Calculating normalization constants of closed queueing networks by numerically inverting their generating functions

Abstract

A new algorithm is developed for calculating normalization constants (partition functions) and moments of product-form steady-state distributions of closed queuing networks and related models. The essential idea is to numerically invert the generating function of the normalization constant and related generating functions appearing in expressions for the moments. 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. For p -dimensional transforms, as occur with queuing networks having p closed chains, the algorithm recursively performs p one-dimensional inversions. The required computation grows exponentially in the dimension, but the dimension can often be reduced by exploiting conditional decomposition based on special structure. For large populations, the inversion algorithm is made more efficient by computing large sums using Euler summation. The inversion algorithm also has a very low storage requirement. A key ingredient in the inversion algorithm is scaling. An effective static scaling is developed for multichain closed queuing networks with only single-server and (optionally) infinite-server queues. An important feature of the inversion algorithm is a self-contained accuracy check, which allows the results to be verified in the absence of alternative algorithms.