Advertisement

Model Checking Two Layers of Mean-Field Models

  • Anna Kolesnichenko
  • Anne Remke
  • Pieter-Tjerk de Boer
  • Boudewijn R. Haverkort
Chapter
Part of the Springer Series in Reliability Engineering book series (RELIABILITY)

Abstract

Recently, many systems that consist of a large number of interacting objects have been analysed using the mean-field method, which allows a quick and accurate analysis of such systems, while avoiding the state-space explosion problem. To date, the mean-field method has primarily been used for classical performance evaluation purposes. In this chapter, we discuss model-checking mean-field models. We define and motivate two logics, called Mean-Field Continuous Stochastic Logic (MF-CSL) and Mean-Field Logic (MFL), to describe properties of systems composed of many identical interacting objects. We present model-checking algorithms and discuss the differences in the expressiveness of these two logics and their combinations.

Keywords

Transition Rate Boolean Function Local Model Individual Object Reachability Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

The work in this chapter has been performed when Anna Kolesnichenko was still at the University of Twente. She has been supported through NWO grant 612.063.918, MATMAN (Mean-Field Approximation Techniques for Markov Models), as well as the FP7 Sensation project (see below). Anne Remke has been supported through an NWO VENI grant on Dependability Analysis of Fluid Critical Infrastructures using Stochastic Hybrid Models. Boudewijn Haverkort and Pieter-Tjerk de Boer have been supported through FP7 STREP 318490, Sensation (Self Energy-Supporting Autonomous Computation).

