Stochastic equivalence for performance analysis of concurrent systems in dtsiPBC

We propose an extension with immediate multiactions of discrete time stochastic Petri Box Calculus (dtsPBC), presented by I.V. Tarasyuk. The resulting algebra dtsiPBC is a discrete time analogue of stochastic Petri Box Calculus (sPBC) with immediate multiactions, designed by H. Maci\`a, V. Valero et al. within a continuous time domain. The step operational semantics is constructed via labeled probabilistic transition systems. The denotational semantics is based on labeled discrete time stochastic Petri nets with immediate transitions. To evaluate performance, the corresponding semi-Markov chains are analyzed. We define step stochastic bisimulation equivalence of expressions that is applied to reduce their transition systems and underlying semi-Markov chains while preserving the functionality and performance characteristics. We explain how this equivalence can be used to simplify performance analysis of the algebraic processes. In a case study, a method of modeling, performance evaluation and behaviour reduction for concurrent systems is outlined and applied to the shared memory system.

[1]  Michael Rettelbach.  Probabilistic Branching in Markovian Process Algebras , 1995, Comput. J..

[2]  Robin Milner,et al.  Communication and concurrency , 1989, PHI Series in computer science.

[3]  Enrico Vicario,et al.  Correctness verification and performance analysis of real-time systems using stochastic preemptive time Petri nets , 2005, IEEE Transactions on Software Engineering.

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

[5]  Robin Milner,et al.  A Calculus of Mobile Processes, II , 1992, Inf. Comput..

[6]  Peter Buchholz.  Iterative Decomposition and Aggregation of Labeled GSPNs , 1998, ICATPN.

[7]  Erik P. de Vink,et al.  Compositionality for Markov reward chains with fast and silent transitions , 2009, Perform. Evaluation.

[8]  Peter Buchholz.  A Notion of Equivalence for Stochastic Petri Nets , 1995, Application and Theory of Petri Nets.

[9]  David de Frutos-Escrig,et al.  SPBC: a Markovian extension of finite Petri box calculus , 2001, Proceedings 9th International Workshop on Petri Nets and Performance Models.

[10]  Diego Latella,et al.  Stochastic Simulation of Event Structures , 1996 .

[11]  Marco Ajmone Marsan,et al.  Stochastic Petri nets: an elementary introduction , 1988, European Workshop on Applications and Theory in Petri Nets.

[12]  Erik P. de Vink,et al.  Reconciling real and stochastic time: the need for probabilistic refinement , 2012, Formal Aspects of Computing.

[13]  Mario Bravetti,et al.  Towards Performance Evaluation with General Distributions in Process Algebras , 1998, CONCUR.

[14]  Robin Milner,et al.  A Calculus of Mobile Processes, II , 1992, Inf. Comput..

[15]  Mario Bravetti,et al.  Reward Based Congruences: Can We Aggregate More? , 2001, PAPM-PROBMIV.

[16]  Falko Bause,et al.  Stochastic Petri Nets: An Introduction to the Theory , 2012, PERV.

[17]  Igor V. Tarasyuk,et al.  Performance analysis of concurrent systems in algebra dtsiPBC , 2014, Programming and Computer Software.

[18]  Jeremy T. Bradley,et al.  Passage-time computation and aggregation strategies for large semi-Markov processes , 2011, Perform. Evaluation.

[19]  Roberto Gorrieri,et al.  A Tutorial on EMPA: A Theory of Concurrent Processes with Nondeterminism, Priorities, Probabilities and Time , 1998, Theor. Comput. Sci..

[20]  Marco Bernardo.  On the tradeoff between compositionality and exactness in weak bisimilarity for integrated-time Markovian process calculi , 2015, Theor. Comput. Sci..

[21]  Holger Hermanns,et al.  Process Algebra and Markov Chains , 2002, European Educational Forum: School on Formal Methods and Performance Analysis.

[22]  van Km Kees Hee,et al.  Analysis of discrete‐time stochastic petri nets , 2000 .

[23]  I. V. Tarasyuk.  Iteration in discrete time stochastic Petri box calculus , 2006 .

[24]  Igor V. Tarasyuk,et al.  Stochastic Petri Box Calculus with Discrete Time , 2007, Fundam. Informaticae.

[25]  David de Frutos-Escrig,et al.  Extending the Petri Box Calculus with Time , 2001, ICATPN.

