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International Conference on Applications and Theory of Petri Nets and Concurrency

PETRI NETS 2016: Application and Theory of Petri Nets and Concurrency pp 281–300Cite as

Efficient Decomposition Algorithm for Stationary Analysis of Complex Stochastic Petri Net Models

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9698))

Abstract

Stochastic Petri nets are widely used for the modeling and analysis of non-functional properties of critical systems. The state space explosion problem often inhibits the numerical analysis of such models. Symbolic techniques exist to explore the discrete behavior of even complex models, while block Kronecker decomposition provides memory-efficient representation of the stochastic behavior. However, the combination of these techniques into a stochastic analysis approach is not straightforward. In this paper we integrate saturation-based symbolic techniques and decomposition-based stochastic analysis methods. Saturation-based exploration is used to build the state space representation and a new algorithm is introduced to efficiently build block Kronecker matrix representation to be used by the stochastic analysis algorithms. Measurements confirm that the presented combination of the two representations can expand the limits of previous approaches.

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References

  1. Ajmone Marsan, M.: Stochastic Petri nets: an elementary introduction. In: Rozenberg, G. (ed.) Advances in Petri Nets 1989, Covers the 9th European Workshop on Applications and Theory in Petri Nets, held in Venice, Italy in June 1988, Selected Paper. LNCS, vol. 424, pp. 1–29. Springer, Heidelberg (1988)

    Google Scholar 

  2. Baier, C., Katoen, J.P., Hermanns, H.: Approximative symbolic model checking of continuous-time Markov chains. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR, 1999 Concurrency Theory. LNCS, vol. 1664, pp. 146–161. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  3. Barrett, R., Berry, M.W., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., Van der Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, vol. 43. SIAM, Philadelphia (1994)

    Book  MATH  Google Scholar 

  4. Bause, F., Buchholz, P., Kemper, P.: A toolbox for functional and quantitative analysis of DEDS. In: Puigjaner, R., Savino, N.N., Serra, B. (eds.) TOOLS 1998. LNCS, vol. 1469, p. 356. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  5. Benoit, A., Plateau, B., Stewart, W.J.: Memory-efficient Kronecker algorithms with applications to the modelling of parallel systems. Future Gener. Comp. Syst. 22(7), 838–847 (2006)

    Article  Google Scholar 

  6. Bianco, A., de Alfaro, L.: Model checking of probabilistic and nondeterministic systems. In: Thiagarajan, P.S. (ed.) FSTTCS 1995. LNCS, vol. 1026, pp. 499–513. Springer, Heidelberg (1995)

    Chapter  MATH  Google Scholar 

  7. Buchholz, P.: Hierarchical structuring of superposed GSPNs. IEEE Trans. Softw. Eng. 25(2), 166–181 (1999)

    Article  Google Scholar 

  8. Buchholz, P., Ciardo, G., Donatelli, S., Kemper, P.: Complexity of memory-efficient Kronecker operations with applications to the solution of Markov models. INFORMS J. Comput. 12(3), 203–222 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Buchholz, P., Kemper, P.: On generating a hierarchy for GSPN analysis. SIGMETRICS Perform. Eval. Rev. 26(2), 5–14 (1998)

    Article  Google Scholar 

  10. Buchholz, P., Kemper, P.: Kronecker based matrix representations for large Markov models. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P., Siegle, M. (eds.) Validation of Stochastic Systems: A Guide to Current Research. LNCS, vol. 2925, pp. 256–295. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  11. Ciardo, G., Jones, R.L., Miner, A.S., Siminiceanu, R.: Logical and stochastic modeling with SMART. In: Kemper, P., Sanders, W.H. (eds.) TOOLS 2003. LNCS, vol. 2794, pp. 78–97. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  12. Ciardo, G., Miner, A.S.: A data structure for the efficient Kronecker solution of GSPNs. In: The 8th International Workshop on Petri Nets and Performance Models, 1999, pp. 22–31. IEEE (1999)

    Google Scholar 

  13. Ciardo, G., Zhao, Y., Jin, X.: Ten years of saturation: a Petri net perspective. In: Jensen, K., Donatelli, S., Kleijn, J. (eds.) Transactions on Petri Nets and Other Models of Concurrency V. LNCS, vol. 6900, pp. 51–95. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  14. Czekster, R.M., Rose, C., Fernandes, P.H.L., de Lima, A.M., Webber, T.: Kronecker descriptor partitioning for parallel algorithms. In: McGraw, R.M., Imsand, E.S., Chinni, M.J. (eds.) Proceedings of the 2010 Spring Simulation Multiconference, SpringSim 2010, Orlando, Florida, USA, 11–15, April 2010, p. 242. SCS/ACM (2010)

