Product-forms in Stochastic Petri nets (SPNs) are obtained by a compositional technique for the first time, by combining small SPNs with product-forms in a hierarchical manner. In this way, performance engineering methodology is enhanced by the greatly improved efficiency endowed to the steady-state solution of a much wider range of Markov models. Previous methods have relied on analysis of the whole net and so are not incremental - hence they are intractable in all but small models. We show that the product-form condition for open nets depends, in general, on the transition rates, whereas closed nets have only structural conditions for a product-form, except in rather pathological cases. Both the "building blocks" formed by the said small SPNs and their compositions are solved for their product-forms using the Reversed Compound Agent Theorem (RCAT), which, to date, has been used exclusively in the context of process-algebraic models. The resulting methodology provides a powerful, general and rigorous route to product-forms in large stochastic models and is illustrated by several detailed examples. © 2012 Elsevier Inc. All rights reserved.
Balsamo, S., Harrison, P. G., & Marin, A. (2012). Methodological construction of product-form stochastic Petri nets for performance evaluation. Journal of Systems and Software, 85(7), 1520–1539. https://doi.org/10.1016/j.jss.2011.11.1042