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

We investigate two classes of multivariate polynomials with variables indexed by the edges of a uniform hypergraph and coefficients depending on certain patterns of unions of edges. These polynomials arise naturally to model job-occupancy in some queuing problems with redundancy scheduling policies. The question, posed by Cardinaels, Borst, and van Leeuwaarden in [Redundancy Scheduling with Locally Stable Compatibility Graphs, arXiv preprint, 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.