A Numerical Algorithm for the Solution of Product-Form Models with Infinite State Spaces

Abstract

Markovian models play a pivotal role in system performance evaluation field. Several high level formalisms are capable to model systems consisting of some interacting sub-models, but often the resulting underlying process has a number of states that makes the computation of the solution unfeasible. Product-form models consist of a set of interacting sub-models and have the property that their steady-state solution is the product of the sub-model solutions considered in isolation and opportunely parametrised. The computation of the steady-state solution of a composition of arbitrary and possibly different types of models in product-form is still an open problem. It consists of two parts: a) deciding whether the model is in product-form and b) in this case, compute the stationary distribution efficiently. In this paper we propose an algorithm to solve these problems that extends that proposed in [14] by allowing the sub-models to have infinite state spaces. This is done without a-priori knowledge of the structure of the stochastic processes underlying the model components. As a consequence, open models consisting of non homogeneous components having infinite state space (e.g., a composition of G-queues, G-queues with catastrophes, Stochastic Petri Nets with product-forms) may be modelled and efficiently studied.