal dimension and fractal index for stationary stochastic processes, Ann. Appl. Probab. 4 (1) (1994) 241–253. [59] G. Chan, P. Hall, D.S. Poskitt, Periodogram-based estimators of fractal properties, Ann. Statist. 23 (5) (1995) 1684–1711. [60] J.T. Kent, A.T. Wood, Estimating the fractal dimension of a locally self-similar Gaussian process by using increments, J. R. Stat. Soc. Ser. B 59 (3) (1997) 679–699. [61] A.J. Adler, The Geometry of Random Fields, Wiley, New York, 1981. [62] M.B. Priestley, H. Tong, On the analysis of bivariate non-stationary processes, J. R. Stat. Soc. Ser. B 35 (2) (1973) 153–166. [63] M. Li, S.C. Lim, Power spectrum of generalized Cauchy process, Telecommun. Syst. 43 (3–4) (2010) 219–222. [64] I.M. Gelfand, K. Vilenkin, Generalized Functions, Vol. 1, Academic Press, New York, 1964. [65] M. Li, W. Zhao, Quantitatively investigating locally weak stationarity of modified multifractional Gaussian noise, Physica A 391 (24) (2012) 6268–6278. [66] R.F. Peltier, J. Levy-Vehel, Multifractional Brownian motion: Definition and preliminaries results, INRIA RR 2645, 1995. [67] R.F. Peltier, J. Levy-Vehel, A new method for estimating the parameter of fractional Brownian motion, INRIA RR 2696, 1994. [68] R. Le Guevel, J. Levy-Vehel, Incremental moments and holder exponents of multifractional multistable processes, INRIA, 2010. [69] A. Ayache, S. Cohen, J. Levy-Vehel, The covariance structure of multifractional Brownian motion, with application to long range dependence, in: ICASSP, 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, Vol. 6, 2000, pp. 3810–3813. [70] K.J. Falconer, J. Levy-Vehel, Multifractional, multistable, and other processes with prescribed local form, J. Theoret. Probab. 22 (2) (2009) 375–401. [71] S.V. Muniandy, S.C. Lim, On some possible generalizations of fractional Brownian motion, Phys. Lett. A 266 (2–3) (2000) 140–145. [72] S.V. Muniandy, S.C. Lim, R. Murugan, Inhomogeneous scaling behaviors in Malaysia foreign currency exchange rates, Physica A 301 (1–4) (2001) 407–428. [73] M. Roughan, D. Veitch, P. Abry, Real time estimation of the parameters of long-range dependence, IEEE/ACM Trans. Netw. 8 (4) (2000) 467–478. [74] P. Abry, D. Veitch, Wavelet analysis of long-range dependent traffic, IEEE Trans. Inform. Theory 44 (1) (1998) 2–15. [75] M. Li, Modeling autocorrelation functions of long-range dependent teletraffic series based on optimal approximation in Hilbert space—A further study, Appl. Math. Model. 31 (3) (2007) 625–631. [76] M. Li, W. Zhao, W. Jia, D.Y. Long, C.-H. Chi, Modeling autocorrelation functions of self-similar teletraffic in communication networks based on optimal approximation in Hilbert space, Appl. Math. Model. 27 (3) (2003) 155–168. [77] M. Li, W.J. Jia, W. Zhao, Correlation form of timestamp increment sequences of self-similar traffic on ethernet, Electron. Lett. 36 (19) (2000) 1668–1669. [78] P. Borgnat, G. Dewaele, K. Fukuda, P. Abry, K. Cho, Seven years and one day: Sketching the evolution of Internet traffic, in: Proceedings of the 28th IEEE INFOCOM 2009, Rio de Janeiro (Brazil), 2009, pp. 711–719. [79] W. Willinger, V. Paxson, Where mathematics meets the Internet, Notices Amer. Math. Soc. 45 (8) (1998) 961–970. [80] X.H. Chang, M.L. Stein, Decorrelation property of discrete wavelet transform under fixed-domain asymptotics, IEEE Trans. Inform. Theory 59 (12) (2013) 8001–8013. [81] I.A. Lubashevsky, Truncated Lévy flights and generalized Cauchy processes, Eur. Phys. J. B 82 (2) (2011) 189–195. [82] S.V. Muniandy, J. Stanslas, Modelling of chromatin morphologies in breast cancer cells undergoing apoptosis using generalized Cauchy field, Comput. Med. Imaging Graph. 32 (7) (2008) 631–637. [83] B. Lacaze, A stochastic model for propagation through tissue, IEEE Trans. Ultrasonics Ferroelectr. Frequency Control 56 (10) (2009) 2180–2186. [84] J. de Coninck, F. Dunlop, T. Huillet, On the correlation structure of some random point processes on the line, Physica A 387 (4) (2008) 725–744. [85] A. Stanislavsky, K. Weron, Subordination scenario of the Cole-Davidson relaxation, Phys. Lett. A 373 (30) (2009) 2520–2524. [86] A. Ayache, F. Roueff, Y.M. Xiao, Linear fractional stable sheets: Wavelet expansion and sample path properties, Stochastic Process. Appl. 119 (4) (2009) 1168–1197. [87] A. Ayache, D. Wu, Y.M. Xiao, Joint continuity of the local times of fractional Brownian sheets, Ann. Inst. Henri Poincare B 44 (3) (2008) 727–748. [88] A.L. Goldberger, L.A.N. Amaral, J.M. Hausdorff, P. Ch. Ivanov, C.-K. Peng, H.E. Stanley, Fractal dynamics in physiology: Alterations with disease and aging, Proc. Natl. Acad. Sci. USA 99 (Suppl. 1) (2002) 2466–2472. [89] B. Podobnik, D.F. Fu, H.E. Stanley, P.Ch. Ivanov, Power-law autocorrelated stochastic processes with long-range cross-correlations, Eur. Phys. J. B 56 (1) (2007) 47–52. [90] C. Cattani, Harmonic wavelet approximation of random, fractal and high frequency signals, Telecommun. Syst. 43 (3–4) (2010) 207–217.

