  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.
