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Large Deviations for Gaussian Queues -- Modelling Communication Networks

高斯序列的大偏差：通信网络建模

ISBN：9780470015230，出版年：2007，中图分类号：TN 被引 63次

Preface and acknowledgments. 1 Introduction. Part A: Gaussian traffic and large deviations. 2 The Gaussian source model. 2.1 Modeling network traffic. 2.2 Notation and preliminaries on Gaussian random variables. 2.3 Gaussian sources. 2.4 Generic examples-long-range dependence and smoothness. 2.5 Other useful Gaussian source models. 2.6 Applicability of Gaussian source models for network traffic. 3 Gaussian sources: validation, justification. 3.1 Validation. 3.2 Convergence of on-off traffic to a Gaussian process. 4 Large deviations for Gaussian processes. 4.1 Cram-er's theorem. 4.2 Schilder's theorem. Part B: Large deviations of Gaussian queues. 5 Gaussian queues: an introduction. 5.1 Lindley's recursion, the steady-state buffer content. 5.2 Gaussian queues. 5.3 Special cases: Brownian motion and Brownian bridge. 5.4 A powerful approximation. 5.5 Asymptotics. 5.6 Large-buffer asymptotics. 6 Logarithmic many-sources asymptotics. 6.1 Many-sources asymptotics: the loss curve. 6.2 Duality between loss curve and variance function. 6.3 The buffer-bandwidth curve is convex. 7 Exact many-sources asymptotics. 7.1 Slotted time: results. 7.2 Slotted time: proofs. 7.3 Continuous time: results. 7.4 Continuous time: proofs. 8 Simulation. 8.1 Determining the simulation horizon. 8.2 Importance sampling algorithms. 8.3 Asymptotic efficiency. 8.4 Efficient estimation of the overflow probability. 9 Tandem and priority queues. 9.1 Tandem: model and preliminaries. 9.2 Tandem: lower bound on the decay rate. 9.3 Tandem: tightness of the decay rate. 9.4 Tandem: properties of the input rate path. 9.5 Tandem: examples. 9.6 Priority queues. 10 Generalized processor sharing. 10.1 Preliminaries on GPS. 10.2 Generic upper and lower bound on the overflow probability. 10.3 Lower bound on the decay rate: class 2 in underload. 10.4 Upper bound on the decay rate: class 2 in underload. 10.5 Analysis of the decay rate: class 2 in overload. 10.6 Discussion of the results. 10.7 Delay asymptotics. 11 Explicit results for short-range dependent inputs. 11.1 Asymptotically linear variance some preliminaries. 11.2 Tandem queue with srd input. 11.3 Priority queue with srd input. 11.4 GPS queue with srd input. 11.5 Concluding remarks. 12 Brownian queues. 12.1 Single queue: detailed results. 12.2 Tandem: distribution of the downstream queue. 12.3 Tandem: joint distribution. Part C: Applications. 13 Weight setting in GPS. 13.1 An optimal partitioning approach to weight setting. 13.2 Approximation of the overflow probabilities. 13.3 Fixed weights. 13.4 Realizable region. 14 A link dimensioning formula and empirical support. 14.1 Objectives, modeling, and analysis. 14.2 Numerical study. 14.3 Empirical study. 14.4 Implementation aspects. 15 Link dimensioning: indirect variance estimation. 15.1 Theoretical foundations. 15.2 Implementation issues. 15.3 Error analysis of the inversion procedure. 15.4 Validation. 16 A framework for bandwidth trading. 16.1 Bandwidth trading. 16.2 Model and preliminaries. 16.3 Single-link network. 16.4 Gaussian traffic utility as a function of loss. 16.5 Sanov's theorem and its inverse. 16.6 Estimation of loss probabilities. 16.7 Numerical example. Bibliography. Index.

Gaussian random field Gaussian process Gaussian function Gaussian Priority queue Large deviations theory Brownian bridge Random variable Applied mathematics Mathematical optimization Mathematics Brownian bridge

Multidimensional Stochastic Processes as Rough Paths: Theory and Applications

多维随机过程的艰难路径

Cambridge University Press Peter K Friz

ISBN：9780511686511，出版年：2010，中图分类号：O21

Rough path analysis provides a fresh perspective on Ito's important theory of stochastic differential equations. Key theorems of modern stochastic analysis (existence and limit theorems for stochastic flows, Freidlin-Wentzell theory, the Stroock-Varadhan support description) can be obtained with dramatic simplifications. Classical approximation results and their limitations (Wong-Zakai, McShane's counterexample) receive 'obvious' rough path explanations. Evidence is building that rough paths will play an important role in the future analysis of stochastic partial differential equations and the authors include some first results in this direction. They also emphasize interactions with other parts of mathematics, including Caratheodory geometry, Dirichlet forms and Malliavin calculus. Based on successful courses at the graduate level, this up-to-date introduction presents the theory of rough paths and its applications to stochastic analysis. Examples, explanations and exercises make the book accessible to graduate students and researchers from a variety of fields.

Symmetric multiprocessing Stochastic theory Asymmetric multiprocessing Stochastic process Large deviations theory Random utility theory Decomposition method (queueing theory) Path (graph theory) Theory of multiple intelligences Statistical learning theory