A Mixed-Fractal Model for Network Traffic

There are numerous explanations and models for the origins and appearances of fractality for network traffic; see e.g. [1]- [11] and the references therein. Under the multi-fractality assumption for Internet traffic, we propose a new flow model to explain the crossover phenomena found in the estimation of Hurst exponent [5], [11] (the crossover phenomenon is defined in the next section). Results indicate that this model help gain more insights into the actual network traffic dynamics. Moreover, we can use this model to better simulate network traffic in a simple way.

In data networks, the traffic flow is often viewed a certain self-similar stochastic process Y (k), which by definition satisfies [12]- [13] Y (ak)

where a > 0, and d = denotes the equality of finite-dimensional distributions. H > 0 is the so called Hurst exponent.

A self-similar process with stationary increments can then be defined by multiplexing the increments X(k) = Y (k + 1) -Y (k) over non-overlapping blocks of size n as

The aggregated process X (n) (k) is called a stationary self-similar H-SSS process with Hurst exponent H. It has finite-dimensional distributions similar to X(k)

There are various ways to study the stochastic properties of X (n) (k). [11] considers the cumulants of the aggregated series, which are defined as the Taylor coefficients of the cumulant-generating function

with cum m (X) = g (m) (0). In [14], [15], it is shown that the m th order cumulants of an aggregated H-sss process usually scales as

As pointed out in [11], Eq.( 5) implies that for each n, m ∈ N the logarithm of the modulus of cum m X (n) (k) scales linearly with log(n) with slope mH(m)

The simplest form of mH(m) is a linear function of m, i.e.

In [11], (7) is called linear-fractal model, where the coefficients A and B are directly estimated (interpolated) during the fittings to the cumulants. A special case of the linear-fractal model is the uni-fractal model [6][11], which has the form

where H is the corresponding Hurst exponent.

[11] compared the linear-fractal model and the uni-fractal model using empirical network flow data. As shown in Fig. 3 and Fig. 4 of [11], although both models catch the main trends of the real data in the estimation of Hurst exponent., neither of them can perfectly match the so-called crossover phenomenon.

Fig. 1 gives an illustration of the crossover phenomenon. In the current case the X-axis stands for log(n) and the Y-axis stands for log cumulant. The slope of the fitting curve crossovers from a small value to a notably larger value. Therefore the curve consists of three parts: a line segment with a gentler slope when log(n) is small, the intermediate transition part, and another line segment with a steeper slope when log(n) is large. The crossover phenomenon can also be observed in the wavelet-based Hurst coefficient estimation for network flow [5], [16]. As shown in [5], a self-similar process has the following power law Var(y j ) = C y j (2H+1) (9) where C y is a constant, octave j = log 2 (scale), and Var(y j ) is a specially defined measuring variance on octave j. Eq.( 9) indicates that the logarithm of Var(y j ) scales linearly with log(j) with slope (2H + 1) as log Var(y j ) = (2H + 1) log(j) + log C y (10) However, real flow data again exhibits the unexpected crossover phenomenon as illustrated in Fig. 1, (in this case the X-axis stands for log(j) and the Y-axis stands for log cumulant). An example can be found in Fig. 2

There are numerous physical models for the origin of the self-similarities in network traffic flows, but few of them can explain the crossover phenomenon. This gap calls for a finer model that accounts for the crossover phenomenon, as it will not only enhance our understanding of the underlying mechanism of the network traffic, but may also lead to improvements of network performance.

Our previous study [17] shows that the PCA eigen-spectrum of the mixed fBm signals with different Hurst exponents may yield bi-scaling/multi-scaling be-havior. Inspired by that finding, we propose a mixed-fractal flow model, which reproduces the above crossover phenomena and sheds light on its origin. We assume that the network flow process W (k) is the sum of two independent self-similar processes X 1 (k) and X 2 (k) with different Hurst exponents H 1 and H 2 , respectively:

It is assumed that Var {X 1 } = Var {X 2 } = 1 and thus λ 1 , λ 2 > 0 control the variance of the two components. Without loss of generality, we assume

Similarly to Eq. ( 2), we can define Z (n) (k) by multiplexing the increments Z(k) = W (k + 1) -W (k) over non-overlapping blocks of size n ∈ N as

For the linear-fractal and uni-fratal models, we obtain by independence

wher