scholarly journals On Rice's Formula for Stationary Multivariate Piecewise Smooth Processes

2012 ◽  
Vol 49 (2) ◽  
pp. 351-363 ◽  
Author(s):  
K. Borovkov ◽  
G. Last
Keyword(s):  
Queueing Networks ◽  
Smooth Surface ◽  
Network Models ◽  
Stress Release ◽  
Continuous Vector ◽  
Rice's Formula ◽  
Continuous Vector Field ◽  
Integral Curves ◽  
Piecewise Continuous ◽  
Rice’S Formula

Let X = {Xt: t ≥ 0} be a stationary piecewise continuous Rd-valued process that moves between jumps along the integral curves of a given continuous vector field, and let S ⊂ Rd be a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the intensity of the point process of (localized) continuous crossings of S by X to the distribution of X0. Our result is illustrated by examples relating to queueing networks and stress release network models.

scholarly journals On Rice's Formula for Stationary Multivariate Piecewise Smooth Processes

2012 ◽  
Vol 49 (02) ◽  
pp. 351-363
Author(s):  
K. Borovkov ◽  
G. Last
Keyword(s):  
Queueing Networks ◽  
Smooth Surface ◽  
Network Models ◽  
Stress Release ◽  
Continuous Vector ◽  
Rice's Formula ◽  
Continuous Vector Field ◽  
Integral Curves ◽  
Piecewise Continuous ◽  
Rice’S Formula

LetX= {Xt:t≥ 0} be a stationary piecewise continuousRd-valued process that moves between jumps along the integral curves of a given continuous vector field, and letS⊂Rdbe a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the intensity of the point process of (localized) continuous crossings ofSbyXto the distribution ofX0. Our result is illustrated by examples relating to queueing networks and stress release network models.

scholarly journals Editorial: Special issue on marine safety and Rice’s formula

Extremes ◽  
2011 ◽  
Vol 14 (2) ◽  
pp. 153-155
Author(s):  
Holger Rootzén ◽  
Ross Leadbetter
Keyword(s):  
Special Issue ◽  
Rice's Formula ◽  
Editorial Special Issue ◽  
Rice’S Formula ◽  
Marine Safety

Artificial Polynomial and Trigonometric Higher Order Neural Network Group Models

2013 ◽  
pp. 78-102 ◽  
Author(s):  
Ming Zhang
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Continuous Function ◽  
Trigonometric Polynomial ◽  
Network Models ◽  
Higher Order ◽  
Financial Data ◽  
Neural Network Models ◽  
Piecewise Continuous ◽  
Discontinuous Data

Real world financial data is often discontinuous and non-smooth. Accuracy will be a problem, if we attempt to use neural networks to simulate such functions. Neural network group models can perform this function with more accuracy. Both Polynomial Higher Order Neural Network Group (PHONNG) and Trigonometric polynomial Higher Order Neural Network Group (THONNG) models are studied in this chapter. These PHONNG and THONNG models are open box, convergent models capable of approximating any kind of piecewise continuous function to any degree of accuracy. Moreover, they are capable of handling higher frequency, higher order nonlinear, and discontinuous data. Results obtained using Polynomial Higher Order Neural Network Group and Trigonometric polynomial Higher Order Neural Network Group financial simulators are presented, which confirm that PHONNG and THONNG group models converge without difficulty, and are considerably more accurate (0.7542% - 1.0715%) than neural network models such as using Polynomial Higher Order Neural Network (PHONN) and Trigonometric polynomial Higher Order Neural Network (THONN) models.

scholarly journals Weak solutions of differential equations in Banach spaces

1986 ◽  
Vol 33 (3) ◽  
pp. 407-418 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Vector Field ◽  
Banach Spaces ◽  
Existence Result ◽  
Measure Of Noncompactness ◽  
Continuous Vector ◽  
Weakly Continuous ◽  
Continuous Vector Field ◽  
The Cauchy Problem

We consider the Cauchy problem x (t) = f (t,x (t)), x (0) = x0 in a nonreflexive Banach space X and for f: T × X → X a weakly continuous vector field. Using a compactness hypothesis involving a weak measure of noncompactness we prove an existence result that generalizes earlier theorems by Chow-Shur, Kato and Cramer-Lakshmikantham-Mitchell.

scholarly journals Heavy traffic analysis of a transportation network model

1996 ◽  
Vol 33 (03) ◽  
pp. 870-885
Author(s):  
William P. Peterson ◽  
Lawrence M. Wein
Keyword(s):  
Queueing Networks ◽  
Transportation Network ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Network Models ◽  
Reflected Brownian Motion ◽  
Traffic Conditions ◽  
Heavy Traffic Analysis ◽  
Functional Central Limit ◽  
Heavy Traffic Conditions

We study a model of a stochastic transportation system introduced by Crane. By adapting constructions of multidimensional reflected Brownian motion (RBM) that have since been developed for feedforward queueing networks, we generalize Crane's original functional central limit theorem results to a full vector setting, giving an explicit development for the case in which all terminals in the model experience heavy traffic conditions. We investigate product form conditions for the stationary distribution of our resulting RBM limit, and contrast our results for transportation networks with those for traditional queueing network models.

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