A note on integrating products of linear forms over the unit simplex
This work connects computational integration to queueing theory, providing an alternative approach for a specific mathematical problem, but it is incremental as it surveys and applies existing methods.
The paper tackles the problem of integrating products of linear forms over the unit simplex by showing its equivalence to computing normalizing constants in queueing network Markov processes, and surveys existing queueing theory algorithms that can perform exact integration by solving N systems of linear equations for total degree N.
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.