Model Retardo de Cola Pesada: Un Modelo Alfa–Estable

David Muñoz–Rodríguez¹, Salvador Villarreal Reyes¹, Cesar Vargas Rosales¹, Marlenne Angulo Bernal^1,2,3, Deni Torres–Román² and Luis Rizo Domínguez²

¹ Center of Electronics and Telecommunications; Monterrey, N.L, 64849, México e–mails: dmunoz@itesm.mx ; cvargas@itesm.mx

² CINVESTAV Research Center; Guadalajara Jal., 45232, México e–mails: mangulo@gdl.cinvestav.mx ; dtorres@gdl.cinvestav.mx ; lrizo@gdl.cinvestav.mx

³ Autonomous University of Baja California, Mexicali B.C., 21280, México e–mail mangulo@uabc.mx

Abstract

Adequate quality of IP services demands low transmission delays. However, packets traveling in a network are subject to a variety of delays that, in real–time applications, severely degrade the quality of service (QoS). This paper presents a general end–to–end delay model suitable for a multi–node path in the presence of heavy–tailed traffic. The proposed methodology is based on an alpha–stable random variable description. This allows us to define a network processing measure that relates the delay spread to the heavy tail characteristics of the traffic, the number of nodes in a route, and the processing speed at the nodes.

Key words: Network delay, Jitter, Alpha stable traffic, QoS.

Resumen

Una calidad de servicio adecuada en redes IP demanda retardos de transmisión bajos. Sin embargo los paquetes que viajan en la red están sujetos a una variedad de retardos que, en el caso de servicios en tiempo real, degrada la calidad de servicio severamente. En este artículo se presenta un modelo general para el retardo de extremo a extremo en trayectorias multinodales y tráfico de cola pesada. La metodología propuesta se basa en una descripción alfa–estable. Esto permite definir un medida del procesamiento de la red que relaciona la variación del retardo con las características de cola pesada del tráfico, el numero de nodos y la velocidad de procesamiento en los nodos.

Palabras clave: Retardo de red, Jitter, Trafico alfa estable, Calidad de Servicio.

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Appendix. Mathematical Concepts

Survival function

A survival function describes the probability that a variable X takes on a value greater than a number x. The survival function S(x) and the distribution function D(x) are related by

Heavy-tailed distributions

A random variable X is a heavy–tailed distribution if its survival function decays as a

Alpha–stable distribution

The alpha–stable PDF and the cumulative distribution function (CDF) are not analytically expressible (a few exceptions are the Gaussian, Cauchy and Levy distributions). However, this family of distributions is represented by their characteristic function, as in the following equation:

µ (– ∞,∞) is known as the location parameter; γ> 0 is the scale or dispersion parameter; β [–1,1] is the symmetry

or skewness parameter defined as , where F(x) is the corresponding distribution; and α is known as the stability index.

Properties of ℑ–stable random variables

1.– Let X₁ and X₂ be independent random variables, then X₁ +X₂ S_ε ( γ, β, µ) with

2.–For any 0 < α < 2,

3.– Let X S_ε ( γ, β, µ) and let α be a real constant. Then

4.– Let X S_ε ( γ, β, µ) and let α be a real constant. Then

The associated PDF and CDF are respectively denoted as f (x ;α , γ, β, µ) and F (x ; α , γ, β, µ) The PDF has correspondingly the following scaling and shifting properties:

For more details of alpha–stable distributions, see [15].