From reality to abstract in stochastic processes applied to telecommunications

The first Chapter contains a study of ATM (asynchronous transfer mode) cell data from Sprint network, considering both the analysis and the modeling of the data. The source modeling is made by a sequence of pairs of random variables (X, Y) where X represents the cell bursts and Y represents the interarrival times between bursts. X and Y are determined by different finite state Markov chains or simplified versions of them. For the main user, the interarrival time between bursts can be considered also dependent on the previous burst size. The source model is used to fit a given rate process obtained on line. The model is validated by comparing the interarrival times and the rates of the model and of the trace data. The analysis of the queue is based on the computation of mean cell delay, cell loss ratio and other statistical aspects. Besides mathematics, the work here relies heavily on C and MATLAB programming and graphing.; The second chapter bridges between the first and the third. A two-component fractional Brownian motion (FBM)-based traffic model for telecommunications is considered. Lower bounds for the overflow probability of its associated queueing system are obtained. Upper bounds for the busy period are determined. Using them a dynamic buffer allocation function is introduced. Upper bounds for the busy period are obtained by both probabilistic and deterministic methods. The latter requires the introduction of a suitable deterministic envelope process. The results are extended to a more general FBM-based model. Computer simulations are used to illustrate the theoretical results.; Vector-valued fractional Brownian motion turns out to be an operator self-similar stochastic process with a simple scaling family of operators acting on real, finite-dimensional spaces. In the third chapter the notion of operator self-similar processes is extended to (possibly infinite-dimensional) Banach space valued processes. Their properties are studied and the theory of semigroups of linear operators is used to derive information about them, such as uniqueness of scaling families for processes with spanning set of expectations, power-growth upper bounds for the norms of the expectations, under various continuity or smoothness assumptions on the process.