Markovian Modeling of Real Data Traffic: Heuristic Phase Type and MAP Fitting of Heavy Tailed and Fractal Like Samples

Abstract

In order to support the effective use of telecommunication infrastructure, the "random" behavior of traffic sources has been studied since the early days of telephony. Strange new features, like fractal like behavior and heavy tailed distributions were observed in high speed packet switched data networks in the early '90s. Since that time a fertile research aims to find proper models to describe these strange traffic features and to establish a robust method to design, dimension and operate such networks. In this paper we give an overview of methods that, on the one hand, allow us to capture important traffic properties like slow decay rate, Hurst parameter, scaling factor, etc., and, on the other hand, makes possible the quantitative analysis of the studied systems using the effective analysis approach called matrix geometric method. The presentation of this analysis approach is associated with a discussion on the properties and limits of Markovian fitting of the typical non-Markovian behavior present in telecommunication networks.

A Novel Approach for Phase-Type Fitting with the EM Algorithm

“…To overcome these problems the class of PH distributions used for fitting has to be restricted which is in principle possible in the basic EM algorithm by initializing only some elements in the matrix with non-zero values, However, the optimization problem for general PH distributions is too complex to yield satisfactory results if the number of phases is larger than two or three. As shown in several papers [2], [3], [13], [14], the fitting problem becomes much easier if acyclic instead of general phase-type distributions are used, because for this type of distributions a canonical representation exists which reduces the number of free parameters to 2N compared to N 2 + N for the general case, where N is then number of phases [5]. On the other hand, the restriction to acyclic PH distributions does not seem to limit the flexibility of the approach.…”

Workload Modeling for Computer Systems Performance Evaluation

“…[340]). For example, the servers can have different rates, clients can enter "directly into the middle" of the network of servers, the network can have loops in it, and so on.…”

“…In addition, a set of m individual stages are added [341,340]. Effectively, the sequence of the first n stages creates a mode that serves as the body of the distribution, and the additional individual stages are actually a hyper-exponential distribution used to construct a long tail for the distribution.…”

“…The simplest (yet quite general) approach is to create a mixture of hyper-exponential and Erlang distributions as shown in Section 3.2.6 [567,340,341]. The hyper-exponential part contributes the tail, and the Erlang part contributes the mode.…”

“…The procedures for doing so were described in Section 4.4.3. Importantly, it is also possible to construct mixtures of exponentials to fit distributions with various modes, not only those with a monotonically decaying tail [97,340,567,568].…”