An Algorithm for Closed Queueing Networks Based on Numerical Transform Inversion

We propose a new algorithm for closed queueing networks and related product-form models based on numerical inversion of the generating function of the normalization constant (or partition function). It is known that the generating function of the normalization constant often has a remarkably simple form, but numerical inversion evidently has not been considered before. We also show that moments of steady-state distributions can be calculated directly by only performing two inversions. For closed queueing networks with p closed chains, the generating function is p dimensional. For these generating functions, the algorithm recursively performs p one-dimensional inversions. The required computation grows exponentially in the dimension, but we show that the dimension can often be reduced dramatically by exploiting special structure. Other key ingredients in the algorithm are scaling and the computation of large sums efficiently by Euler summation. Numerical examples indicate that this new algorithm can usefully complement previous algorithms.