Please cite this article as: M. Li, Multi-fractional generalized Cauchy process and its application to teletraffic, Physica A (2020) 123982, https://doi.org/10.1016/j.physa.2019.123982.

14

M. Li / Physica A xxx (xxxx) xxx

[91] C. Cattani, A. Ciancio, E. Laserra, I. Bochicchio, On the fractal distribution of primes and prime-indexed primes by the binary image analysis, Physica A 460 (2016) 222–229. [92] C.G. Gu, C.P. Coomans, K. Hu, F.A.J.L. Scheer, H.E. Stanley, J.H. Meijer, Lack of exercise leads to significant and reversible loss of scale invariance in both aged and young mice, Proc. Natl. Acad. Sci. USA 112 (8) (2015) 2320–2324. [93] C. Toma, Wavelets-computational aspects of Sterian-realistic approach to uncertainty principle in high energy physics-a transient approach, Adv. High Energy Phys. 2013 (2013) 735452. [94] E.G. Bakhoum, C. Toma, Transient aspects of wave propagation connected with spatial coherence, Math. Probl. Eng. 2013 (2013). [95] J. Lévy-Véhel, Beyond multifractional Brownian motion: New stochastic models for geophysical modeling, Nonlinear Process. Geophys. 20 (5) (2013) 643–655. [96] K. Kiyono, Establishing a direct connection between detrended fluctuation analysis and Fourier analysis, Phys. Rev. E 92 (4) (2015) 042925. [97] M. Pinchas, Residual ISI obtained by blind adaptive equalizers and fractional noise, Math. Probl. Eng. 2013 (2013) 972174, 11 pages. [98] M. Pinchas, Symbol error rate for non-blind adaptive equalizers applicable for the SIMO and fGn case, Math. Probl. Eng. 2014 (2014) 606843, 11 pages. [99] S.V. Muniandy, W.X. Chew, C.S. Wong, Fractional dynamics in the light scattering intensity fluctuation in dusty plasma, Phys. Plasmas 18 (1) (2011) 013701. [100] X. Gabaix, P. Gopikrishnan, V. Plerou, H.E. Stanley, A theory of power-law distributions in financial market fluctuations, Nature 423 (6937) (2003) 267–270. [101] I. Tejado, S.H. Hosseinnia, B.M. Vinagre, X. Song, Y.Q. Chen, Dealing with fractional dynamics of IP network delays, Int. J. Bifurcation Chaos 22 (4) (2012) 1250089, (13 pp). [102] Y. Gao, F. Villecco, M. Li, W. Song, Multi-scale permutation entropy based on improved LMD and HMM for rolling bearing diagnosis, Entropy 19 (4) (2017) 176, (10 pp). [103] W.E. Leland, D.V. Wilson, High time-resolution measurement and analysis of LAN traffic: Implications for LAN interconnection, in: Proc. IEEE INFOCOM ’91, Bal Harbour, FL, USA, 7–11 April 1991, pp. 1360–1366. [104] H.J. Fowler, W.E. Leland, Characteristics, with implications for broadband network congestion management, IEEE J. Sel. Areas Commun. 9 (7) (1991) 1139–1149. [105] V. Paxson, Fast, approximate synthesis of fractional Gaussian noise for generating self-similar network traffic, ACM SIGCOMM Comput. Commun. Rev. 27 (5) (1997) 5–18. [106] K. Cho, Recursive lattice search: Hierarchical heavy hitters revisited, in: ACM IMC 2017, London, UK, 1–3 Nov. 2017, pp. 283–289. [107] M. Kato, K. Cho, M. Honda, H. Tokuda, Monitoring the dynamics of network traffic by recursive multi-dimensional aggregation, in: OSDI2012 MAD Workshop, Hollywood, CA, 8–10 Oct. 2012, 7 pp.

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