Deterministic and Stochastic Petri Nets (DSPNs) are a widely used high-level formalism for modeling discreteevent systems where events may occur either without consuming time, after a deterministic time, or after an exponentially distributed time. CSL (Continuous Stochastic Logic) is a (branching) temporal logic developed to express probabilistic properties in continuous time Markov chains (CTMCs). In this paper we present a Mathematica-based tool that implements recent developments for model checking CSL style properties on DSPNs. Furthermore, as a consequence of the type of process underlying DSPNs (a superset of Markovian processes), we are also able to check CSL properties of Generalized Stochastic Petri Nets (GSPNs) and labeled CTMCs

Haverkort, Boudewijn R.

Martinez, José M

English

Martinez Verdugo, J.M.

Haverkort, Boudewijn R.H.M.

University of Twente Research Information

MathMC: A Mathematica-Based Tool for CSL Model Checkingof Deterministic and Stochastic Petri NetsJosé M. Martı́nez Boudewijn R. HaverkortFaculty of Electrical Engineering, Mathematics and Computer ScienceUniversity of Twente, the Netherlands{martinez,brh}@cs.utwente.nlAbstractDeterministic and Stochastic Petri Nets (DSPNs) are awidely used high-level formalism for modeling discrete-event systems where events may occur either without con-suming time, after a deterministic time, or after an ex-ponentially distributed time. CSL (Continuous StochasticLogic) is a (branching) temporal logic developed to expressprobabilistic properties in continuous time Markov chains(CTMCs). In this paper we present a Mathematica-basedtool that implements recent developments for model check-ing CSL style properties on DSPNs. Furthermore, as a con-sequence of the type of process underlying DSPNs (a super-set of Markovian processes), we are also able to check CSLproperties of Generalized Stochastic Petri Nets (GSPNs)and labeled CTMCs.Keywords: DSPNs, CSL, model checking, Markov pro-cess, Markov regenerative process.1 IntroductionCSL is a widely used logic for the specification of sys-tem performance and dependability properties for Marko-vian models. DSPNs have been used widely for modelingand evaluation of all kinds of performance and dependabil-ity aspects of computer and communication systems.MathMC is a Mathematica-based tool that implementsrecent developments [4] to extend the use of CSL in the con-text of so-called Markov regenerative processes (MRGPs),a class of stochastic processes that arises from DSPNs. Thisclass of process is more general than CTMCs. Togetherwith the CSL implementation for DSPNs, fully Markovianmodel checking techniques are also implemented, thus al-lowing to check properties for GSPNs and labeled CTMCsas well.This paper is organized as follows. In Section 2 we givean overview of the tool and its capabilities. In Section 3 wepresent some information related to the algorithms imple-mented. Section 4 presents current and future work.2 Tool overviewMathMC comprises two parts, one based on Mathemat-ica [8], and another one implemented in C++. Mathemat-ica was chosen since it provides a flexible model applica-tion, evaluation, and result presentation framework due tolist processing, symbolic manipulation, functional program-ming, pattern matching, and graphics facilities.Part of the numerical analysis of DSPNs, as well as thestate space generation (reachability graph) is carried out bythe algorithms implemented in SPNica, a prototype tool de-veloped by German [2, 7] for the analysis of DSPNs. Be-sides the above implementation, algorithms for transientand steady-state analysis of CTMCs, as well as part of theanalysis of DSPN are implemented in C++ (together withthe data structures needed to store the relevant information,i.e., vectors and sparse matrices). The communication be-tween Mathematica and the C++ compiled code is accom-plished via MathLink, a toolkit for interfacing external codewith Mathematica.DSPNCSLFormulaSolverMRGPCHECKERMODELSatisfactionSetInterfaceProbabilitiesRRGCTMC analysis algorithmsFigure 1. Block Diagram of the ToolThe structure of MathMC is summarized in the block di-agram of Figure 1. DSPN is the specification of the modelfollowing the SPNica syntax, for a detailed explanation see[2]. RRG is the set of functions that generate the reducedreachability graph (RRG) of a Petri net. The explorationof states is carried out through a breadth-first-search algo-rithm. The information of the process is acquired via theRRG. Solver MRGP comprises the set of functions for thenumerical evaluation of DSPNs, using either steady state ortransient analysis algorithms. The logic formulas, in CSLFormula, are specified using the names of places definedin the DSPN model together with logical functions takenfrom Mathematica. CTMC Analyisis Algorithms imple-ments the numerical methods involved in the analysis ofMarkovian processes, which are also part of the analysisof DSPNs. MODEL CHECKER is the set of functionsfor evaluating the operators involved in the model checkingprocedure. MODEL CHECKER functions interact with theThird International Conference on the Quantitative Evaluation of Systems (QEST'06)0-7695-2665-9/06 $20.00  © 2006Authorized licensed use limited to: UNIVERSITEIT TWENTE. Downloaded on January 22, 2010 at 08:51 from IEEE Xplore.  