Analysis of a three node queueing network

The three node Jackson queueing network is the simplest acyclic network in which in equilibrium the sojourn times of a customer at each of the nodes are dependent. We show that assuming the individual sojourn times are independent provides a good approximation to the total sojourn time. This is done by simulating the network and showing that the sojourn times generally pass a Kolmogorov-Smirnov test as having come from the approximating distribution. Since the sum of dependent random variables may have the same distribution as the sum of independent random variables with the same marginal distributions, it is conceivable that our approximation is exact. However, we numerically compute upper and lower bounds for the distribution of the total sojourn time; these bounds are so close that the approximating distribution lies outside of the bounds. Thus, the bounds are accurate enough to distinguish between the two distributions even though the Kolmogorov-Smirnov test generally cannot.     

A NEW METHOD FOR THE UNCERTAIN RESPONSE ANALYSIS OF STRUCTURES WITH UNCERTAIN PARAMETERS

A computing method for estimating the upper and lower bounds of the response ofstructures with uncertainties is presented.The uncertain parameters are described by the convexmodel.A numerical example of the frame structure is given to illustrate the effectiveness of thismethod.     

Recursive evaluation of aggregate claims distributions

In this paper, we give a survey of recursive methods presented in the actuarial literature for exact and approximate evaluation of univariate and multivariate aggregate claims distributions. For the approximations, we present error bounds.     

The Container Transshipment Problem: Searching Representation Landscapes with Metaheuristic Algorithms

The container transshipment problem involves scheduling a fleet of lorries to collect and deliver containers of various sizes while minimizing the total distance travelled. The problem originates in the need for logistics companies to solve the problem on a regular basis as part of their daily operations. In this paper, we compare two genetic algorithms tailored to solve this problem based on permutation and bin-packing inspired encodings. Results are presented and analysed in order to evaluate the validity and robustness of the two approaches. As part of the analysis, bounds were calculated to determine how well both GAs perform in absolute terms as well as relative to each other. Of the two GA there is one clear winner, although it is not the one that would have been indicated by previous research. Whilst the winning GA is able to generate significant savings in practice, compared to the optimum there remains room for further improvement.     

Tight frame expansions of multiscale reproducing kernels in Sobolev spaces

Multiscale kernels are a new type of positive definite reproducing kernels in Hilbert spaces. They are constructed by a superposition of shifts and scales of a single refinable function and were introduced in the paper of R. Opfer [Multiscale kernels, Adv. Comput. Math. (2004), in press]. By applying standard reconstruction techniques occurring in radial basis function- or machine learning theory, multiscale kernels can be used to reconstruct multivariate functions from scattered data. The multiscale structure of the kernel allows to represent the approximant on several levels of detail or accuracy. In this paper we prove that multiscale kernels are often reproducing kernels in Sobolev spaces. We use this fact to derive error bounds. The set of functions used for the construction of the multiscale kernel will turn out to be a frame in a Sobolev space of certain smoothness. We will establish that the frame coefficients of approximants can be computed explicitly. In our case there is neither a need to compute the inverse of the frame operator nor is there a need to compute inner products in the Sobolev space. Moreover we will prove that a recursion formula between the frame coefficients of different levels holds. We present a bivariate numerical example illustrating the mutiresolution and data compression effect.     

Extended degree functions and monomial modules

The arithmetic degree, the smallest extended degree, and the homological degree are invariants that have been proposed as alternatives of the degree of a module if this module is not Cohen-Macaulay. We compare these degree functions and study their behavior when passing to the generic initial or the lexicographic submodule. This leads to various bounds and to counterexamples to a conjecture of Gunston and Vasconcelos, respectively. Particular attention is given to the class of sequentially Cohen-Macaulay modules. The results in this case lead to an algorithm that computes the smallest extended degree.

     

Merit functions for general variational inequalities

In this paper, we consider some classes of merit functions for general variational inequalities. Using these functions, we obtain error bounds for the solution of general variational inequalities under some mild conditions. Since the general variational inequalities include variational inequalities, quasivariational inequalities and complementarity problems as special cases, results proved in this paper hold for these problems. In this respect, results obtained in this paper represent a refinement of previously known results for classical variational inequalities.     

The mixing efficiency of open flows

Mixing in open incompressible flows is studied in a model problem with inhomogeneous passive scalar injection on an inlet boundary. As a measure of the efficiency of stirring, the bulk scalar concentration variance is bounded, and the bound is shown to be sharp at low Péclet number. Although no specific flow saturating the bound at high Péclet number is produced here, the estimate is conjectured to be approached for flows possessing sufficiently sustained chaotic regions.     

Convolutions of singular measures and applications to the Zakharov system

Uniform L²-estimates for the convolution of singular measures with respect to transversal submanifolds are proved in arbitrary space dimension. The results of Bennett-Bez are used to extend previous work of Bejenaru-Herr-Tataru. As an application, it is shown that the 3D Zakharov system is locally well-posed in the full subcritical regime.     

Statistical ensemble error bounds for homogenized microheterogeneous solids

Typically, in order to characterize the homogenized effective macroscopic response of new materials possessing random heterogeneous microstructure, a relation between averages is sought, where and where and are the stress and strain tensor fields within a statistically representative volume element (SRVE) of volume ||. The quantity, is known as the effective property, and is the elasticity tensor used in usual macroscale analyses. In order to generate homogenized responses computationally, a series of detailed boundary value representations resolving the heterogeneous microstructure, posed over the SRVEs domain, must be solved. This requires an enormous numerical effort that can overwhelm most computational facilities. A natural way of generating an approximation to the SRVEs response is by first computing the response of smaller (subrepresentative) samples, each with a different random realization of the microstructural type under investigation, and then to ensemble average the results afterwards. Compared to a direct simulation of an SRVE, testing many small samples is a computationally inexpensive process since the number of floating point operations is greatly reduced, as well as the fact that the samples responses can be computed trivially in parallel. However, there is an inherent error in this process. Clearly the populations ensemble average is not the SRVEs response. However, as shown in this work, the moments on the distribution of the population can be used to generate rigorous upper and lower error bounds on the quality of the ensemble-generated response. Two-sided bounds are given on the SRVE response in terms of the ensemble average, its standard deviation and its skewness.Received: December 11, 2001