Buzen's algorithm

Buzen's algorithm is an algorithm related to queueing theory used to calculate the normalization constant $G(N)$ for a closed network. (Is this true?)

It can be used when $G(N)$ is of the form:

$G(N)=\sum _{n_{1}+n_{2}+\cdot \cdot \cdot +n_{M}=N}\prod _{i=1}^{M}f_{i}(n_{i})$ with $f_{i}(n)=y_{i}^{n}$ for $1<i<=M$

This sum can be calculated efficiently in O(MN) time by utilizing a definition with a recursive structure:

let $G(N)=g(m,n)$

$g(m,n)=\sum _{n_{1}+n_{2}+\cdot \cdot \cdot +n_{m}=n}\prod _{i=1}^{m}f_{i}(n_{i})=\sum _{n_{1}+\cdot \cdot \cdot +n_{m-1}=n,n_{m}=0}\prod _{i=1}^{m-1}f_{i}(n_{i})+\sum _{n_{1}+\cdot \cdot \cdot +n_{m}=n,n_{m}>0}\prod _{i=1}^{m}f_{i}(n_{i})=g(m-1,n)+{\frac {f_{m}(n)}{f_{m}(n-1)}}g(m,n-1)$ (Is this last bit true?)

$g(m,0)=1$, $g(1,n)=f_{1}(n)$