In a previous work we have introduced a multifractal traffic model based on so-called stochastic L-Systems, which were introduced by biologist A. Lindenmayer as a method to model plant growth. L-Systems are string rewriting techniques, characterized by an alphabet, an axiom (initial string) and a set of production rules. In this paper, we propose a novel traffic model, and an associated parameter fitting procedure, which describes jointly the packet arrival and the packet size processes. The packet arrival process is modeled through a L-System, where the alphabet elements are packet arrival rates. The packet size process is modeled through a set of discrete distributions (of packet sizes), one for each arrival rate. In this way the model is able to capture correlations between arrivals and sizes. We applied the model to measured traffic data: the well-known pOct Bellcore, a trace of aggregate WAN traffic and two traces of specific applications (Kazaa and Operation Flashing Point). We assess the multifractality of these traces using Linear Multiscale Diagrams. The suitability of the traffic model is evaluated by comparing the empirical and fitted probability mass and autocovariance functions; we also compare the packet loss ratio and average packet delay obtained with the measured traces and with traces generated from the fitted model. Our results show that our L-System based traffic model can achieve very good fitting performance in terms of first and second order statistics and queuing behavior.
In this paper we compare two traffic models based on Markov modulated Poisson processes (MMPPs), that were designed to capture self-similar behavior over multiple time scales. These models are both constructed by fitting the distribution of packet counts in a number of time scales. The first model is a superposition of MMPPs where each MMPP describes a different time scale. The second one is obtained as the equivalent to an hierarchical construction process that, starting at the coarsest time scale, successively decomposes MMPP states into new MMPPs to incorporate the characteristics offered by finner time scales. We evaluate the accuracy of the models by comparing the probability mass function at each time scale, as well as the loss probability and average waiting time in queue, corresponding to measured traces and to traces synthesized according to the proposed models. The analysis is based on three measured traffic traces exhibiting self-similar behavior: the well-known pOct Bellcore trace and two traces measured in a Portuguese ISP. Based on the obtained results, we conclude that both Markovian models have good and very similar performances in matching the characteristics of the data traces over the relevant time scales. However, one advantage of the hierarchical approach is that the number of states of the corresponding MMPP can be much smaller.
Network traffic processes can exhibit properties of self-similarity and long-range dependence, i.e., correlations over a wide range of time scales. However, as already shown by several authors for the case of a single queue, the second-order behavior at time scales beyond the so-called correlation horizon or critical time scale does not significantly affect network performance. In this work, we extend previous studies to the case of a network with two queuing stages, using discrete event simulation. Results show that the second stage provokes a decrease in the correlation horizon, meaning that the range of time scales that need to be considered for accurate network performance evaluation is lower than predicted by a single stage model. We also used simulation to evaluate the single queue model. In this case, the estimated correlation horizon values are compared with those predicted by a formula derived by Grossglauser and Bolot, which presumes the approximation of the input data by a traffic model that enables to control the autocorrelation function independently of first-order statistics. Results indicate that although the correlation horizon increases linearly with the buffer size in both methods, the simulation ones predict a lower increase rate.
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