# A note on integrating products of linear forms over the unit simplex

## TL;DR

This paper reveals the equivalence between integrating products of linear forms over the unit simplex and computing normalizing constants in queueing networks, highlighting existing algorithms for exact integration.

## Contribution

It establishes a novel connection between polynomial integration and queueing theory, enabling the use of queueing algorithms for exact integration tasks.

## Key findings

- Integration can be performed in polynomial time for fixed variables
- Queueing algorithms can compute exact integrals under certain conditions
- N systems of linear equations suffice for degree N polynomial integration

## Abstract

Integrating a product of linear forms over the unit simplex can be done in polynomial time if the number of variables n is fixed (V. Baldoni et al., 2011). In this note, we highlight that this problem is equivalent to obtaining the normalizing constant of state probabilities for a popular class of Markov processes used in queueing network theory. In light of this equivalence, we survey existing computational algorithms developed in queueing theory that can be used for exact integration. For example, under some regularity conditions, queueing theory algorithms can exactly integrate a product of linear forms of total degree N by solving N systems of linear equations.