Restrictions apply. Solver MRGP and the CTMC Analysis Algorithms in orderto calculate those probabilities needed to determine the log-ical value of the expression being checked. Satisfaction Setis the set of states that satisfies the formula being checked.Probabilities corresponds to the probabilities associated toeach state for which the CSL formula is evaluated.Besides the description given above, two other formal-ism are implemented: GSPNs and labeled CTMCs. Effi-cient implementation of the algorithms, as well as the datastructures involved in model checking of these formalismsis accomplished via compiled code linked to MathMC. Im-porting functions are also available for exchanging modelinformation with other tools, specifically, MRMC [6],ETMCC [1], and Möbius [5].3 CSL model checking of DSPNsCSL expressions are defined in terms of atomic proper-ties which refer to the number of tokens in the places ofthe DSPN. CSL model checking algorithms (originally in-tended for continuous-time Markov processes) have beenextended to Markov regenerative processes, specificallythose underlying a DSPN. The following examples illus-trate the type of properties that can be expressed:• S>0[num processors up > 2]: “in the long-run, theprobability there are more than two processors work-ing properly is greater than 0”.• P[0.5,0.8][(buffer < 10) U≤5 (buffer = 10)]: “theprobability of the system reaching a buffer occupancyof 10 jobs within 5 time units when the buffer fills uplies in the interval [0.5, 0.8]”.The analysis of DSPNs is based on the method of sup-plementary variables [2], where a state of the underlyingprocess (MRGP) is characterized by both a marking and theelapsed time of the deterministic transition that is enabledin that marking (if there is any).The evaluation techniques implemented in our tool re-quire some restrictions with respect to deterministic tran-sitions: (1) at most one deterministic transition is enabledin each marking and (2) the firing policy is preemptive re-peat different, i.e., when a timed transition is preempted, itsalready performed work is lost.With respect to CSL, MathMC implements the steady-state and time-bounded until operator, following the proce-dure described in [4]. In the case of obtaining the satisfac-tion set for a steady-state formula, Sp(φ), we compute thesteady-state probabilities (or long-term proportion of time,if there are no limiting probabilities) for those states satisfy-ing the formula φ. Steady-state probabilities are calculatedaccording to the procedure described in [2].In the case of the satisfaction set for a time-bounded untilformula, Pp(φ1 U≤t φ2), we follow the same procedurethat for the case of CTMCs, i.e., making absorbing thosestates that satisfy (¬φ1 ∨ φ2) and considering the transientprobability of being in states that satisfy φ2 at time t, givenwe start in a state that satisfies φ1. In order to calculate thetransient probabilities we use the methodology developed in[2, 3]. Special attention must be taken when we make somestates absorbing, since, it may be the case that a transitionchanges from being a non-preemtive one to be a preemtiveone. For more details on this issue we refer to [4]. Further-more, for the verification of the time-bounded until operatorwe had to assume that a deterministic transition may be ini-tially enabled, but its elapsed time must be assumed zero.This assumption is required in order to be able to use thewell-established transient algorithms.From a practical point of view, the applicability of modelchecking techniques in DSPNs lags behind the CTMC case.Verifying a CSL formula for DSPNs is more time/spaceconsuming than for CTMCs due to the fact we are deal-ing with non-memoryless transitions. The time-boundednext operator has been left out for efficiency reasons, as dis-cussed in [4].4 Future developmentsFurther work must concentrate on overcoming some lim-itations imposed by the assumptions for the time-boundeduntil operator, i.e., information about the elapsed time thata deterministic transition has been enabled must be consid-ered.Furthermore, the implementation of some numerical al-gorithms for the analysis of DSPNs must be changed fromMathematica code to C++ compiled code to increase effi-ciency.References[1] ETMCC. Erlangen-Twente Markov Chain Checker.http://www7.informatik.uni-erlangen.de/etmcc.[2] R. German. Performance Analysis of Communication Sys-tems: Modeling with Non-Markovian Stochastic Petri Nets.John Wiley & Sons Ltd., 2000.[3] A. Heindl and R. German. A fourth-order algorithm with au-tomatic stepsize control for the transient analysis of DSPNs.IEEE Transactions on Software Engineering, 25(2):194–206,March 1999.[4] J. M. Martı́nez and B. R. Haverkort. CSL model checking ofdeterministic and stochastic Petri nets. In Proceedings of the13th GI/ITG Conference in Measuring, Modelling and Eval-uation of Computer and Communication Systems, pages 265–282, Nürnberg, March 2006. VDE Verlag.[5] Möbius. http://www.mobius.uiuc.edu.[6] MRMC. The Markov Reward Model Checker.http://www.cs.utwente.nl/∼zapreev/mrmc.[7] SPNica. ftp://ftp.wiley.co.uk/pub/books/german/.[8] Wolfram Research. Mathematica 5.2.http://www.wolfram.com.Third International Conference on the Quantitative Evaluation of Systems (QEST'06)0-7695-2665-9/06 $20.00  © 2006Authorized licensed use limited to: UNIVERSITEIT TWENTE. Downloaded on January 22, 2010 at 08:51 from IEEE Xplore.  Restrictions apply. 

MathMC: A mathematica-based tool for CSL model checking of deterministic and stochastic Petri nets

NARCIS 

A fourth-order algorithm with automatic stepsize control for the transient analysis of DSPNs.

Erlangen-Twente Markov Chain Checker.

M a r t ´ ınez

Mathematica 5.2. http://www.wolfram.com.

Performance Analysis of Communication Systems: Modeling with Non-Markovian Stochastic Petri Nets.

The Markov Reward Model Checker.

https://research.utwente.nl/en/publications/528eb185-0919-4adf-9984-f42e1f13661e

MathMC: A mathematica-based tool for CSL model checking of deterministic and stochastic Petri nets

Abstract

Similar works

Full text

Available Versions

University of Twente Research Information

NARCIS