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Bounds on the rate of convergence for one class of inhomogeneous Markovian queueing models with possible batch arrivals and services

Zeifman A., Razumchik R., Satin Y., Kiseleva K., Korotysheva A., Korolev V.

International Journal of Applied Mathematics and Computer Science

2018

Vol. 28, no. 1

141--154

In this paper we present a method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically, we consider an inhomogeneous M/M/S queueing system with possible state-dependent arrival and service intensities, and additionally possible batch arrivals and batch service. A unified approach based on a logarithmic norm of linear operators for obtaining sharp upper and lower bounds on the rate of convergence and corresponding sharp perturbation bounds is described. As a side effect, we show, by virtue of numerical examples, that the approach based on a logarithmic norm can also be used to approximate limiting characteristics (the idle probability and the mean number of customers in the system) of the systems considered with a given approximation error.

Certain generalized q-operators

Prakash O., Sharma D., Maheshwari P.

Demonstratio Mathematica

2015

Vol. 48, nr 3

404--412

The applications of q-calculus in the approximation theory is a very interesting area of research in the recent years, several new q-operators were introduced and their behaviour were discussed by many researchers. This paper is the extension of the paper [15], in which Durrmeyer type generalization of q-Baskakov–Stancu type operators were discussed by using the concept of q-integral operators. Here, we propose to study the Stancu variant of q-Baskakov–Stancu type operators. We establish an estimate for the rate of convergence in terms of modulus of continuity and weighted approximation properties of these operators.

Contributions to the Horn-Schunck Optical Flow Equations -Part III: Alternating Iteration Algorithms

Dong G., An X., Hu D.

Przegląd Elektrotechniczny

2013

R. 89, nr 3b

193--197

The Horn-Schunck equations are a coupled system of two partial differential equations which aim at finding motion information in a given image sequence. Recent results [1, 2] asserted that this system is well-posed and can not be decoupled under any linear transformations. In this paper, two alternating iterative algorithms are proposed to solve this system. These algorithms have three properties: first, at each single iteration, both algorithms consist of two decoupled, scalar equations of elliptic type, driven by the last approximate solutions; second, the particular form of iterations allows analytical solutions expressed via potential integral, Poisson integral, conformal mapping and Feynman-Kac formula; third, exponential convergence of these algorithms are established under mild conditions and the rates of convergence are given with the help of energy inequalities and Banach fixed point principle for contraction mappings. Limitations of these algorithms are discussed.

W artykule zaproponowano sposób analizy i rozwiązywania równań Horn’a-Schunck’a. Rozwiązanie polega na zastosowaniu dwóch algorytmów o przemiennej iteracji. W każdym kroku obydwa algorytmy składają się z dwóch niezależnych eliptycznych równań skalarnych, bazujących na ostatnim przybliżonym rozwiązaniu. Otrzymane rozwiązanie może być wyrażone poprzez całkę potencjału, całkę Poisson’a, odwzorowanie wiernokątne, formułę Feynman’a-Kac’a. Przedstawiono i omówiono ograniczenia stosowania proponowanych algorytmów.