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Stochastic Evaluation of Large Interdependent Composed Models Through Kronecker Algebra and Exponential Sums

  • Giulio MasettiEmail author
  • Leonardo Robol
  • Silvano Chiaradonna
  • Felicita Di Giandomenico
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11522)

Abstract

The KAES methodology for efficient evaluation of dependability-related properties is proposed. KAES targets systems representable by Stochastic Petri Nets-based models, composed by a large number of submodels where interconnections are managed through synchronization at action level. The core of KAES is a new numerical solution of the underlying CTMC process, based on powerful mathematical techniques, including Kronecker algebra, Tensor Trains and Exponential Sums. Specifically, advancing on existing literature, KAES addresses efficient evaluation of the Mean-Time-To-Absorption in CTMC with absorbing states, exploiting the basic idea to further pursue the symbolic representation of the elements involved in the evaluation process, so to better cope with the problem of state explosion. As a result, computation efficiency is improved, especially when the submodels are loosely interconnected and have small number of states. An instrumental case study is adopted, to show the feasibility of KAES, in particular from memory consumption point of view.

Keywords

Stochastic Petri Nets Stochastic Automata Networks Markov chains Mean Time To Absorption Kronecker algebra Exponential sums Tensor Train 

References

  1. 1.
    Ajmone Marsan, M., Balbo, G., Chiola, G., Conte, G., Donatelli, S., Franceschinis, G.: An introduction to generalized stochastic Petri nets. Microelectron. Reliab. 31(4), 699–725 (1991)CrossRefGoogle Scholar
  2. 2.
    Babar, J., Beccuti, M., Donatelli, S., Miner, A.: GreatSPN enhanced with decision diagram data structures. In: Applications and Theory of Petri Nets, pp. 308–317 (2010)CrossRefGoogle Scholar
  3. 3.
    Braess, D., Hackbusch, W.: Approximation of \(1/x\) by exponential sums in \([1,\infty )\). IMA J. Numer. Anal. 25(4), 685–697 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brenner, L., Fernandes, P., Sales, A., Webber, T.: A framework to decompose GSPN models. In: Ciardo, G., Darondeau, P. (eds.) ICATPN 2005. LNCS, vol. 3536, pp. 128–147. Springer, Heidelberg (2005).  https://doi.org/10.1007/11494744_9CrossRefGoogle Scholar
  5. 5.
    Buchholz, P., Dayar, T., Kriege, J., Orhan, M.C.: On compact solution vectors in Kronecker-based Markovian analysis. Perform. Eval. 115, 132–149 (2017)CrossRefGoogle Scholar
  6. 6.
    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. LNCS, vol. 2925, pp. 256–295. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-24611-4_8CrossRefGoogle Scholar
  7. 7.
    Ciardo, G.: Data representation and efficient solution: a decision diagram approach. In: Bernardo, M., Hillston, J. (eds.) SFM 2007. LNCS, vol. 4486, pp. 371–394. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-72522-0_9CrossRefGoogle Scholar
  8. 8.
    Ciardo, G., Miner, A.S.: A data structure for the efficient Kronecker solution of GSPNs. In: Proceedings 8th International Workshop on Petri Nets and Performance Models (Cat. No. PR00331), pp. 22–31 (1999)Google Scholar
  9. 9.
    Ciardo, G., Tilgner, M.: On the use of Kronecker operators for the solution of generalized stochastic Petri nets. Nasa Technical Report Server 20040110963 (1996)Google Scholar
  10. 10.
    Czekster, R.M., Fernandes, P., Vincent, J.-M., Webber, T.: Split: a flexible and efficient algorithm to vector-descriptor product. In: VALUETOOLS, pp. 83:1–83:8 (2007)Google Scholar
  11. 11.
