modeling network traffic using cauchy correlation ...

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Modern Physics Letters B, Vol. 19, No. 17 (2005) 829–840 c World Scientific Publishing Company

MODELING NETWORK TRAFFIC USING CAUCHY CORRELATION MODEL WITH LONG-RANGE DEPENDENCE

Mod. Phys. Lett. B 2005.19:829-840. Downloaded from www.worldscientific.com by NATIONAL UNIVERSITY OF SINGAPORE on 02/08/16. For personal use only.

MING LI School of Information Science & Technology, East China Normal University, Shanghai 200026, P. R. China [email protected] ming [email protected] S. C. LIM Faculty of Engineering, Multimedia University, 63100 Cyberjaya, Selangor, Malaysia [email protected] Received 3 February 2005 Much attention has been given to the long-range dependence and fractal properties in network traffic engineering, and these properties are also widely observed in many fields of science and technologies. Traffic time series is conventionally characterized by its fractal dimension D, which is a measure for roughness, and by the Hurst parameter H, which is a measure for long-range dependence, see for examples (Refs. 10–12). Each property has been traditionally modeled and explained by self-affine random functions, such as fractional Gaussian noise (FGN)1,10 – 13,18,22 – 28 and fractional Brownian motion (FBM),6,7 where a linear relationship between D and H, say D = 2 − H for onedimensional series, links the two properties. The limitation of single parameter models (e.g., FGN) in long-range dependent (LRD) traffic modeling has been noticed as can be seen from Refs. 1, 18 and 25. Hence, models which can provide good fitting of LRD traffic for both short-term lags and long-term ones are worth studying due to the importance of accurate models of traffic in network communications.13 This letter utilizes a statistical model called the Cauchy correlation model to model LRD traffic. This model characterizes D and H separately, and it allows any combination of two within the constraint of LRD condition. It is a new power-law correlation model for LRD traffic modeling with its local and global behavior decoupling. Its flexibility in data modeling in comparison with a single parameter model of FGN is briefly discussed, and applications to LRD traffic modeling demonstrated. Keywords: Long-range dependence; fractal dimension; Hurst parameter; Cauchy correlation model; network traffic; time series. PACS Number(s): 02.50Ey, 05.45Tp

1. Introduction Long-range dependent time series analysis gains applications in many fields of science and technologies, see for examples (Refs. 1–9), including network traffic time 829


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series (traffic for short).10 – 13,17,18,22 – 30,56 – 59 An accurate model of traffic is essential to network communications for performance analysis.13 For instance, an accurate correlation model is crucial to queuing analysis of systems.14 – 16 As is well known, long-range dependence is characterized by the Hurst parameter H. The fractal dimension D ∈ [n, n+1) for a surface in Rn conventionally relates to H with the linear relationship D +H = n+1. In what follows, only one-dimensional series is considered. Hence,


H = 2−D,

(1.1)

see for examples, Refs. 4, 6, 10, 12, 20, 36, 40–50. Based on Eq. (1.1), local properties are reflected in global ones. Recent work20,21,30,38 discusses a correlation structure with two parameters. With the Cauchy correlation model explained in Refs. 20 and 21, D and H of a given series are separately characterized instead of being linearly related as that of (1.1). This makes it become more flexible to accurately fit LRD series for both shortterm and long-terms lags. To the best of our knowledge, this is the first attempt in applying it to model LRD traffic from a view of the Cauchy class. This letter is organized as follows. Section 2 gives a brief summary of the research background. Section 3 discusses the Cauchy correlation model with longrange dependence. Section 4 demonstrates its application to LRD traffic modeling. Conclusions are given in Sec. 5. 2. Background A stationary series x is LRD if its autocorrelation function (ACF) r(τ ) has the asymptotic property given by r(τ ) ∼ cτ 2H−2 (τ → ∞),

H ∈ (0.5, 1)

(2.1a)

where c > 0 is a constant, ∼ stands for asymptotic equivalence and the parameter H is called the Hurst parameter (Ref. 1, p. 42). As r(τ ) is an even function, (2.1a) can be expressed by r(τ ) ∼ c|τ |2H−2 (τ → ∞),

H ∈ (0.5, 1) .

