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Serial Disk-Based Analysis of Large Stochastic Models

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Validation of Stochastic Systems

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2925))

Abstract

The paper presents a survey of out-of-core methods available for the analysis of large Markov chains on single workstations. First, we discuss the main sparse matrix storage schemes and review iterative methods for the solution of systems of linear equations typically used in disk-based methods. Next, various out-of-core approaches for the steady state solution of CTMCs are described. In this context, serial out-of-core algorithms are outlined and analysed with the help of their implementations. A comparison of time and memory requirements for typical benchmark models is given.

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References

  1. Bahar, I., Frohm, E., Gaona, C., Hachtel, G., Macii, E., Pardo, A., Somenzi, F.: Algebraic Decision Diagrams and their Applications. In: ICCAD 1993, Santa Clara, pp. 188–191 (1993)

    Google Scholar 

  2. Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J.M., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., van der Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. Philadalphia: Society for Industrial and Applied Mathematics (1994)

    Google Scholar 

  3. Baskett, F., Chandy, K.M., Muntz, R.R., Palacios, F.G.: Open, Closed, and Mixed Networks of Queues with Different Classes of Customers. Journal of ACM 22(2) (1975)

    Google Scholar 

  4. Bause, F.: Queueing Petri Nets: A Formalism for the Combined Qualitative and Quantitative Analysis of Systems. In: Proc. PNPM 1993. IEEE Computer Society Press, Los Alamitos (1993)

    Google Scholar 

  5. Bell, A.: Verteilte Bewertung Stochastischer Petrinetze, Diploma thesis, RWTH, Aachen, Department of Computer Science (March 1999)

    Google Scholar 

  6. Bell, A., Haverkort, B.R.: Serial and Parallel Out-of-Core Solution of Linear Systems arising from Generalised Stochastic Petri Nets. In: High Performance Computing 2001, Seattle, USA (April 2001)

    Google Scholar 

  7. Bernardo, M., Gorrieri, R.: Extended Markovian Process Algebra. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119. Springer, Heidelberg (1996)

    Google Scholar 

  8. Buchholz, P., Kemper, P.: Kronecker based Matrix Representations for Large Markov Models. In: This Proceedings (2003)

    Google Scholar 

  9. Ciardo, G., Miner, A.: A Data Structure for the Efficient Kronecker Solution of GSPNs. In: Proc. PNPM 1999, Zaragoza (1999)

    Google Scholar 

  10. Ciardo, G., Tilgner, M.: On the use of Kronecker Operators for the Solution of Generalized Stochastic Petri Nets. ICASE Report 96-35, Institute for Computer Applications in Science and Engineering (1996)

    Google Scholar 

  11. Ciardo, G., Trivedi, K.S.: A Decomposition Approach for Stochastic Reward Net Models. Performance Evaluation 18(1), 37–59 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  12. Clarke, E., Fujita, M., McGeer, P., Yang, J., Zhao, X.: Multi-Terminal Binary Decision Diagrams: An Effificient Data Structure for Matrix Representation. In: International Workshop on Logic Synthesis, IWLS 1993 (May 1993)

    Google Scholar 

  13. Conway, A.E., Georganas, N.D.: Queueing Networks - Exact Computational Algorithms: A Unified Theory based on Decomposition and Aggregation. MIT Press, Cambridge (1989)

    Google Scholar 

  14. Deavours, D.D., Sanders, W.H.: An Efficient Disk-based Tool for Solving Very Large Markov Models. In: Marie, R., Plateau, B., Calzarossa, M.C., Rubino, G.J. (eds.) TOOLS 1997. LNCS, vol. 1245, pp. 58–71. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  15. Deavours, D.D., Sanders, W.H.: An Efficient Disk-based Tool for Solving Large Markov Models. Performance Evaluation 33(1), 67–84 (1998)

    Article  Google Scholar 

  16. Deavours, D.D., Sanders, W.H.: “On-the-fly”Solution Techniques for Stochastic Petri Nets and Extensions. IEEE Transactions on Software Engineering 24(10), 889–902 (1998)

    Article  Google Scholar 

  17. Hermanns, H., Meyer-Kayser, J., Siegle, M.: Multi Terminal Binary Decision Diagrams to Represent and Analyse Continuous Time Markov Chains. In: Proc. NSMC 1999, Zaragoza (1999)

    Google Scholar 

  18. Hermanns, H., Rettelbach, M.: Syntax, Semantics, Equivalences, and Axioms for MTIPP. In: Proc. PAPM 1994, Germany (1994)

    Google Scholar 

  19. Heroux, M.: A proposal for a sparse BLAS Toolkit, Technical Report TR/PA/92/90, Cray Research, Inc., USA (December 1992)