    Google Scholar 

  15. Donatelli, S.: Superposed generalized stochastic Petri nets: Definition and efficient solution. In: Valette, R. (ed.) Application and Theory of Petri Nets 1994. LNCS, vol. 815, pp. 258–277. Springer, Heidelberg (1994)

    Chapter  Google Scholar 

  16. Fernandes, P., Presotto, R., Sales, A., Webber, T.: An alternative algorithm to multiply a vector by a Kronecker represented descriptor. In: 21st UK Performance Engineering Workshop, pp. 57–67 (2005)

    Google Scholar 

  17. Ghosh, R.: Scalable stochastic models for cloud services. Ph.D. thesis, Duke University (2012)

    Google Scholar 

  18. Kemper, P.: Numerical analysis of superposed GSPNs. IEEE Trans. Softw. Eng. 22(9), 615–628 (1996)

    Article  Google Scholar 

  19. Kwiatkowska, M., Mehmood, R., Norman, G., Parker, D.: A symbolic out-of-core solution method for Markov models. Electron. Notes Theoret. Comput. Sci. 68(4), 589–604 (2002)

    Article  MATH  Google Scholar 

  20. Kwiatkowska, M., Norman, G., Parker, D.: Probabilistic symbolic model checking with PRISM: a hybrid approach. Int. J. Softw. Tools Technol. Transfer 6(2), 128–142 (2004)

    Article  MATH  Google Scholar 

  21. Miner, A.S.: Efficient solution of GSPNs using canonical matrix diagrams. In: 9th International Workshop on Petri Nets and Performance Models, 2001, pp. 101–110. IEEE (2001)

    Google Scholar 

  22. Roux, P., Siminiceanu, R.: Model checking with edge-valued decision diagrams. In: Muñoz, C.A. (ed.) Second NASA Formal Methods Symposium - NFM 2010, Washington D.C., USA, 13–15 April 2010. Proceedings. NASA Conference Proceedings, vol. NASA/CP-2010-216215, pp. 222–226 (2010)

    Google Scholar 

  23. Stewart, W.J.: Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling. Princeton University Press, Princeton (2009)

    Book  MATH  Google Scholar 

  24. Van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Scientific Stat. Comput. 13(2), 631–644 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wan, M., Ciardo, G., Miner, A.S.: Approximate steady-state analysis of large Markov models based on the structure of their decision diagram encoding. Performance Eval. 68(5), 463–486 (2011)

    Article  Google Scholar 

  26. Zhao, Y., Ciardo, G.: A two-phase Gauss-Seidel algorithm for the stationary solution of EVMDD-encoded CTMCs. In: Ninth International Conference on Quantitative Evaluation of Systems, QEST 2012, London, United Kingdom, 17–20 September 2012, pp. 74–83. IEEE Computer Society (2012)

    Google Scholar 

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Acknowledgement

This work was partially supported by the ARTEMIS JU and the Hungarian National Research, Development and Innovation Fund in the frame of the R5-COP project. This research was partially performed within the framework of the grant of the Hungarian Scientific Resarch Fund (grant no. OTKA K101150).

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Authors and Affiliations

  1. Department of Measurement and Information Systems, Budapest University of Technology and Economics, Budapest, Hungary

    Kristóf Marussy, Attila Klenik & István Majzik

  2. MTA-BME Lendület Cyber-Physical Systems Research Group, Budapest, Hungary

    Vince Molnár & András Vörös

  3. MTA-BME Information Systems Research Group, Budapest, Hungary

    Miklós Telek

Authors
  1. Kristóf Marussy
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  2. Attila Klenik
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  3. Vince Molnár
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  4. András Vörös
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  5. István Majzik
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  6. Miklós Telek
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Corresponding author

Correspondence to András Vörös .

Editor information

Editors and Affiliations

  1. Université Pierre and Marie Curie, Paris, France

    Fabrice Kordon

  2. Department Informatik, Universität Hamburg Department Informatik, Hamburg, Germany

    Daniel Moldt

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Marussy, K., Klenik, A., Molnár, V., Vörös, A., Majzik, I., Telek, M. (2016). Efficient Decomposition Algorithm for Stationary Analysis of Complex Stochastic Petri Net Models. In: Kordon, F., Moldt, D. (eds) Application and Theory of Petri Nets and Concurrency. PETRI NETS 2016. Lecture Notes in Computer Science(), vol 9698. Springer, Cham. https://doi.org/10.1007/978-3-319-39086-4_17

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  • DOI: https://doi.org/10.1007/978-3-319-39086-4_17

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-39085-7

  • Online ISBN: 978-3-319-39086-4

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