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.

翻译：若变量数n固定，单位单纯形上线性形式乘积的积分可在多项式时间内完成（V. Baldoni 等人，2011）。本文指出，该问题等价于排队网络理论中一类常用马尔可夫过程状态概率归一化常数的求解。基于该等价关系，我们系统梳理了排队理论中可用于精确积分的现有计算算法。例如，在特定正则性条件下，排队理论算法可通过求解N个线性方程组，精确计算总次数为N的线性形式乘积的积分。