Generalized Convexity Properties and Shape Based Approximation in Networks Reliability

TL;DR

This paper introduces a shape-preserving approximation method for reliability polynomials of dual minimal networks using quadratic splines, enhancing accuracy and efficiency while maintaining key convexity properties.

Contribution

It develops a novel approximation technique based on generalized convexity and shape knowledge, specifically for reliability polynomials of dual minimal networks.

Findings

The method preserves the shape properties of the original reliability polynomials.

Numerical examples demonstrate low complexity and small approximation errors.

The approach can be refined to increase accuracy.

Abstract

Some properties of generalized convexity for sets and for functions are identified in case of the reliability polynomials of two dual minimal networks. A method of approximating the reliability polynomials of two dual minimal network is developed based on their mutual complementarity properties. The approximating objects are from the class of quadratic spline functions, constructed based both on interpolation conditions and on shape knowledge. It is proved that the approximant objects preserve the shape properties of the exact reliability polynomials. Numerical examples and simulations show the performance of the algorithm, both in terms of low complexity, small error and shape preserving. Possibilities of increasing the accuracy of approximation are discussed.