Servicios Personalizados
Revista
Articulo
Indicadores
- Citado por SciELO
- Accesos
Links relacionados
- Similares en SciELO
Compartir
Computación y Sistemas
versión On-line ISSN 2007-9737versión impresa ISSN 1405-5546
Comp. y Sist. vol.10 no.1 Ciudad de México jul./sep. 2006
Heavy Tailed Network Delay: An AlphaStable
Model Retardo de Cola Pesada: Un Modelo AlfaEstable
David MuñozRodríguez1, Salvador Villarreal Reyes1, Cesar Vargas Rosales1, Marlenne Angulo Bernal1,2,3, Deni TorresRomán2 and Luis Rizo Domínguez2
1 Center of Electronics and Telecommunications; Monterrey, N.L, 64849, México emails: dmunoz@itesm.mx ; cvargas@itesm.mx
2 CINVESTAV Research Center; Guadalajara Jal., 45232, México emails: mangulo@gdl.cinvestav.mx ; dtorres@gdl.cinvestav.mx ; lrizo@gdl.cinvestav.mx
3 Autonomous University of Baja California, Mexicali B.C., 21280, México email mangulo@uabc.mx
Article received on June 27, 2005
Accepted on October 31, 2006
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 realtime applications, severely degrade the quality of service (QoS). This paper presents a general endtoend delay model suitable for a multinode path in the presence of heavytailed traffic. The proposed methodology is based on an alphastable 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 alfaestable. 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.
DESCARGAR ARTÍCULO EN FORMATO PDF
References
1. Reynolds, R., Rix, A.: Quality VOIP "An Engineering Challenge", BT Technology Journal. Vol. 19, No. 2 (2001) 2332. [ Links ]
2. Paxson, V., Floyd, S.: "Wide Area Traffic: The failure of Poisson Modeling", IEEE/ACM Transaction on Networking, Vol. 3, No. 3 (1995) 226244. [ Links ]
3. Downey, A.: Evidence for Longtailed Distributions in the Internet. In ACM SIGCOM Internet Measurement Workshop (2001). [ Links ]
4. Crovella, M., Taqqu, M., Bestavros A.: Heavy Tailed Probability Distributions in the WWW. In R. Adler, R. Feldman and M. Taqqu. A Practical Guide to Heavy Tails Statistical Techniques and Applications, Birkhäuser (1998). [ Links ]
5. Park, K., Willinger, W.: Selfsimilar Network Traffic and Performance Evaluation, John Wiley & Sons, Chichester, England (2000). [ Links ]
6. Gallardo, J. R., Makrakis, D., OrozcoBarbosa, L.: "Use of alphastable selfsimilar stochastic processes for modeling traffic in broadband networks", Perform. Eval., vol. 40, no. 13 (2000) 7198. [ Links ]
7. Karasaridis, A., Hatzinakos, D.: "Network heavy traffic modeling using astable selfsimilar processes", IEEE Transactions on Communications, Vol. 49, No. 7 (2001) 1203 1214. [ Links ]
8. Harmantzis, F.C., Hatzinakos, D., Lambadaris I.: "Effective Bandwidths and tail probabilities for Gaussian and Stable SelfSimilar Traffic". IEEE International Conference on Communications 2003, Vol. 3, No. 1115 (2003)1515 1520. [ Links ]
9. Matragi, W., Sohraby, K., Bisdikian, C.: "Jitter Calculus in ATM Networks: Multiple Nodes". IEEE/ACM Trans. on Networking, Vol. 5 (1997) 122133. [ Links ]
10. Arnold, B.: Pareto Distributions. Int. Coop Publishing House. (1981). [ Links ]
11. Blum, M.: "On the Sum of Independent Distributed Pareto Variables". J. Siam: Applied Math, Vol 19, No. 1 (1970) 191198. [ Links ]
12. MuñozRodriguez, D., Villarreal, S., Campos, G., Vargas, C., RodríguezCruz, J. R., Donis, G.: "EndtoEnd Network Delay Model for HeavyTailed Environments," European Transactions on Telecommunications, vol. 14 (2003) 391398. [ Links ]
13. Johnson, N., Kots S., Balakrishnan N.: Continuous Univariate Distributions, John Wiley & Sons, Chichester, England (1994). [ Links ]
14. Uchaikin, V., Zolotarev, V.: Chance and Stability: Stable Distributions and their Applications, VSP, Utrecht, Netherlands (1999). [ Links ]
15. Samorodnitsky, G., Taqqu, M.: Stable NonGaussian Random Processes: Stochastic Models with Infinite Variance, Chapman and Hall (1994). [ Links ]
16. Ahonen, J., Laine, A.: Realtime Speech and voice transmission on the Internet. In Proceedings of the HUT Internetworking Seminar. Helsinki (1997) Tik110.551. (http://www.tml.hut.fi/Opinnot/Tik-110.551/1997/seminar_paper.html). [ Links ]
17. Karlsson, G.: "Asynchronous transfer of video". IEEE Communications Magazine, Vol. 34, No. 8 (1996)118126. [ Links ]
18. Zolotarev, V.: "One dimensional Stable Distributions", Translations of Mathematical Monographs, American Mathematical Society Vol. 65 (1986). [ Links ]
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 heavytailed distribution if its survival function decays as a
Heavytailed distributions have the following properties: If ξ <2, then the distribution has infinite variance. If ξ < 1, then the distribution has infinite mean. Thus, as ξ decreases, a probability mass increases at the tail of the distribution.
Alphastable distribution
The alphastable 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 ℑstable random variables
1. Let X1 and X2 be independent random variables, then X1 +X2 Sε ( γ, β, µ) with
2.For any 0 < α < 2,
3. Let X Sε ( γ, β, µ) and let α be a real constant. Then
4. Let X Sε ( 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 alphastable distributions, see [15].