Fluid Stochastic Petri Nets: An Extended Formalism to Include non-Markovian Models

A Fluid Stochastic Petri Net (FSPN) formalism, where there are two kind of places, one which carries discrete tokens and the other which contains continuous quantity, is presented and discussed. In the proposed formulation, a new primitive is introduced, called flush-out arc. A flush-out arc connects a transition to a continuous place and has the effect of instantaneously empty the place when the transition fires. With this extension, FSPNs can be viewed as a graphical formalism to represent stochastic models with reward rates that can be dependent on the discrete as well as the continuous component of the state space descriptor. First the model is formally introduced and the integro-differential equations representing the dynamic of the system are fully derived in the case of a single continuous place. However, the goal of the paper is to propose a first step towards the automatic solution of a general FSPN model starting from its graphical description. Furthermore, in order to illustrate the potentiality of the approach, we show that the proposed formalism is suited to convert a non-Markovian SPN, of the type considered up to now in the literature, into a FSPN. Since, however, the FSPN formalism is more general and flexible, various modeling extensions, not conceivable in the non-Markovian SPN setting, are investigated.