Piecewise deterministic Markov processes represented by dynamically coloured Petri nets

Piecewise deterministic Markov processes (PDPs) are known as the largest class of Markov processes virtually describing all continuous-time processes not involving diffusions. For PDPs, a substantial amount of powerful analysis and control results are available. As such, PDPs are attractive for use in modelling complex distributed systems. However, the specification of an appropriate PDP model for complex distributed systems that exist in practice is far from trivial. This difficulty already applies for the specification of a continuous-time Markov chain (CTMC). For a compositional specification of a CTMC model, Petri Nets have proven to be extremely useful. In order to realise a similar situation for PDP, this paper develops a novel type of Petri Net, named dynamically coloured Petri Net (DCPN), and proves that there exist into-mappings between PDPs and DCPNs.† †Part of this research has been performed within the project HYBRIDGE of the European Commission, contract number IST-2001-32460

[1]  Mark H. Davis.  Markov Models and Optimization , 1995 .

[2]  Henk A. P. Blom,et al.  Air traffic collision risk modelling , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[3]  John Lygeros,et al.  Synthesizing Controllers for Nonlinear Hybrid Systems , 1998, HSCC.

[4]  Mark H. A. Davis.  Piecewise‐Deterministic Markov Processes: A General Class of Non‐Diffusion Stochastic Models , 1984 .

[5]  Kishor S. Trivedi,et al.  Power-hierarchy of dependability-model types , 1994 .

[6]  Christos G. Cassandras,et al.  Introduction to Discrete Event Systems , 1999, The Kluwer International Series on Discrete Event Dynamic Systems.

[7]  Nancy A. Lynch,et al.  Verification of Automated Vehicle Protection Systems , 1995, Hybrid Systems.

[8]  Stephen P. Banks,et al.  Modelling of Hybrid Systems Based on Extended Coloured Petri Nets , 1994, Hybrid Systems.

[9]  Kurt Jensen,et al.  Coloured Petri Nets: Basic Concepts, Analysis Methods and Practical Use. Vol. 2, Analysis Methods , 1992 .

[10]  René David,et al.  Petri nets for modeling of dynamic systems: A survey , 1994, Autom..

[11]  Kishor S. Trivedi,et al.  Techniques for System Dependability Evaluation , 2000 .

[12]  Jan H. van Schuppen,et al.  A Sufficient Condition for Controllability of a Class of Hybrid Systems , 1998, HSCC.

[13]  Isabel Demongodin,et al.  Differential Petri nets: representing continuous systems in a discrete-event world , 1998 .

[14]  Henk A. P. Blom,et al.  Petri-Nets and Hybrid-State Markov Processes in a Power-Hierarchy of Dependability Models , 2003, ADHS.

[15]  Giovanni Chiola,et al.  An introduction to generalized stochastic Petri nets , 1991 .

[16]  G. J. Bakker,et al.  ACCIDENT RISK ASSESSMENT FOR ADVANCED ATM , 1999 .

[17]  Kishor S. Trivedi,et al.  FSPNs: Fluid Stochastic Petri Nets , 1993, Application and Theory of Petri Nets.

[18]  Thomas A. Henzinger,et al.  Hybrid Automata: An Algorithmic Approach to the Specification and Verification of Hybrid Systems , 1992, Hybrid Systems.

[19]  Gianfranco Balbo,et al.  Introduction to Generalized Stochastic Petri Nets , 2007, SFM.

[20]  R. Valette,et al.  Modeling and simulation of a hybrid system through PR/TR PN-DAE model , 1998 .

[21]  John Lygeros,et al.  An Approach to the Verification of the Center-TRACON Automation System , 1998, HSCC.

[22]  Michael S. Branicky,et al.  Studies in hybrid systems: modeling, analysis, and control , 1996 .

[23]  Henk A. P. Blom,et al.  Dynamically Coloured Petri Nets for Air Traffic Management Safety Purposes , 1997 .

[24]  P. Protter.  Stochastic integration and differential equations : a new approach , 1990 .

[25]  Alessandro Giua,et al.  Hybrid Petri Nets: a , 1996 .