Skip to main content

Explicit State Space and Markov Chain Generation Using Decision Diagrams

  • Conference paper
Book cover Computer Performance Engineering (EPEW 2014)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 8721))

Included in the following conference series:

  • 916 Accesses

Abstract

Various forms of decision diagrams have been successfully used for quite some time to generate the state space and Markov chain from models expressed in some high-level formalism. A variety of efficient, “symbolic” algorithms, which manipulate sets of states instead of individual states, are known for this purpose. However, there are cases where explicit generation algorithms are still used. This paper seeks to efficiently use decision diagrams as replacement data structures within an existing explicit generation implementation. The necessary decision diagram algorithms are presented, and small changes to the explicit generation algorithm are suggested to improve the overall generation process. The efficiency of the new algorithms is illustrated using several models.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Babar, J., Beccuti, M., Donatelli, S., Miner, A.: GreatSPN enhanced with decision diagram data structures. In: Lilius, J., Penczek, W. (eds.) PETRI NETS 2010. LNCS, vol. 6128, pp. 308–317. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  2. Babar, J., Miner, A.: Meddly: Multi–terminal and Edge–valued Decision Diagram LibrarY. In: 7th Int. Conf. on Quantitative Evaluation of Systems (QEST 2010), pp. 195–196 (September 2010)

    Google Scholar 

  3. Baier, C., Haverkort, B., Hermanns, H., Katoen, J.P.: Model checking algorithms for continuous-time Markov chains. IEEE Trans. Softw. Eng. 29(6), 524–541 (2003)

    Article  Google Scholar 

  4. Bryant, R.E.: Graph–based algorithms for boolean function manipulation. IEEE Trans. Comp. C-35(8), 677–691 (1986)

    Google Scholar 

  5. Burch, J.R., Clarke, E.M., McMillan, K.L., Dill, D.L., Hwang, L.J.: Symbolic model checking: 1020 states and beyond. Information and Computation 98(2), 142–170 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chiola, G., Franceschinis, G., Gaeta, R., Ribaudo, M.: GreatSPN 1.7: Graphical Editor and Analyzer for Timed and Stochastic Petri Nets. Perf. Eval. 24(1-2), 47–68 (1995)

    Article  MATH  Google Scholar 

  7. Chiola, G.: Compiling techniques for the analysis of stochastic Petri nets. In: Proc. 4th Int. Conf. on Modelling Techniques and Tools for Performance Evaluation, pp. 13–27 (1989)

    Google Scholar 

  8. Chiola, G., Dutheillet, C., Franceschinis, G., Haddad, S.: Stochastic well-formed colored nets and symmetric modeling applications. IEEE Trans. Comp. 42(11), 1343–1360 (1993)

    Article  Google Scholar 

  9. Ciardo, G., Lüttgen, G., Miner, A.S.: Exploiting interleaving semantics in symbolic state–space generation. Formal Methods in System Design 31(1), 63–100 (2007)

    Article  MATH  Google Scholar 

  10. PCiardo, G., Miner, A.S., Wan, M.: Advanced features in SMART: the Stochastic Model checking Analyzer for Reliability and Timing. SIGMETRICS Perform. Eval. Rev. 36(4), 58–63 (2009)

    Article  Google Scholar 

  11. Ciardo, G., Trivedi, K.S.: A decomposition approach for stochastic reward net models. Perf. Eval. 18(1), 37–59 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  12. Couvreur, J.-M., Thierry-Mieg, Y.: Hierarchical decision diagrams to exploit model structure. In: Wang, F. (ed.) FORTE 2005. LNCS, vol. 3731, pp. 443–457. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  13. Delamare, C., Gardan, Y., Moreaux, P.: Performance evaluation with asynchronously decomposable SWN: Implementation and case study. In: 10th Int. Workshop on Petri Nets and Performance Models (PNPM 2003), pp. 20–29. IEEE Comp. Soc. Press (September 2003)

