From Systems to Components: Constructive Methods for Product-Form Solutions

Product-form models are a class of Markovian models whose steady-state solutions can be computed efficiently thanks to separable equilibrium probability distributions. They first appeared in queuing theory with the well-know result of Jackson networks. Many research efforts have been devoted to the identification of new product-form models both for queueing networks and for other formalisms (e.g. for Markovian process algebra or for stochastic Petri nets etc.). The results presented in [1], [2], [3] allow the computation of the product-form solutions of Markovian models via structural analysis of their underlying processes and constraints on the rates of the corresponding reversed processes. These results greatly simplify the derivation of steady-state distributions, as they offer a constructive way to establish whether a Continuous Time Markov Chain (CTMC) enjoys product-form solution. Furthermore, the results provide a practical methodology to transform models which are not in product-form, into approximated ones that do enjoy product-form solutions [4]. The methodology relies on representation of CTMCs as Labelled Markov Automata. The latter formalism appears to be simple and expressive enough for the purpose of our work.