(2.1b)

From (2.1), we see that H characterizes the long-range dependence (a global property) of x. Let S(ω) be the power spectrum of x. Then, Z ∞ r(τ )e−jωτ dτ , ω ∈ (−∞, ∞) S(ω) = F(r) = −∞

where F is the operator of the Fourier transform. In the domain of generalized functions (Ref. 31, p. 43), we obtain F(|τ |−(2−2H) ) = 2 cos

π(2H − 1) (2H − 2)!|ω|−(2H−1) . 2

(2.2)


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Reference 32 introduces fractional Brownian motions (FBM) and fractional Gaussian noise (FGN). Let B(t), t ∈ (0, ∞), be Wiener Brownian motion.33 Let BH (t) be FBM. Let Γ(·) be Gamma function. Then, BH (t) is a function transformed from B(t) such that Z 0    H−0.5 H−0.5  [(t − u) − (−u) ]dB(u)      1 −∞ BH (t) − BH (0) = . (2.3) Z t  Γ(H + 1/2)      H−0.5  +  (t − u) dB(u)


0

Let G(t) be the increment series of BH (t):

G(t) = BH (t + a) − BH (t)

(2.4)

where a is a real number. Then, G(t) is called FGN. The ACF of FGN in the continuous case is given by σ2 |τ + 1|2H − 2|τ |2H + |τ − 1|2H , 2

where σ 2 =

Γ(1−2H) cos(Hπ) Hπ

. For τ ≥ 0, one has

σ2 [(τ + 1)2H − 2τ 2H + (τ − 1)2H ] . 2 In the discrete time case,

and

(2.5)

σ2 |k + 1|2H − 2|k|2H + |k − 1|2H , 2

σ2 [(k + 1)2H − 2k 2H + (k − 1)2H ], 2 In the normalized case,

(2.6)

k ∈ I (the integer set)

(2.6a)

k = 0, 1, 2, · · · .

(2.6b)

0.5[(k + 1)2H − 2k 2H + (k − 1)2H ] , G(k) .

(2.7)

This letter considers LRD traffic and H ∈ (0.5, 1) is assumed accordingly. Though ACF of FGN is a model widely used in traffic engineering, see e.g., (Refs. 10–13, 18, 19, 22–28, 56–59), the limitation of FGN in accurately modeling LRD traffic has been noticed. For example, it is noted that “it might be difficult to characterize the correlations over the entire traffic traces with a single Hurst parameter” and considered that “further work is required to fully understand the correlational structure of wide-area traffic (Ref. 18, last sentence, paragraph 4, §7.4)”. It is also remarked that FGN is too narrow for modeling actual traffic (Ref. 25, Sec. II). In the field of LRD series modeling, the weakness by using models characterized by one single parameter for distinguishing short-term and long-term correlation effects is investigated in Ref. 34, while the limitation of models with a single parameter for accurately modeling is explained in Ref. 35. As a matter of fact, “fractal


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dimension is originally a local property, notwithstanding the fact that in the essay 37 the local properties are reflected in the global ones”. Aiming at achieving a whole correlation structure to model LRD series, (Ref. 1, pp. 101–102) suggests developing a sufficiently flexible class of parametric correlation models. In this regard, Ref. 30 proposes a two-parameter correlation form given by (|k|α + 1)2H−2 ,

α ∈ (0, 1),

H ∈ (0.5, 1) ,

(2.8)


which is further written in Ref. 38 as (|k|a2 + 1)−a1 ,

a1 , a2 ∈ (0, 1] .

(2.9)

Equation (2.9) in fact belongs to the class of the Cauchy correlation model as that in Refs. 20 and 21.

3. Cauchy Correlation Model with Long-Range Dependence and Brief Discussions 3.1. Cauchy correlation model with long-range dependence Rewrite the expression (2.1b) by r(τ ) ∼ c|τ |−b (τ → ∞),

b ∈ (0, 1) .

