Optimizing hypergraph-based polynomials modeling job-occupancy in queueing with redundancy scheduling

TL;DR

This paper studies specific hypergraph-based polynomials modeling job-occupancy in queueing systems with redundancy scheduling, providing conditions under which their minima are achieved at uniform distributions, thus aiding optimization in such systems.

Contribution

It introduces and analyzes two classes of hypergraph polynomials related to queueing models, proving their convexity and optimality properties over the simplex.

Findings

Positive results for the first class of polynomials' minima at uniform distribution.

Partial results and stronger convexity properties for the second class.

Insights into the structure of polynomials modeling redundancy scheduling in queueing theory.

Abstract

We investigate two classes of multivariate polynomials with variables indexed by the edges of a uniform hypergraph and coefficients depending on certain patterns of union of edges. These polynomials arise naturally to model job-occupancy in some queuing problems with redundancy scheduling policy. The question, posed by Cardinaels, Borst and van Leeuwaarden (arXiv:2005.14566, 2020), is to decide whether their global minimum over the standard simplex is attained at the uniform probability distribution. By exploiting symmetry properties of these polynomials we can give a positive answer for the first class and partial results for the second one, where we in fact show a stronger convexity property of these polynomials over the simplex.