    Dolgov, S.V., Savostyanov, D.V.: Alternating minimal energy methods for linear systems in higher dimensions. SIAM J. Sci. Comput. 36(5), A2248–A2271 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Donatelli, S.: Superposed stochastic automata: a class of stochastic Petri nets with parallel solution and distributed state space. Perform. Eval. 18(1), 21–36 (1993)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Donatelli, S.: Superposed generalized stochastic Petri nets: definition and efficient solution. In: Application and Theory of Petri Nets 1994, pp. 258–277 (1994)Google Scholar
  14. 14.
    Goševa-Popstojanova, K., Trivedi, K.: Stochastic modeling formalisms for dependability, performance and performability. In: Haring, G., Lindemann, C., Reiser, M. (eds.) Performance Evaluation: Origins and Directions. LNCS, vol. 1769, pp. 403–422. Springer, Heidelberg (2000).  https://doi.org/10.1007/3-540-46506-5_17CrossRefGoogle Scholar
  15. 15.
    Grasedyck, L.: Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure. Computing 72(3–4), 247–265 (2004)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kemper, P.: Numerical analysis of superposed GSPNs. IEEE Trans. Softw. Eng. 22(9), 615–628 (1996)CrossRefGoogle Scholar
  17. 17.
    Kressner, D., Macedo, F.: Low-rank tensor methods for communicating Markov processes. In: Norman, G., Sanders, W. (eds.) QEST 2014. LNCS, vol. 8657, pp. 25–40. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-10696-0_4CrossRefGoogle Scholar
  18. 18.
    Kressner, D., Tobler, C.: Krylov subspace methods for linear systems with tensor product structure. SIAM J. Matrix Anal. Appl. 31(4), 1688–1714 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kutzelnigg, W.: Theory of the expansion of wave functions in a Gaussian basis. Int. J. Quant. Chem. 51(6), 447–463 (1994)CrossRefGoogle Scholar
  20. 20.
    Masetti, G., Robol, L.: Tensor methods for the computation of MTTF in large systems of loosely interconnected components. Technical report, ISTI-CNR Open Portal (2019). http://dcl.isti.cnr.it/tmp/tchrep-RtCv-63_CtAx_Ol19_jEN5.pdf
  21. 21.
    MathWorks. MATLAB R2018a. The Mathworks Inc. (2018)Google Scholar
  22. 22.
    Murata, T.: Petri nets: properties, analysis and applications. Proc. IEEE 77(4), 541–580 (1989)CrossRefGoogle Scholar
  23. 23.
    Oseledets, I.: DMRG approach to fast linear algebra in the TT-format. Comput. Methods Appl. Math. 11(3), 382–393 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Oseledets, I.V.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Plateau, B., Fourneau, J.-M., Lee, K.-H.: Peps: a package for solving complex Markov models of parallel systems. In: Puigjaner, R., Potier, D. (eds.) Modeling Techniques and Tools for Computer Performance Evaluation, pp. 291–305. Springer, Boston (1989).  https://doi.org/10.1007/978-1-4613-0533-0_19CrossRefGoogle Scholar
  26. 26.
    Sanders, W.H., Meyer, J.F.: A unified approach for specifying measures of performance, dependability and performability. Dependable Computing for Critical Applications. Dependable Computing and Fault-Tolerant Systems, vol. 4, pp. 215–237. Springer, Vienna (1991).  https://doi.org/10.1007/978-3-7091-9123-1_10CrossRefGoogle Scholar
  27. 27.
    Trivedi, K.S., Bobbio, A.: Reliability and Availability Engineering: Modeling, Analysis, and Applications (2017)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Giulio Masetti
    • 1
    • 3
    Email author
  • Leonardo Robol
    • 2
    • 3
  • Silvano Chiaradonna
    • 3
  • Felicita Di Giandomenico
    • 3
  1. 1.Department of Computer SciencePisaItaly
  2. 2.Department of MathematicsPisaItaly
  3. 3.Institute of Science and Technology “A. Faedo”PisaItaly

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