(3.1)

Then, the process is said to have long-range dependence with the Hurst parameter H =1−

b . 2

(3.2)

On the other hand, the local properties of realizations can be determined by the behavior of r(τ ) for r → 0. Describe this by 1 − r(τ ) ∼ |τ |−a (τ → 0),

a ∈ (0, 2] .

(3.3)

Then, with the probability one,51,52 we have D =2−

a . 2

(3.4)

Combining (3.1) with (3.3) yields the following Cauchy correlation model with long-range dependence r(τ ) = (1 + |τ |a )−b/a ,

a ∈ (0, 2],

b ∈ (0, 1)

(3.5)

where the fractal dimension and the Hurst parameter are separately characterized by (3.2) and (3.4), respectively, see Refs. 20 and 21 for details.


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3.2. Brief discussions for the flexibility of Cauchy correlation model The expression 0.5[(τ + 1)2H − 2τ 2H + (τ − 1)2H ] is the finite second-order difference of 0.5τ 2H . As can be seen from Ref. 55, approximating it with the second-order differential of 0.5τ 2H yields 0.5[(τ + 1)2H − 2τ 2H + (τ − 1)2H ] ≈ H(2H − 1)τ 2H−2 .

(3.6)


Without loss of generality, we consider the discrete and normalized ACF structure below. As |k|2H−2 ∼ (|k| + 1)2H−2 ,

k → ∞,

(3.7)

we say that x is a stationary LRD process if r(k) ∼ (|k| + 1)2H−2 ,

k → ∞,

H ∈ (0.5, 1) .

(3.8)

On the other hand, for k 6= 0, we have |k|2H−2 ≈ (|k| + 1)2H−2 , g(k) .

(3.9)

The above expression shows that g(k) is an approximation of G(k). A physically measured series is usually a single history and has finite length. Without loss of generality, the maximum possible length of a series is assumed to 2 be a space containing all ACFs, including ACFs of be N ∈ I+ (= 1, 2, . . .). Let lN real LRD traffic. Let r be an ACF ofqa real LRD traffic series. Define the norm of p PN −1 2 r by inner product krk = hr, ri = k=0 |r| . Then, 2 lN = {r; krk < ∞}

(3.10)

is a Hilbert space. Lemma 3.1. (Refs. 28, 53, 54) (Existence of a unique minimizing element in Hilbert space). Let H be a Hilbert space and M be a closed convex subset of H. Let x ∈ H, x ∈ / M. Then there exists a unique element xˆ ∈ M satisfying kx − x ˆk = inf y∈M kx − yk. 2 Statement 3.1. Let r ∈ lN be an ACF of a real-traffic series. Let S be a closed sub2 space of lN . Then, there exists a unique R ∈ S such that kr − Rk = inf s∈S kr − sk. 2 Proof. lN is a Hilbert space and S is its closed convex subset. According to 2 Lemma 3.1, for any ACF of real-traffic series r ∈ lN , there exists a unique ACF 2 rˆ ∈ S ⊆ lN such that

kr − rˆk = inf kr − sk . s∈S

q P p N −1 2 Let e = R−r be error. Denote kek = he, ei = N1 k=0 |e| . Let F (e) = kek. Then, F (e) is convex. The optimal approximation of r in S can be expressed by R = arg min F (e),

2 r ∈ lN ,

R∈S.

(3.11)


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Suppose R has two parameters R(k) = R(k; a1 , a2 ) .

(3.12)

Then, the following cost function is of two dimensions: 1 X J(a1 , a2 ) = [R(k) − r(k)]2 . N

(3.13)


k

The partial derivative of J, with respect to two parameters, will be zero when J is minimum and has components ∂R 2 X ∂J (R − r) = , j = 1, 2 . ∂aj N ∂aj k

∂J = 0. Then, R(k; a10 , a20 ) is the optimal apLet (a10 , a20 ) be the solution of ∂a j proximation of r in S. Now, let G1 be the set containing ACFs of LRD series in the form (|τ | + 1)2H−2 . Then,

G1 = {r; r = (|k| + 1)2H−2 , krk < ∞} .