    Google Scholar 

  20. Hillston, J.: A Compositional Approach to Performance Modelling. PhD thesis, University of Edinburgh (1994)

    Google Scholar 

  21. Ibe, O., Trivedi, K.: Stochastic Petri Net Models of Polling Systems. IEEE Journal on Selected Areas in Communications 8(9), 1649–1657 (1990)

    Article  Google Scholar 

  22. Kahan, W.: Gauss-Seidel methods of solving large systems of linear equations. PhD thesis, University of Toronto (1958)

    Google Scholar 

  23. Knottenbelt, W.J., Harrison, P.G.: Distributed Disk-based Solution Techniques for Large Markov Models. In: Proc. NSMC 1999 (1999)

    Google Scholar 

  24. Kwiatkowska, M., Mehmood, R.: Out-of-core solution of large linear systems of equations arising from stochastic modelling. In: Hermanns, H., Segala, R. (eds.) PROBMIV 2002, PAPM-PROBMIV 2002, and PAPM 2002. LNCS, vol. 2399, p. 135. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  25. Kwiatkowska, M., Mehmood, R., Norman, G., Parker, D.: A Symbolic Out-of- Core Solution Method for Markov Models. In: Proc. Parallel and Distributed Model Checking, PDMC 2002 (August 2002); Appeared in ENTCS 68(4), http://www.elsevier.nl/locate/entcs

  26. Kwiatkowska, M., Norman, G., Parker, D.: PRISM: Probabilistic symbolic model checker. In: Field, T., Harrison, P.G., Bradley, J., Harder, U. (eds.) TOOLS 2002. LNCS, vol. 2324, p. 200. Springer, Heidelberg (2002)

    Google Scholar 

  27. Kwiatkowska, M., Norman, G., Parker, D.: Probabilistic symbolic model checking with PRISM: A hybrid approach. In: Katoen, J.-P., Stevens, P. (eds.) TACAS 2002. LNCS, vol. 2280, p. 52. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  28. Marsan, M.A., Balbo, G., Conte, G.: A Class of Generalized Stochastic Petri Nets for the Performance Analysis of Multiprocessor Systems. ACM Transactions on Computer Systems 2(2) (1984)

    Google Scholar 

  29. Marsan, M.A., Balbo, G., Conte, G., Donatelli, S., Franceschinis, G., Kartson, D.: Modelling With Generalized Stochastic Petri Nets. John Wiley & Son Ltd, Chichester (1995)

    MATH  Google Scholar 

  30. Mehmood, R.: On the Development of Techniques for the Analysis of Large Markov Models. PhD thesis, University of Birmingham (2003) (to appear)

    Google Scholar 

  31. Miner, A., Parker, D.: Symbolic Representations and Analysis of Large State Spaces. In: This Proceedings (2003)

    Google Scholar 

  32. Molloy, M.K.: Performance Analysis using Stochastic Petri Nets. IEEE Trans. Comput. 31, 913–917 (1982)

    Article  Google Scholar 

  33. Parker, D.: Implementation of Symbolic Model Checking for Probabilistic Systems. PhD thesis, University of Birmingham (August. 2002)

    Google Scholar 

  34. Plateau, B.: On the Stochastic Structure of Parallelism and Synchronisation Models for Distributed Algorithms. In: Proc. 1985 ACM SIGMETRICS Conference on Measurement and Modeling of Computer Systems (1985)

    Google Scholar 

  35. Plateau, B., Atif, K.: Stochastic Automata Network for Modeling Parallel Systems. IEEE Transactions on Software Engineering 17(10) (1991)

    Google Scholar 

  36. PRISM Web Page. http://www.cs.bham.ac.uk/~dxp/prism/

  37. Saad, Y.: SPARSKIT: A basic tool kit for sparse matrix computations. Technical Report RIACS-90-20, NASA Ames Research Center, CA (1990)

    Google Scholar 

  38. Siegle, M.: Advances in model representations. In: de Luca, L., Gilmore, S. (eds.) PROBMIV 2001, PAPM-PROBMIV 2001, and PAPM 2001. LNCS, vol. 2165, p. 1. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  39. Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton (1994)

    MATH  Google Scholar 

  40. Toledo, S.: A Survey of Out-of-Core Algorithms in Numerical Linear Algebra. In: External Memory Algorithms and Visualization, DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society Press, Providence (1999)

    Google Scholar 

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Mehmood, R. (2004). Serial Disk-Based Analysis of Large Stochastic Models. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, JP., Siegle, M. (eds) Validation of Stochastic Systems. Lecture Notes in Computer Science, vol 2925. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24611-4_7

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  • DOI: https://doi.org/10.1007/978-3-540-24611-4_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22265-1

  • Online ISBN: 978-3-540-24611-4

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