[26]  Norbert Götz,et al.  Multiprocessor and Distributed System Design: The Integration of Functional Specification and Performance Analysis Using Stochastic Process Algebras , 1993, Performance/SIGMETRICS Tutorials.

[27]  David de Frutos-Escrig,et al.  A congruence relation for sPBC , 2008, Formal Methods Syst. Des..

[28]  Gilles Kahn,et al.  Natural Semantics , 1987, STACS.

[29]  Lorenzo Ridi,et al.  Transient analysis of non-Markovian models using stochastic state classes , 2012, Perform. Evaluation.

[30]  Joost-Pieter Katoen,et al.  Approximate Parameter Synthesis for Probabilistic Time-Bounded Reachability , 2008, 2008 Real-Time Systems Symposium.

[31]  Michael K. Molloy.  Discrete Time Stochastic Petri Nets , 1985, IEEE Transactions on Software Engineering.

[32]  Roberto Gorrieri,et al.  A Formal Approach to the Integration of Performance Aspects in the Modeling and Analysis of Concurrent Systems , 1998, Inf. Comput..

[33]  Joost-Pieter Katoen,et al.  General Distributions in Process Algebra , 2002, European Educational Forum: School on Formal Methods and Performance Analysis.

[34]  Marco Bernardo,et al.  A Survey of Markovian Behavioral Equivalences , 2007, SFM.

[35]  Albert R. Meyer,et al.  Deciding True Concurrency Equivalences on Safe, Finite Nets , 1996, Theor. Comput. Sci..

[36]  Stephen Gilmore,et al.  PEPA nets: a structured performance modelling formalism , 2002, Perform. Evaluation.

[37]  Christel Baier,et al.  Deciding Bisimilarity and Similarity for Probabilistic Processes , 2000, J. Comput. Syst. Sci..

[38]  Gianfranco Ciardo,et al.  Discrete Deterministic and Stochastic Petri Nets , 1997, MMB.

[39]  Jane Hillston,et al.  A compositional approach to performance modelling , 1996 .

[40]  Peter Buchholz.  Markovian process algebra: Composition and equiva-lence , 1994 .

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

[42]  Igor V. Tarasyuk,et al.  Equivalence relations for modular performance evaluation in dtsPBC , 2013, Mathematical Structures in Computer Science.

[43]  Valentín Valero Ruiz,et al.  Introducing the Iteration in sPBC , 2004, FORTE.

[44]  Allan Clark,et al.  Performance Specification and Evaluation with Unified Stochastic Probes and Fluid Analysis , 2013, IEEE Transactions on Software Engineering.

[45]  Andrea Maggiolo-Schettini,et al.  Parametric probabilistic transition systems for system design and analysis , 2007, Formal Aspects of Computing.

[46]  Maciej Koutny,et al.  Petri Net Algebra , 2001, Monographs in Theoretical Computer Science An EATCS Series.

[47]  Corrado Priami.  Language-based Performance Prediction for Distributed and Mobile Systems , 2002, Inf. Comput..

[48]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[49]  Lijun Zhang,et al.  The Quest for Minimal Quotients for Probabilistic Automata , 2013, TACAS.

[50]  P. Buchholz.  Exact and ordinary lumpability in finite Markov chains , 1994, Journal of Applied Probability.

[51]  Scott A. Smolka,et al.  Equivalences, Congruences, and Complete Axiomatizations for Probabilistic Processes , 1990, CONCUR.

[52]  J. Bradley.  SEMI-MARKOV PEPA : MODELLING WITH GENERALLY DISTRIBUTED ACTIONS , 2005 .

[53]  Corrado Priami,et al.  Non-Interleaving Semantics for Mobile Processes , 1999, Theor. Comput. Sci..

[54]  H. Hermanns,et al.  Syntax , Semantics , Equivalences , and Axioms for MTIPP y , 1994 .

[55]  Maciej Koutny,et al.  A Refined View of the Box Algebra , 1995, Application and Theory of Petri Nets.

[56]  Conrado Daws.  Symbolic and Parametric Model Checking of Discrete-Time Markov Chains , 2004, ICTAC.

[57]  Erik P. de Vink,et al.  Extending Timed Process Algebra with Discrete Stochastic Time , 2008, AMAST.

[58]  Jean-Michel Fourneau,et al.  Collaboration of discrete-time Markov chains: Tensor and product form , 2010, Perform. Evaluation.