(3.14)

Therefore, an element in G1 is an approximation of an ACF of FGN. Let a = a1 and a2 = ab . We can then construct an extension of G1 as follows. Statement 3.2. The following G2 is an extension of G1 : G2 = {r; r = (|k|a + 1)−b/a , a ∈ (0, 2], b ∈ (0, 1), krk < ∞} .

(3.15)

Proof. (|k|a + 1)b/a equals to (|k| + 1)2H−2 for a = 1 and b = 2 − 2H. Thus, G2 ⊃ G 1 . 2 2 Let K be a subset of lN . Denote the distance from r ∈ lN to K by d(r, K) = inf s∈K d(r, s). Then, one has

d(r; G2 ) ≤ d(r; G1 ) .

(3.16)

The expression (3.16) exhibits the flexibility of the Cauchy correlation model in LRD traffic modeling. Since (|k|a + 1)−b/a ∼ (|k|)−b as k → ∞, it is non-summable and obeys power law. 4. Application of Cauchy Correlation Model to LRD Traffic Modeling Considering an approximation of r in G2 , we construct the cost function below 1 X 2 (R − r)2 , R ∈ G 2, r ∈ lN . J(a, b) = N k

Denote (a0 , b0 ) as the solution of arg min J(a, b).

∂J ∂a

= 0 and

∂J ∂b

= 0. Then, (a0 , b0 ) =

Fig. 5. 4. Fitting the data the Cauchy class: class. dot line, r(k); solid line, R1(k). Fig. Modeled ACFin R1(k) in the Cauchy September 14, 2005 9:30 WSPC/147-MPLB

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Fig. 6. 5. Fitting FGN:class: dot line, r(k); r(k); solidsolid line,line, G1(k). Fig. Fittingthe thedata databased in theon Cauchy dot line, R1(k). Table Fig. 1: 6. Modeling Fitting the results. data based on FGN: dot line, r(k); solid line, G1(k). Table 1: Modeling results.

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Fig. 1. Traffic series: NUS-TCP-1. Fig. 1.

Traffic series: NUS-TCP-1.

Fig. 1. Traffic series: NUS-TCP-1. 1

r(k) (log)

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r(k) (log)


i

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Fig. 2.

Measured ACF r(k).

Fig. 2. Measured ACF r(k). Fig. 2. Measured ACF r(k).

For the purpose of data modeling, we consider discrete series. Therefore, (3.5) can be written by R(k) = (1 + |k|a )−b/a .

(4.1)

Let x[t(i)] be a traffic series, indicating the number of bytes in a packet at time t(i) (i = 0, 1, 2, . . .). Then, x(i) is a series representing the number of bytes in the ith packet. We carry out the ACF modeling of x(i) of real traffic DEC-PKT-n.tcp (n = 1, 2) collected at Digital Equipment Corporation in March 1995, and NUSTCP-n (n = 1, 2) recorded at National University of Singapore in August 2003, then compare the modeling results based on (4.1) and FGN. Denote the measured ACF of x(i) by r(k). Let R1(k) and G1(k) be the modeled ACFs of x(i) based on (4.1) and (2.7), respectively. Let M 2 (R1) = E[(R1 − r)2 ] and M 2 (G1) = E[(G1 − r)2 ] be the 16 mean square errors by using R(k) and G(k), 16 respectively. Figure 1 shows x(i) of NUS-TCP-1.tcp. Figure 2 is the plot of measured ACF r(k). By least squares fitting, we obtain the estimate (a0 , b0 ) = (0.18, 0.36) (Fig. 3). Thus, we have R1(k) = (k 0.18 + 1)−0.36/0.18 ,

k = 0, 1, 2, . . .

