Individuals, populations and fluid approximations: A Petri net based perspective

Abstract

Discrete Event Systems (DES) theory and engineering are mainly driven by needs that arise in many different human-made systems (manufacturing, communications, logistics, workflow management, traffic, etc.). With the accelerated increase in the complexity and size of new technological constructions, the state explosion problem in DES analysis and synthesis becomes more and more acute.Two traditional conceptual and complementary ways of dealing with the computational complexities in the Petri nets (PN) framework are structure theory (that investigates the relationship between the behavior of a net system and its structure) and fluid relaxations, here leading to particular classes of hybrid systems. In the second case, the expected computational gains for analysis and synthesis problems are usually achieved at the expense of the fidelity or accuracy of the relaxed model. This invited overview will mainly focus on the second strategy, nevertheless always interspersed with basic structural concepts and methods. Using an example-driven approach, starting with a DES “view of the system”, the legitimization and improvement of fluidization process, the aggregation of local states by symmetries and the decolorization of models will be briefly addressed, together with reflections about the analysis of the new models obtained.As the linearization of a continuous dynamical system, the fluidization of a DES is a relaxation that has to be used with care, depending on the problem at hand. This abstraction is here considered from two complementary perspectives: at logical and at performance levels, both for untimed and timed PNs. On the one hand, the expressive power of timed fluid PNs under infinite server semantics is such that the simulation of Turing machines is possible. From a complementary perspective, the expression of modeling capabilities such as non-monotonicities and bifurcations may also be revealed for steady-state behaviors. Symmetries (more generally, lumping) seek to group together “equivalent” behaviors and decolorization seeks to abstract identities, in order to create new collectivities of processes and resources. The synergy between symmetry-decolorization state-aggregation approaches and fluid relaxations is highlighted. In fact, the first approaches not only reduce the state space, but also “produce” populations, thus proceed upgrading the applicability of fluidization. Opening the window, related issues such as control, optimization, observation or diagnosis are briefly pointed out. For conciseness, this work is limited to fully fluid (or continuous) PN models and their relationships with the corresponding discrete systems.