[59]  Trevor N. Mudge,et al.  A semi-Markov model for the performance of multiple-bus systems , 1985, IEEE Transactions on Computers.

[60]  Christel Baier,et al.  Polynomial Time Algorithms for Testing Probabilistic Bisimulation and Simulation , 1996, CAV.

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

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

[63]  Roberto Gorrieri,et al.  Theory and Application of Extended Markovian Process Algebra , 2011 .

[64]  Diego Latella,et al.  A Stochastic Causality-Based Process Algebra , 1995, Comput. J..

[65]  J. Hillston.  The nature of synchronisation , 1994 .

[66]  Gianfranco Balbo,et al.  Introduction to Stochastic Petri Nets , 2002, European Educational Forum: School on Formal Methods and Performance Analysis.

[67]  Bernhard Steffen,et al.  Reactive, Generative and Stratified Models of Probabilistic Processes , 1995, Inf. Comput..

[68]  Jan A. Bergstra,et al.  Algebra of Communicating Processes with Abstraction , 1985, Theor. Comput. Sci..

[69]  C. Priami.  Stochastic -calculus with General Distributions , 1996 .

[70]  Marco Pistore,et al.  Efficient Minimization up to Location Equivalence , 1996, ESOP.

[71]  Boudewijn R. Haverkort,et al.  Markovian Models for Performance and Dependability Evaluation , 2002, European Educational Forum: School on Formal Methods and Performance Analysis.

[72]  Ivan Christoff,et al.  Testing Equivalences and Fully Abstract Models for Probabilistic Processes , 1990, CONCUR.

[73]  Holger Hermanns,et al.  Symbolic partition refinement with automatic balancing of time and space , 2010, Perform. Evaluation.

[74]  William H. Sanders,et al.  Optimal state-space lumping in Markov chains , 2003, Inf. Process. Lett..

[75]  Diego Latella,et al.  A Stochastic Causality-Based Process , 1995 .

[76]  P. Buchholz.  Net and algebraic approaches to probabilistic modeling , 2001 .

[77]  Valentín Valero,et al.  sPBC: A Markovian Extension of Petri Box Calculus with Immediate Multiactions , 2008, Fundam. Informaticae.

[78]  Gordon D. Plotkin,et al.  A structural approach to operational semantics , 2004, J. Log. Algebraic Methods Program..

[79]  A. Ershov.  Discrete time stochastic Petri box calculus ∗ , 2005 .

[80]  J. Ben Atkinson,et al.  Modeling and Analysis of Stochastic Systems , 1996 .

[81]  Apostolos Niaouris,et al.  An Algebra of Petri Nets with Arc-Based Time Restrictions , 2004, ICTAC.

[82]  Maciej Koutny,et al.  A Compositional Model of Time Petri Nets , 2000, ICATPN.

[83]  Kim G. Larsen,et al.  Bisimulation through Probabilistic Testing , 1991, Inf. Comput..

[84]  Robert E. Tarjan,et al.  Three Partition Refinement Algorithms , 1987, SIAM J. Comput..

[85]  Roberto Segala,et al.  Decision Algorithms for Probabilistic Bisimulation , 2002, CONCUR.

[86]  Günter Hommel,et al.  Discrete time stochastic petri nets for modeling and evaluation of real-time systems , 2001, Proceedings 15th International Parallel and Distributed Processing Symposium. IPDPS 2001.

[87]  Raymond R. Devillers,et al.  The box calculus: a new causal algebra with multi-label communication , 1992, Advances in Petri Nets: The DEMON Project.

[88]  P. Merlin,et al.  Recoverability of Communication Protocols - Implications of a Theoretical Study , 1976, IEEE Transactions on Communications.

[89]  C. A. R. Hoare,et al.  Communicating sequential processes , 1978, CACM.

[90]  Igor V. Tarasyuk,et al.  Discrete Time Stochastic Petri Box Calculus with Immediate Multiactions dtsiPBC , 2013, Electron. Notes Theor. Comput. Sci..

[91]  Erik P. de Vink,et al.  Performance Evaluation of Distributed Systems Based on a Discrete Real- and Stochastic-Time Process Algebra , 2009, Fundam. Informaticae.

[92]  Philippe Schnoebelen,et al.  Place Bisimulations in Petri Nets , 1992, Application and Theory of Petri Nets.

[93]  Rob J. van Glabbeek,et al.  The Linear Time - Branching Time Spectrum II , 1993, CONCUR.