Continuous and Hybrid Petri Nets

Abstract

Petri nets are widely used to model discrete event dynamic systems (computer systems, manufacturing systems, communication systems…). When a PN contains a large number of tokens, the number of reachable states explodes. This is a practical limitation to the use of Petri nets. Continuous models may provide very good approximations for discrete event systems: this is the basic idea leading to the definition of continuous Petri nets. A continuous PN is a model in which the marking of each place is a real number. In a timed continuous PN, a firing speed is associated with each transition (this basic model is unique although the firing speed may be defined in several ways). Various timed continuous PN models have been defined and they correspond to a specific calculation of the firing speeds. They provide good approximations for performance evaluation when a PN contains a large number of tokens. Modeling a number of parts in a buffer by a real number may generally be an acceptable approximation. However, the state of a machine, up or down, cannot be modeled by a real number. This naturally leads to the idea of hybrid PNs, containing a discrete part and a continuous part. Autonomous and timed hybrid PNs are presented. A large of the presentation is informal and intuitive. Three examples illustrate the application of the different models.

Efficient steady-state analysis of second-order fluid stochastic Petri nets

“…This CPU time should be compared to 12, 995 seconds as reported in [10]. 2 This implies that the new second-order FSPN model and the solution technique we developed almost halved the total CPU time required to obtain all the model results without lowering their accuracy. Furthermore, it should also be noted that the complexity of the solution for the second-order FSPN model is only dependent on the number of discrete states that, in our case, is equal to K + 1 where K is the number of TCP connections.…”

“…Figure 1 represents all the graphic primitives of the second-order FSPN used in this paper. Note that, similarly to [2], we have chosen to use fluid transitions instead of fluid arcs. In this case, the flow rates and flow variances are associated to transitions rather than to fluid arcs.…”

“…For this reason, higher level mathematical formalisms have been introduced in the literature for the specification of such models. For first-order fluid models, an example of such higher-level formalisms is first-order Fluid Stochastic Petri Nets (FSPN) [22,15,7,2,14]; FSPN models are analyzed by writing the equations of the underlying stochastic process (a first-order fluid model) and then by solving them numerically. The equations are written by decoupling the discrete part of the model from the continuous one.…”

Operational Management of Supply Chains: A Hybrid Petri Net Approach

“…This assumption is not realistic in large systems with a huge amount of material flow: the state space of the SC model generally turns out to be excessively large, so that inconveniences in the simulation and performance optimization often arise, leading to large computational efforts. Since SCs are DEDSs whose number of reachable states is very large, PN formalisms using fluid approximations provide an 138 aggregate formulation to reduce the state space dimension (Alla & David, 1998;Silva & Recalde, 2004). Hence, hybrid PNs may be employed to describe SCs efficiently and effectively at the operational level.…”

On Controllability of Timed Continuous Petri Nets

“…Using fluid models, more analytical techniques can be used for the analysis of some interesting properties. In Petri Nets, fluidification has been introduced from different perspectives ( [2,3]). Here we consider the approach adopted in [4].…”