# Model Description

Following the generalized central limit theorem, $$\alpha$$-Stable models manifest themselves in the capability to approximate the distribution of normalized sums of a relatively large number of independent identically distributed random variables. Besides, $$\alpha$$-Stable models produce strong bursty results with properties of heavy-tailed distributions and long range dependence [1]. Therefore, they arose in a natural way to characterize the traffic in fixed broadband networks [2,5] and have been exploited in resource management analyses [8,9].

$$\alpha$$-Stable models, with few exceptions, lack a closed-form expression of the PDF, and are generally specified by their characteristic functions.

Definition:

A random variable $$X$$ is said to obey $$\alpha$$-Stable models if there are parameters $$0<\alpha \leq 2$$, $$c \geq 0$$, $$-1\leq \beta \leq 1$$, and $$\mu \in \mathcal{R}$$ such that its characteristic function is of the following form:

\Phi(\omega)= E(\exp j\omega X)\\ =\left\{ \begin{aligned} &\exp\left\{-\sigma^{\alpha} \vert c \vert^{\alpha} \left(1-j\beta(\text{sgn} (c)) \tan \frac{\pi \alpha}{2} \right) + j\mu c \right\}, \alpha\neq 1;\\ &\exp\left\{-\sigma \vert c \vert \left(1+j\beta(\text{sgn} (c)) \ln\vert c\vert \right) + j\mu c \right\}, \alpha= 1.\\ \end{aligned} \right.

Here, the function $$E(\cdot)$$ represents the expectation operation with respect to a random variable. $$\alpha$$ is called the characteristic exponent and indicates the index of stability, while $$\beta$$ is identified as the skewness parameter. $$\alpha$$ and $$\beta$$ together determine the shape of the models. Moreover, $$c$$ and $$\mu$$ are called scale and shift parameters, respectively. Specifically, if $$\alpha=2$$, $$\alpha$$-Stable models reduce to Gaussian distributions.

Furthermore, for an $$\alpha$$-Stable modeled random variable $$X$$, there exists a linear relationship between the parameter $$\alpha$$ and the function $$\Psi(\omega) = \ln\left\{- \text{Re} \left[ \ln \left(\Phi(\omega) \right)\right] \right\}$$ as
$$\Psi(\omega) = \ln\left\{- \text{Re} \left[ \ln \left(\Phi(\omega) \right)\right] \right\} =\alpha \ln (\omega) + \alpha \ln(\sigma),$$
where the function $$\text{Re}(\cdot)$$ calculates the real part of the input variable.

Figure Illustrations:

Symmetric $$\alpha$$-Stable distributions with unit scale factor. Courtesy to Wikipedia.

Skewed centered Stable distributions with unit scale factor. Courtesy to Wikipedia.

Validation Methodology:

Usually, it’s challenging to prove whether a dataset follows a specific distribution, especially for $$\alpha$$-Stable models without a closed-form expression for their PDF. Therefore, when a dataset is said to satisfy $$\alpha$$-Stable models, it usually means the dataset is consistent with the hypothetical distribution and the corresponding properties. In other words, the validation needs to firstly estimate parameters of $$\alpha$$-Stable models from the given dataset, and then compare the real distribution of the dataset with the estimated $$\alpha$$-Stable model. Specifically, the corresponding parameters in $$\alpha$$-Stable models can be determined by quantile methods, or sample characteristic function methods.

Useful references:

1. G. Samorodnitsky, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. New York: Chapman and Hall/CRC, 1994.
2. J. R. Gallardo, D. Makrakis, and L. Orozco-Barbosa, “Use of alpha-Stable self-similar stochastic processes for modeling traffic in broadband networks,” in Proc. SPIE Conf. P. Soc. Photo-Opt. Ins, Boston. Massachusetts, Nov. 1998, vol. 3530, pp. 281–296.
3. S. M. Koyon and D. B. Williams, “On the characterization of impulsive noise with $$\alpha$$-Stable distributions using Fourier techniques,” in Proc. Asilomar Conf. Signals, Systems, Computers, Oct. 1995.
4. J. B. Hill, “Minimum Dispersion and Unbiasedness: ‘Best’ Linear Predictors for Stationary ARMA a-Stable Processes,” University of Colorado at Boulder, Discussion Papers in Economics Working Paper No. 00-06, Sep. 2000.
5. X. Ge, G. Zhu, and Y. Zhu, “On the testing for alpha-Stable distributions of network traffic,” Comput. Commun., vol. 27, no. 5, pp. 447–457, Mar. 2004.
6. A. Karasaridis and D. Hatzinakos, “Network heavy traffic modeling using alpha-Stable self-similar processes,” IEEE Trans. Commun., vol. 49, no. 7, pp. 1203–1214, Jul. 2001.
7. P. Zagaglia, “Estimation of alpha-Stable distribution parameters using a quantile method,” 25-Jan-2012. [Online]. Available: http://www.mathworks.com/matlabcentral/fileexchange/34783-estimation-of-alpha-Stable-distribution-parameters-using-a-quantile-method. [Accessed: 09-Oct-2014].
8. W. Song and W. Zhuang, “Resource Reservation for Self-Similar Data Traffic in Cellular/WLAN Integrated Mobile Hotspots,” in Proc. IEEE ICC 2010, Cape Town, South Africa, May 2010.
9. J. C.-I. Chuang and N. R. Sollenberger, “Spectrum resource allocation for wireless packet access with application to advanced cellular Internet service,” IEEE J. Sel. Area. Comm., vol. 16, no. 6, pp. 820–829, Aug. 1998.
10. Rongpeng Li, Zhifeng Zhao, Chen Qi, Xuan Zhou, Yifan Zhou, and Honggang Zhang, “Understanding the Traffic Nature of Mobile Instantaneous Messaging in Cellular Networks: A Revisiting to alpha-Stable Models” , IEEE Access, vol. 3, pp. 1416-1422, 2015.
11. Luca Chiaraviglio, Francesca Cuomo, Maurizio Maisto, Andrea Gigli, Josip Lorincz, Yifan Zhou, Zhifeng Zhao, Chen Qi, Honggang Zhang, “What is the Best Spatial Distribution to Model Base Station Density? A Deep Dive in Two European Mobile Networks”, IEEE Access, Apr. 2016.
12. Yifan Zhou, Rongpeng Li, Zhifeng Zhao, Xuan Zhou, and Honggang Zhang, “On the $$\alpha$$-Stable Distribution of Base Stations in Cellular Networks”, IEEE Communications Letters, vol. 19, no. 10, pp. 1750-1753, Aug. 2015.