els
In this section, we present a critique on the applicability of count models based on self- similar processes and renewal processes for Internet traffic count data modelling.
4.5.1
Applicability of Self-Similar Count Models
Taking into account the scope of research presented in this thesis, we outline the issues concerning the applicability of self-similar count models in Internet traffic as follows:
• The FGN model allows the possibility of negative values which is not realistic in modelling of packet or byte counts unless the mean of data is significantly greater than its standard deviation [Norros,1995].
4.5 Self-Similar Count Models versus Renewal Count Models 119
• Regarding the convergence to a pure Gaussian process under superposition, the following remark is noteworthy. Namely, a distribution with heavy tail index greater than two, that is, α > 2, even with a million convolutions of it with itself it will eventually behave like Gaussian up to around three standard deviations but will retain the heavy-tailed attributes outside of such a regime [Taleb,2009] .
• The fractal properties introduced by the FGN process degenerate. The computational efficiency of FGN for traffic generation decreases as the value of the Hurst parameter increases to one [Riedi et al.,1999].
• The heavy tailed ON/OFF superposition models, leading to FBM in the limit, are accurate only in the regime of coarser time scales, and they do not account for actual queueing and multiplexing occurring in a network at fine time scales [Riedi et al., 1999].
• The FBM model abandons the classical framework of point arrival processes in favour of modelling the “net” work or cumulative input process. That is, instead of defining packet arrival times, FBM models the cumulative traffic arriving up to a certain defined time instant. Therefore, the superposition models using FBM or FGN as a limit counting process cannot be used in studies of the converged stochastic behaviour of interarrival times [Gordon,1996].
• The FBM and Gaussian processes can be expressed in discrete or continuous time. The sampling rate required in a discrete time Gaussian model makes it quite different from the corresponding continuous time Gaussian model [Addie, 1999]. For the correlated interarrival times (of packets, flows or sessions), continuous time Gaussian models are inappropriate due to fast tail decay. For discrete time Gaussian models, there is a trade off in the choice of sampling interval. Long sampling intervals ensures that discrete values (for example, packet counts per unit sampling interval) are not negative but are not suitable for performance evaluation at the time scales concerning traffic queueing or switching, and vice versa.
• Although the phenomena of burstiness and LRD are complementary to each other, they in fact are different. A LRD process may or may not be bursty at fine time scales [Tian et al.,2002]. Therefore, care should be taken in using self-similar count models if one wants to generate bursty traffic data only.
• Traffic modelling based on self-similar count models should also specify an appro- priate estimator for the degree of self-similarity. This is necessary because different
estimators of Hurst parameter can provide different and even conflicting estimates for the strength of self-similarity in traffic count data [Rea et al.,2013].
• The sudden changes in mean levels of count data can also cause LRD estimators to provides false positives regarding strength of long-range dependence [Beran et al., 2013a]. Therefore, data should be carefully assessed with the help of appropriate qualitative estimators of LRD (for example, the wavelet based estimator proposed in [Abry et al.,1998]) to confirm the presence of self-similarity, and therefore, to justify the applicability of self-similar count models.
• In [Addie et al.,1999], it has been shown that fitting the mean, variance and Hurst parameter is not sufficient to consistently characterize Internet traffic count data which exhibits long-range dependence. A fourth parameter defining the level of aggregation is also required. This affects the parsimony of self-similar count models.
• Autoregressive models like FARIMA can capture the empirical second-order corre- lations in data, but they do not generally fit the empirical marginal distribution of traffic count data [Frost & Melamed,1994].
4.5.2
Applicability of Renewal Count Models
Here, we outline some of the advantages which the renewal process framework offers if one uses them in modelling Internet traffic count data.
• A traffic modelling framework has strong physical meanings if it is based on point processes [Veitch et al.,2005]. Such a modelling framework can account for various statistical characteristics of Internet traffic in a consistent manner. Renewal processes can play an important role in establishing such a modelling framework. The self- similar count modelling framework suffers from detection and estimation issues regarding the strength of self-similarity. On the other hand, a traffic modelling framework based on renewal processes has consistent methodologies to assess renewal behaviour of traffic data and how much it deviates from it; see Section2.2.
• Long-range dependence can be physically justified as the result of superposition of renewal processes with heavy-tailed interarrival times, which in turn can be physically justified by the access patterns of users. The superposition of such renewal processes can generate long-range dependence or induce strong temporal correlations in the superposed count process [Beran et al.,2013a].
4.5 Self-Similar Count Models versus Renewal Count Models 121
• Non-renewal processes can be approximated by appropriate renewal processes by fitting a few lower order moments [Kuehn,1979;Whitt,1982].
• Point process based analysis offers a unified framework for modelling of the different structural components of Internet traffic at packet, flow and session levels [Arfeen
et al.,2013].
• Count models based on various renewal processes can generate different types of dispersion in count data. This is useful in modelling Internet traffic count data under various load conditions. For example, renewal processes based on Weibull interarrival times can generate underdispersed, equidispersed and overdispersed data, depending on the value of the Weibull shape parameter.
• A renewal process based on heavy-tailed Weibull interarrival times has similar sta- tistical properties as those of doubly stochastic Poisson processes or Cox processes (Poisson processes with stochastic intensity) which is suitable for modelling variations
in Internet traffic count data [Yannaros,1994].
• Renewal processes have also been used to develop switched Poisson processes with time varying rates and dependencies between interarrival times. These models are shown to exhibit a similar long-range dependence behaviour as that of Internet traffic. They include Markov-modulated Poisson process (MMPP) and Pareto-modulated Poisson process (PMPP); see [Fischer & Meier-Hellstern, 1993; Muscariello et al., 2005] and [Le-Ngoc & Subramanian,2000], for example. These models are multi-state and highly parametrized to fit properly to Internet traffic [Mallor et al.,2007]. Pure renewal processes with heavy-tailed interarrival times provide a simple alternative to such models.
• Renewal processes can be considered as a generalization of ON/OFF processes where each ON period consists of one event (packet, flow or session). In such a case, they are known as singular renewal processes in [Erramilli et al., 1996a]. Hence, theorems regarding superposition of ON/OFF processes can also be generalized to assess properties of superposition of pure renewal processes under relevant conditions; see [Gaigalas & Kaj,2003;Kaj,1999], for example.
• Renewal processes can also be aggregated to implement approximate self-similar processes with normal marginal distribution. An example of such a method is based on spatial renewal process (SRP) which is a mixture of two independent renewal processes with one of them having desired marginal distribution of traffic [Taralp
et al.,1998].