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Computación y Sistemas
versión On-line ISSN 2007-9737versión impresa ISSN 1405-5546
Comp. y Sist. vol.8 no.2 Ciudad de México oct./dic. 2004
Routing with WaveletBased SelfSimilarity Estimation
Ruteo con Estimación de AutoSimilaridad Utilizando Onduletas
Cesar VargasRosales and Luis J. Manzanero
Center for Electronics and Telecommunications ITESM Campus Monterrey, Monterrey, N.L., C.P. 64849, México, Email: cvargas@itesm.mx ; lmanzanero@alestra.com.mx
Article received on June 20, 2002
Accepted on August 09, 2004
Abstract
The discovery of selfsimilar behavior in data traffic has initiated strong research in the area of traffic modeling. However, the way it affects the routing process is a subject not yet studied. This work presents the idea of providing intelligence to routers by estimating the Hurst Parameter using wavelets in a data link and uses this value as part of the routing metric. The motivation is to keep packets from using paths with high values of the Hurst Parameter, since in those paths the traffic tends to be bursty and therefore being more probable to drop packets and increase delay due to congestion. The algorithm is implemented for the Open Shortest Path First (OSPF) routing protocol in a simulation environment. Numerical results show that this technique is useful to provide traffic with low values of endtoend delay and with smaller variance than those obtained when using traditional routing protocols.
Keywords: Selfsimilarity, Routing, Traffic.
Resumen
El descubrimiento de comportamiento autosimilar en el tráfico de datos, ha iniciado una enorme investigación en el área de modelado de tráfico. Sin embargo, la forma en la que afecta al ruteo en redes es un tema que todavía no se ha estudiado. Este trabajo presenta la idea de proporcionar inteligencia a los ruteadores por medio de la estimación del parámetro de Hurst utilizando onduletas en un enlace de datos, y utilizando ésta en la métrica de ruteo. La motivación es que los paquetes no utilicen trayectorias con altos valores del parámetro de Hurst, puesto que en esas trayectorias el tráfico tiende a contener ráfagas y a ser más probable que paquetes se pierdan y a que se incremente el retardo por congestión. El algoritmo es implementado para el protocolo de ruteo Open Shortest Path First (OSPF) en un ambiente se simulación. Resultados numéricos muestran que esta técnica es útil ya que proporciona tráfico con bajos niveles de retardo de extremoaextremo y con menor variación que aquellos valores obtenidos con el protocolo tradicional.
Palabras Clave: Autosimilaridad, Ruteo, Tráfico.
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Appendix. AV Estimator Algorithm
In this Appendix, we provide a brief description of the online AV wavelet estimator algorithm in Abry and Veitch (1998) and Roughan et. al., (1998). Let x(t) be the selfsimilar signal from which the parameters are to be estimated. In this case, the outgoing traffic in bits per second.
• Pass the input data by the filter bank shown in Figure 1. The number of stages of the filter is known as the number of octaves. The length of the response function of the bandpass function is the same as that of the low pass filter and it is known as the number of moments; the values of this vector are obtained from the coefficients of the wavelet family used, and for the band pass filter the coefficients are those of the wavelet with the sign of the elements in the odd position exchanged, while for the low pass filter the elements are the original wavelet coefficients transformed by flipping the original vector (i.e., swap the element in the nth position with that at position one, the element in the (n2)nd position with that at position two and so on).
• The result of the convolution of the bandpass function and the signal is downsampled (that is, the elements in the even positions are discarded) and the resulting vector is called the detail of the octave j, dx (j, i) with length len j . Make Sj S j + dx (j, i)2 where i = 1...., K., lenj. Make nj nj . + lenj . Continue introducing samples from the signal. The previous steps are performed on the arrival of any sample of the signal, while the next are only performed on a longer time scale, when the Hurst parameter wants to be estimated, in our case 60 seconds.
• Calculate for all the stages the following expressions
where ς(x) is the RiemannZeta function ς (x) = see Zwillinger, (1987); ψ(x) = Γ'(x) / Γ(x) is the Psi function (also called the digamma function) see Korn, page 823, (1968) and Γ(x) = dt is the gamma function with derivate Γ(x) .
•Calculate the next sums:
• For each value of j calculate
• Calculate
• And finally, the estimation of the Hurst Parameter is given by
Table 1: Numerical Results for Medium Loaded Network: Topology One.
Table 2: Numerical Results for Highly Loaded Network: Topology One.