    Google Scholar 

  14. Derisavi, S.: A symbolic algorithm for optimal Markov chain lumping. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 139–154. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  15. Hermanns, H., Kwiatkowska, M., Norman, G., Parker, D., Siegle, M.: On the use of MTBDDs for performability analysis and verification of stochastic systems. Journal of Logic and Algebraic Programming: Special Issue on Probabilistic Techniques for the Design and Analysis of Systems 56(1-2), 23–67 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kam, T., Villa, T., Brayton, R., Sangiovanni-Vincentelli, A.: Multi–valued decision diagrams: theory and applications. Multiple-Valued Logic 4(1-2), 9–62 (1998)

    MATH  MathSciNet  Google Scholar 

  17. Lai, Y.T., Pedram, M., Vrudhula, S.B.K.: Formal verification using edge–valued binary decision diagrams. IEEE Trans. Comp. 45, 247–255 (1996)

    Article  MATH  Google Scholar 

  18. Miner, A., Parker, D.: Symbolic representations and analysis of large state spaces. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P., Siegle, M. (eds.) AUTONOMY 2003. LNCS (LNAI), vol. 2925, pp. 296–338. Springer, Heidelberg (2004)

    Google Scholar 

  19. Miner, A.S.: Efficient solution of GSPNs using Canonical Matrix Diagrams. In: 9th Int. Workshop on Petri Nets and Performance Models (PNPM 2001), pp. 101–110. IEEE Comp. Soc. Press (September 2001)

    Google Scholar 

  20. Miner, A.S.: Implicit GSPN reachability set generation using decision diagrams. Perf. Eval. 56(1-4), 145–165 (2004)

    Article  Google Scholar 

  21. Model Checking Contest at Petri Nets (2014), http://mcc.lip6.fr/models.php

  22. Molloy, M.K.: Performance analysis using stochastic Petri nets. IEEE Trans. Comp. 31(9), 913–917 (1982)

    Article  Google Scholar 

  23. Muppala, J.K., Ciardo, G., Trivedi, K.S.: Modeling using Stochastic Reward Nets. In: Proc. 1st Int. Workshop on Modeling, Analysis and Simulation of Computer and Telecommunication Systems (MASCOTS 1993), pp. 367–372. IEEE Comp. Soc. Press, San Diego (1993)

    Google Scholar 

  24. Pastor, E., Roig, O., Cortadella, J., Badia, R.M.: Petri net analysis using boolean manipulation. In: Valette, R. (ed.) ICATPN 1994. LNCS, vol. 815, pp. 416–435. Springer, Heidelberg (1994)

    Chapter  Google Scholar 

  25. Tilgner, M., Takahashi, Y., Ciardo, G.: SNS 1.0: Synchronized Network Solver. In: 1st International Workshop on Manufacturing and Petri Nets, Osaka, Japan, pp. 215–234 (June 1996)

    Google Scholar 

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

    Article  Google Scholar 

  27. Wimmer, R., Braitling, B., Becker, B., Hahn, E.M., Crouzen, P., Hermanns, H., Dhama, A., Theel, O.: Symblicit calculation of long–run averages for concurrent probabilistic systems. In: 7th Int. Conf. on Quantitative Evaluation of Systems (QEST 2010), pp. 27–36 (2010)

    Google Scholar 

  28. Woodside, C.M., Li, Y.: Performance Petri net analysis of communications protocol software by delay-equivalent aggregation. In: 4th Int. Workshop on Petri Nets and Performance Models (PNPM 1991), pp. 64–73. IEEE Comp. Soc. Press, Melbourne (1991)

    Google Scholar 

  29. Xie, A., Beerel, P.A.: Efficient state classification of finite state Markov chains. IEEE Trans. Computer-Aided Design 17(12), 1334–1339 (1998)

    Article  Google Scholar 

  30. Yoneda, T., Hatori, H., Takahara, A., Minato, S.: BDDs vs. zero–suppressed BDDs: for CTL symbolic model checking of Petri nets. In: Srivas, M., Camilleri, A. (eds.) FMCAD 1996. LNCS, vol. 1166, pp. 435–449. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Babar, J., Miner, A.S. (2014). Explicit State Space and Markov Chain Generation Using Decision Diagrams. In: Horváth, A., Wolter, K. (eds) Computer Performance Engineering. EPEW 2014. Lecture Notes in Computer Science, vol 8721. Springer, Cham. https://doi.org/10.1007/978-3-319-10885-8_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-10885-8_17

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-10884-1

  • Online ISBN: 978-3-319-10885-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics