Multi-time-Scale Traffic Modeling Using Markovian and L-Systems Models

  • Paulo Salvador
  • António Nogueira
  • Rui Valadas
  • António Pacheco
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3262)


Traffic engineering of IP networks requires the characterization and modeling of network traffic on multiple time scales due to the existence of several statistical properties that are invariant across a range of time scales, such as self-similarity, LRD and multifractality. These properties have a significant impact on network performance and, therefore, traffic models must be able to incorporate them in their mathematical structure and parameter inference procedures.

In this work, we address the modeling of network traffic using a multi-time-scale framework. We evaluate the performance of two classes of traffic models (Markovian and Lindenmayer-Systems based traffic models) that incorporate the notion of time scale using different approaches: directly in the model structure, in the case of the Lindenmayer-Systems based models, or indirectly through a fitting of the second-order statistics, in the case of the Markovian models. In addition, we address the importance of modeling packet size for IP traffic, an issue that is frequently misregarded. Thus, in each class we evaluate models that are intended to describe only the packet arrival process and models that are intended to describe both the packet arrival and packet size processes: specifically, we consider a Markov modulated Poisson process and a batch Markovian arrival process as examples of Markovian models and a set of four Lindenmayer-Systems based models as examples of non Markovian models that are able to perform a multi-time-scale modeling of network traffic. All models are evaluated by comparing the density function, the autocovariance function, the loss ratio and the average waiting time in queue corresponding to measured traces and to traces synthesized from the fitted models. We resort to the well known Bellcore pOct traffic trace and to a trace measured at the University of Aveiro.

The results obtained show that (i) both the packet arrival and packet size processes need to be modeled for an accurate characterization of IP traffic and (ii) despite the differences in the ways Markovian and L-System models incorporate multiple time scales in their mathematical framework, both can achieve very good performance.


Traffic modeling Markovian arrival processes L-Systems 


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  1. 1.
    Lindenmayer, A.: Mathematical models for cellular interactions in development II. Simple and branching filaments with two-sided inputs. Journal of Theoretical Biology 18, 300–315 (1968)CrossRefGoogle Scholar
  2. 2.
    Peitgen, H., Jurgens, H., Saupe, D.: Chaos and Fractals: New Frontiers of Science. Springer, Heidelberg (1992)Google Scholar
  3. 3.
    Salvador, P., Pacheco, A., Valadas, R.: Multiscale fitting procedure using Markov modulated Poisson processes. Telecommunications Systems 23, 123–148 (2003)CrossRefGoogle Scholar
  4. 4.
    Salvador, P., Pacheco, A., Valadas, R.: Modeling IP traffic: Joint characterization of packet arrivals and packet sizes using BMAPs. Computer Networks Journal 44, 335–352 (2004)zbMATHCrossRefGoogle Scholar
  5. 5.
    Salvador, P., Nogueira,A.,Valadas, R.: Modeling multifractal traffic with stochastic L-Systems. In: Proceedings of GLOBECOM 2002 (2002) Google Scholar
  6. 6.
    Abry, P., Flandrin, P., Taqqu, M., Veitch, D.: Wavelets for the analysis, estimation and synthesis of scaling data. In: Park, K., Willinger, W. (eds.) Self-Similar Network Traffic Analysis and Performance Evaluation (1999)Google Scholar
  7. 7.
    Salvador, P., Nogueira, A., Valadas, R.: Modeling multifractal IP traffic: Characterization of packet arrivals and packet sizes using stochastic L-Systems. In: 10th International Conference on Telecommunication Systems, Modeling and Analysis, pp. 577–587 (2002)Google Scholar
  8. 8.
    Salvador, P., Nogueira, A., Valadas, R.: Joint characterization of the packet arrival and packet size processes of multifractal traffic based on stochastic L-Systems. In: 18th International Teletraffic Congress, ITC 18 (2003)Google Scholar
  9. 9.
    Salvador, P., Nogueira, A., Valadas, R.: Framework based on stochastic L-Systems for modeling IP traffic with multifractal behavior. In: SPIE Conference on Performance and Control of Next Generation Communication Networks, ITCom (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Paulo Salvador
    • 1
  • António Nogueira
    • 1
  • Rui Valadas
    • 1
  • António Pacheco
    • 2
  1. 1.Institute of Telecommunications AveiroUniversity of AveiroAveiroPortugal
  2. 2.Department of Mathematics and CEMATInstituto Superior Técnico – UTLLisboaPortugal

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