Modeling of transient processes in Markov chains with an application to the Internet traffic description

This thesis is on Internet traffic modeling. This thesis examines and critiques long-range dependent characteristics of Internet traffic that are widely accepted in the literature. Based on the information presented in the key articles, this thesis concludes that there is not enough evidence to assume the long-range dependency of Internet traffic. This thesis suggests the more analytically tractable model of a nonstationary Markov process. The Markov assumption of the behavior of the system leads to a system of difference or differential equations. Under the nonstationary assumption, the most important question raised is the question of transient behavior of the system. This is especially true when the length of a period during which traffic can be considered stationary is of the same order as the length of the transient period of the system. An analytic solution for the transient period is not practical for large systems; therefore a numerical solution needs to be obtained. Traditional numerical methods use one of the available integration schemes. These schemes produce solution by numerically integrating system of equations with some integration step. One of the most prominent features of the system of equations considered in this thesis is that they are stiff or ill-defined. This makes their numerical solution using traditional methods unfeasible due to the limitations on the possible integration step. This thesis develops and uses a modified Pseudo Stationary Derivatives (PSD) method to obtain a numerical solution. The method modifies the original system of equations in a way that the solution of the modified system is close to the solution of the original, while the modified system allows a significant increase of the integration step. This thesis includes significant modifications to the PSD method, guaranteeing its applicability for Markov chain modeling. The original PSD method was designed for the solution of differential equations. In this thesis it is proved that this method can be applied for difference equations as well.