Институт вычислительной математики и математической геофизики СОРАН

Тезисы докладов

Статистическое моделирование и методы Монте-Карло

This work deals with the efficient simulation of the overflow probability in a single server queue fed by Long Range Dependent (LRD) Gaussian traffic. This issue, strictly connected with the classical ruin problem in risk probability, is of primary importance for the dimensioning of modern broadband networks in order to provide stringent Quality of Service (QoS) guarantees. Indeed, the overflow probability, defined as the probability that the steady-state queue-length Q exceeds a given threshold b, gives an upper bound for the loss probability in the corresponding finite-buffer system. For a general Gaussian process this probability has not a closed form and in many cases of interest it is very small, which renders usual Monte Carlo (MC) estimation ineffective. To tackle the limits of traditional MC simulation in dealing with rare events, Importance Sampling (IS) techniques can be applied. The efficiency of the resulting estimator strongly depends on the change of measure and the problem has been thoroughly studied in the framework of Large Deviation Theory (LDT). In particular, single-twist IS (i.e., within the class of pdfs that differ from the original one only by a shift in the mean value) has emerged as a quite general approach and asymptotic results are known for the first and second moments of the corresponding IS estimator (and hence asymptotic efficiency can be analytically estimated). The goal of this work is to test the above-mentioned LDT asymptotics for finite values of the buffer size. Although the theory is quite general (it can be applied to any centered Gaussian process), the analysis focuses on fractional Brownian Motion (fBM). Indeed, fBM is one of the most widely used models for LRD traffic because of its ability to capture in a parsimonious way the key features (in terms of statistical analysis as well as queuing performance) of actual traffic flows. The talk will discuss the application of various changes of measure (for sake of precision, linear paths and most likely path) and different levels of self-similarity will be taken into account (including, as a limit case, the standard Brownian Motion). As a side product, the simulation results will be used as a basis for the estimation of the well-known Pickands constants, which play a relevant role in the study of Gaussian processes and, for a generic fBM, are not known in closed form.

Примечание. Тезисы докладов публикуются в авторской редакции

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