Buzen's algorithm

In queueing theory, a discipline within the mathematical theory of probability, Buzen's algorithm (or convolution algorithm) is an algorithm for calculating the normalization constant G(N) in the Gordon–Newell theorem. This method was first proposed by Jeffrey P. Buzen in 1973. Computing G(N) is required to compute the stationary probability distribution of a closed queueing network.

Performing a naïve computation of the normalising constant requires enumeration of all states. For a system with N jobs and M states there are ${\displaystyle {\tbinom {N+M-1}{M-1}}}$ states. Buzen's algorithm "computes G(1), G(2), ..., G(N) using a total of NM multiplications and NM additions." This is a significant improvement and allows for computations to be performed with much larger networks.

Consider a closed queueing network with M service facilities and N circulating customers. Write n_i(t) for the number of customers present at the ith facility at time t, such that ${\displaystyle \scriptstyle \sum _{i=1}^{M}n_{i}=N}$. We assume that the service time for a customer at the ith facility is given by an exponentially distributed random variable with parameter μ_i and that after completing service at the ith facility a customer will proceed to the jth facility with probability p_ij.

...
Wikipedia