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.
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.
This paper proposes a novel fitting procedure for Markov Modulated Poisson Processes (MMPPs), consisting of the superposition of N 2-MMPPs, that is capable of capturing the long-range characteristics of the traffic. The procedure matches both the autocovariance and marginal distribution functions of the rate process. We start by matching each 2-MMPP to a different component of the autocovariance function. We then map the parameters of the model with N individual 2-MMPPs (termed superposed MMPP) to the parameters of the equivalent MMPP with 2N states that results from the superposition of the N individual 2-MMPPs (termed generic MMPP). Finally, the parameters of the generic MMPP are fitted to the marginal distribution, subject to the constraints imposed by the autocovariance matching. Specifically, the matching of the distribution will be restricted by the fact that it may not be possible to decompose a generic MMPP back into individual 2-MMPPs. Overall, our procedure is motivated by the fact that direct relationships can be established between the autocovariance and the parameters of the superposed MMPP and between the marginal distribution and the parameters of the generic MMPP. We apply the fitting procedure to traffic traces exhibiting LRD including (i) IP traffic measured at our institution and (ii) IP traffic traces available in the Internet such as the well known, publicly available, Bellcore traces. The selected traces are representative of a wide range of services/protocols used in the Internet. We assess the fitting procedure by comparing the measured and fitted traces (traces generated from the fitted models) in terms of (i) Hurst parameter; (ii) degree of approximation between the autocovariance and marginal distribution curves; (iii) range of time scales where LRD is observed using a wavelet based estimator and (iv) packet loss ratio suffered in a single buffer for different values of the buffer capacity. Results are very clear in showing that MMPPs, when used in conjunction with the proposed fitting procedure, can be used to model efficiently Internet traffic in the relevant time scales, even when exhibiting LRD behavior.