Classic and non-prophetic model checking for hybrid Petri nets with stochastic firings

Nondeterminism occurs naturally in Petri nets whenever multiple events are enabled at the same time. Traditionally, it is resolved at specification time using probability weights and priorities. In this paper, we focus on model checking for hybrid Petri nets with an arbitrary but finite number of stochastic firings (HPnGs) while preserving the inherent nondeterminism as a first-class modelling and analysis feature. We present two algorithms to compute optimal non-prophetic and prophetic schedulers. The former can be applied to all HPnG models while the latter is only applicable if information on the firing times of general transitions is specifically encoded in the model. Both algorithms make use of recent work on the parametric location tree, which symbolically unfolds the state space of an HPnG. A running example illustrates the approach and confirms the feasibility of the presented algorithm.

[1]  Joost-Pieter Katoen,et al.  A theory of stochastic systems part I: Stochastic automata , 2005, Inf. Comput..

[2]  René David,et al.  Continuous and Hybrid Petri Nets , 1998, J. Circuits Syst. Comput..

[3]  Boudewijn R. Haverkort,et al.  Survivability analysis of a sewage treatment facility using hybrid Petri nets , 2016, Perform. Evaluation.

[4]  Holger Hermanns,et al.  Schedulers are no Prophets , 2015, Semantics, Logics, and Calculi.

[5]  Kousha Etessami,et al.  Multi-Objective Model Checking of Markov Decision Processes , 2007, Log. Methods Comput. Sci..

[6]  Marco Ajmone Marsan,et al.  Modelling with Generalized Stochastic Petri Nets , 1995, PERV.

[7]  C. Petri.  Kommunikation mit Automaten , 1962 .

[8]  Joost-Pieter Katoen,et al.  A compositional modelling and analysis framework for stochastic hybrid systems , 2012, Formal Methods in System Design.

[9]  Arie Hordijk,et al.  Constrained Undiscounted Stochastic Dynamic Programming , 1984, Math. Oper. Res..

[10]  Mario Bravetti,et al.  Tutte le Algebre Insieme: Concepts, Discussions and Relations of Stochastic Process Algebras with General Distributions , 2004, Validation of Stochastic Systems.

[11]  Lars Michael Kristensen,et al.  Coloured Petri Nets - Modelling and Validation of Concurrent Systems , 2009 .

[12]  Lijun Zhang,et al.  On Probabilistic Automata in Continuous Time , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[13]  Anne Remke,et al.  Model Checking HPnGs in Multiple Dimensions: Representing State Sets as Convex Polytopes , 2019, FORTE.

[14]  Boudewijn R. Haverkort,et al.  Energy Resilience Modelling for Smart Houses , 2015, 2015 45th Annual IEEE/IFIP International Conference on Dependable Systems and Networks.

[15]  Anne Remke,et al.  Statistical Model Checking for Hybrid Petri Nets with Multiple General Transitions , 2017, 2017 47th Annual IEEE/IFIP International Conference on Dependable Systems and Networks (DSN).

[16]  H.A.P. Blom,et al.  Enhancing hybrid state Petri nets with the analysis power of stochastic hybrid processes , 2008, 2008 9th International Workshop on Discrete Event Systems.

[17]  Marco Gribaudo,et al.  Hybrid Petri nets with general one-shot transitions , 2016, Perform. Evaluation.

[18]  Sean Sedwards,et al.  A Hierarchy of Scheduler Classes for Stochastic Automata , 2017, FoSSaCS.

[19]  Lijun Zhang,et al.  A Semantics for Every GSPN , 2013, Petri Nets.

[20]  Karl Johan Åström,et al.  Optimal control of Markov processes with incomplete state information , 1965 .

[21]  Pedro R. D'Argenio,et al.  Distributed probabilistic input/output automata: Expressiveness, (un)decidability and algorithms , 2014, Theor. Comput. Sci..

[22]  Anne Condon,et al.  On the Undecidability of Probabilistic Planning and Infinite-Horizon Partially Observable Markov Decision Problems , 1999, AAAI/IAAI.

[23]  Anne Remke,et al.  Does Your Domestic Photovoltaic Energy System Survive Grid Outages , 2016 .

[24]  Lijun Zhang,et al.  Measurability and safety verification for stochastic hybrid systems , 2011, HSCC '11.

[25]  Anne Remke,et al.  State-space Construction of Hybrid Petri Nets with Multiple Stochastic Firings , 2019, QEST.

[26]  David M. Nicol,et al.  Fluid stochastic Petri nets: Theory, applications, and solution techniques , 1998, Eur. J. Oper. Res..

[27]  G. Lepage.  A new algorithm for adaptive multidimensional integration , 1978 .