On Single-Class Load-Dependent Normalizing Constant Equations

Normalizing constant recurrence equations play an important role in the exact analysis of load-independent (LI) product-form queueing networks. However, they have not been extended to the load-dependent (LD) case, and this is a limitation for new solution techniques based on linear systems of recurrence equations. In this paper, we define LD generalizations of existing LI single-class normalizing constant equations. We first extend Buzen´s convolution expression by introducing the new concept of station rate shift. This also leads us to derive a LD extension of the MVA queue-length recursion that does not involve probabilities. Moreover, we propose a technique for the mean value analysis of queue-dependent functions, which provides a generalization of the network population constraint and new exact formulas for LI models