Queueing networks and conditional product-forms

Product-forms are well-known in the community of performance evaluation because they allow the computation of the stationary state probabilities of large models that would otherwise be intractable. Roughly speaking, a product-form model consists of several interacting components. Under some conditions, the steady-state probabilities of these components can be derived in isolation as if the interactions with the remaining parts of the system are modelled by independent Poisson processes. The steady-state distribution of the joint model can be derived as the (normalised) product of the distributions of the isolated components. In the last few years some authors have introduced the idea of higher order product-forms or conditional product-forms that differ from ordinary product-forms because once the components of a model are isolated, the interactions with the rest of the system are not anymore seen as independent Poisson processes. However, to the best of our knowledge, up to now these methods have been applied only to approximate non-product-form models. In this paper we propose for the first time two classes of feed-forward queueing network models whose stationary distributions have conditional product-forms but not ordinary product-forms.