(4.2)

with M 2 (R1) = 4.24 × 10−5 (see Fig. 4). According to (3.2) and (3.4), the Hurst


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(a) (b) 0.25 0.38 0.5 0.25 0.38 0.5 0 0.13 (a)a (b) b Fig. 3. Parameter estimates (a , b ). (a) Estimate a (a) (b) Estimate b0. 0 0 0. (b) Fig. 3.Fig. Parameter (a , b ). (a) Estimate a0(b) . (b) Estimate b0. (a) estimates 3. Parameter estimates (a0 , b0). (a) Estimate a . (b) Estimate b . 0.13

0

0

0

0

1

R1(k) (log)

R1(k)R1(k) (log) (log)

Fig. 3. 1 Parameter estimates (a0, b0). (a) Estimate a0. (b) Estimate b0. 10.1 0.1

0.01 0.1 0 0.01 0

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Fig. Fig. 4. Modeled ACF R1(k) in the Cauchy class. 4. Modeled ACF R1(k) in the Cauchy class. 128 256 384 512 Fig. 4. Modeled ACF R1(k) in the 640 Cauchy 768 class.

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Fitting the data in the Cauchy class: dot line, r(k); solid line, R1(k).

Fig. 5. Fitting the data in the Cauchy class: dot line, r(k); solid line, R1(k). 0.01 Fig. 5.0 Fitting the data in the equal Cauchy class: dot line, r(k); line, parameter and fractal dimension 1.86, respectively. 5 shows 128 256 384 to 0.78 512and 640 768solidFigure 896 R1(k). 1024 the result of fitting the data. k On the other hand, with least squares fitting based on G(k), we have

Fig. G1(k) 5. Fitting the + data in−the Cauchy dot line, r(k); solid = 0.5[(k 1)2H 2k 2H + (k −class: 1)2H ]H=0.83 , k = 0, 1, 2, line, . . . R1(k). (4.3)

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r(k), G1(k) (log)

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Fig. 6.

Fitting the data based on FGN: dot line, r(k); solid line, G1(k).

Fig. 6. Fitting the data based on FGN: dot line, r(k); solid line, G1(k). Table 1. Data series

Data series

DEC-PKT-1 DEC-PKT-2 (a, b) NUS-TCP-1 NUS-TCP-2

Modeling results.

Table 1: Modeling results. (a, b) M 2 (R) H = 1 − b/2 (0.29, (0.28, (0.18, (0.18,

0.45) 0.65) 0.36) 0.73)

8.84 × 10−5 2.442 × 10−5 M (R) 4.24 × 10−5 7.89 × 10−5 −5

0.78 0.68 H 0.82 0.64

D = 2 − a/2

= 1− b

1.86 1.86 ⁄1.91 2 1.91

D=2−a⁄2

DEC-PKT-1 (0.29, 0.45) 0.78 1.86 8.84×10 −5 0.68 1.86 DEC-PKT-2 2 (0.28, 0.65) −3 2.44×10 with M (G1) = 1.02 × 10 . Figure 6 illustrates the data fitting based on FGN. −5 NUS-TCP-1 (0.18, 0.36) 1.91 4.24×10 Figure 5 shows clearly that the Cauchy correlation model is0.82 in far better agreement −5 with the real data 0.73) for both short-term and long-term lags0.64 as compared with the NUS-TCP-2 (0.18, 1.91 7.89×10 FGN model. For DEC-PKT-n.tcp (n = 1, 2) and NUS-TCP-2, we have G1(k) for H = 0.85, 0.77, and 0.72 with M 2 (G1) = 1.79 × 10−3, 3.84 × 10−3, and 2.05 × 10−3, respectively. Table 1 summarizes the modeling results for these four series.

5. Conclusions The Cauchy correlation model with long-range dependence and its flexibility in LRD traffic modeling have been discussed. Its application to LRD traffic modeling has been demonstrated. It is a power law correlation model in which the fractal dimension (i.e. local behavior) and the Hurst parameter (i.e. global properties) of LRD time series are characterized separately. It provides a flexible model for LRD traffic analysis in both short-term and long-term lags.

Acknowledgments The research work is partly sponsored by the Malaysian Ministry of Science, Technology and Environment for the grant IRPA 04-02-02-0033-EA033.


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