References

  1. 1.
    Kurtz TG (1970) Solutions of ordinary differential equations as limits of pure jump Markov processes. J Appl Probab 7(1):49–58MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bortolussi L, Hillston J, Latella D, Massink M (2013) Continuous approximation of collective systems behaviour: a tutorial. Perform Eval 70(5):317–349CrossRefGoogle Scholar
  3. 3.
    Bakhshi R, Cloth L, Fokkink W, Haverkort BR (2009) Mean-field analysis for the evaluation of Gossip protocols. In: QEST, IEEE CS Press, pp 247–256Google Scholar
  4. 4.
    Bakhshi R, Endrullis J, Endrullis S, Fokkink W, Haverkort BR (2010) Automating the mean-field method for large dynamic gossip networks. In: QEST, IEEE CS Press, pp 241–250Google Scholar
  5. 5.
    Kolesnichenko A, Remke A, de Boer PT, Haverkort BR (2011) Comparison of the mean-field approach and simulation in a peer-to-peer botnet case study. In: EPEW, vol 6977. LNCS, Springer, pp 133–147Google Scholar
  6. 6.
    Kolesnichenko A, Remke A, de Boer PT, Haverkort BR (2013) A logic for model-checking mean-field models. In: DSN/PDF, IEEE CS Press, pp 1–12Google Scholar
  7. 7.
    Maler O, Nickovic D (2004) Monitoring temporal properties of continuous signals. In: FORMATS, vol 3253. LNCS, Springer, pp 152–166Google Scholar
  8. 8.
    Donzé A, Clermont G, Legay A, Langmead CJ (2010) Parameter synthesis in nonlinear dynamical systems: application to systems biology. J Comput Biol 17(3):325–336MathSciNetCrossRefGoogle Scholar
  9. 9.
    Donzé A, Ferrère T, Maler O (2013) Efficient robust monitoring for STL. In: CAV, vol 8044. LNCS, Springer, pp 264–279Google Scholar
  10. 10.
    Hillston J (2014) The benefits of sometimes not being discrete. In: CONCUR, vol 8704. LNCS, Springer, pp 7–22Google Scholar
  11. 11.
    Bortolussi L, Hillston J (2012) Fluid model checking. In: CONCUR, vol 7454. LNCS, Springer, pp 333–347Google Scholar
  12. 12.
    Bortolussi L, Hillston J (2013) Checking individual agent behaviours in markov population models by fluid approximation. In: SFM, vol 7938. LNCS, Springer, pp 113–149Google Scholar
  13. 13.
    Bortolussi L, Lanciani R (2013) Model checking Markov population models by central limit approximation. In: QEST, vol 8054. LNCS, Springer, pp 123–138Google Scholar
  14. 14.
    Latella D, Loreti M, Massink M (2014) On-the-fly fast mean-field model-checking. In: TGC, LNCS, Springer, pp 297–314Google Scholar
  15. 15.
    Donzé A, Maler O (2010) Robust satisfaction of temporal logic over real-valued signals. In: FORMATS, vol 6246. LNCS, Springer, pp 92–106Google Scholar
  16. 16.
    Fainekos GE, Pappas GJ (2009) Robustness of temporal logic specifications for continuous-time signals. Theor Comput Sci 410(42):4262–4291MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rizk A, Batt G, Fages F, Soliman S (2008) On a continuous degree of satisfaction of temporal logic formulae with applications to systems biology. In: CMSB, vol 5307. LNCS, Springer, pp 251–268Google Scholar
  18. 18.
    Bartocci E, Bortolussi L, Nenzi L, Sanguinetti G (2013) On the robustness of temporal properties for stochastic models. In: HSB, vol 125. EPTCS, Open Publishing Association, pp 3–19Google Scholar
  19. 19.
    Donzé A, Breach A (2010) Toolbox for verification and parameter synthesis of hybrid systems. In: CAV, vol 6174. LNCS, Springer, pp 167–170Google Scholar
  20. 20.
    Annpureddy Y, Liu C, Fainekos G, Sankaranarayanan S (2011) S-TaLiRo: a tool for temporal logic falsification for hybrid systems. In: TACAS, vol 6605. LNCS, Springer, pp 254–257Google Scholar
  21. 21.
    Calzone L, Fages F, Soliman S (2006) Biocham: an environment for modeling biological systems and formalizing experimental knowledge. Bioinformatics 22(14):1805–1807CrossRefGoogle Scholar
  22. 22.
    Chaintreau A, Le Boudec JY, Ristanovic N (2009) The age of gossip: spatial mean field regime. In: ACM SIGMETRICS/Performance, pp 109–120Google Scholar
  23. 23.
    Bobbio A, Gribaudo M, Telek M (2008) Analysis of large scale interacting systems by mean field method. In: QEST, IEEE Computer Society, pp 215–224Google Scholar
  24. 24.
    Darling RWR, Fluid limits of pure jump Markov processes: a practical guide, ArXiv mathematics e-prints arXiv:arXiv:math/0210109
  25. 25.
    Darling RWR, Norris JR (2008) Differential equation approximations for Markov chains. Probab Surv 5:37–79MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    van Ruitenbeek E, Sanders WH (2008) Modeling peer-to-peer botnets. In: QEST, IEEE CS Press, pp 307–316Google Scholar
  27. 27.
    Kurtz TG (1970) Solutions of ordinary differential equations as limits of pure jump Markov processes. J Appl Probab 7(1):49–58MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Billingsley P (1995) Probability and measure. 3rd edn. Wiley-InterscienceGoogle Scholar
  29. 29.
    Wolfram Research Inc (2010) Mathematica tutorial. http://www.wolfram.com/mathematica/
  30. 30.
    Gast N, Gaujal B (2010) A mean field model of work stealing in large-scale systems. In: ACM SIGMETRICS, ACM, pp 13–24Google Scholar
  31. 31.
    Bortolussi L (2011) Hybrid limits of continuous time Markov chains. In: QEST, IEEE Computer Society, pp 3–12Google Scholar
  32. 32.
    Hayden RA (2012) Mean field for performance models with deterministically-timed transitions. In: QEST, IEEE Computer Society, pp 63–73Google Scholar
  33. 33.
    Hayden RA, Horvàth I, Telek M (2014) Mean field for performance models with generally-distributed timed transitions. In: QEST, vol 8657. LNCS, Springer, pp 90–105Google Scholar
  34. 34.
    Stefanek A, Hayden RA, Gonagle MM, Bradley JT (2012) Mean-field analysis of Markov models with reward feedback. In: ASMTA, vol 7314. LNCS, pp 193–211Google Scholar
  35. 35.
    Stefanek A, Hayden RA, Bradley JT (2014) Mean-field analysis of hybrid Markov population models with time-inhomogeneous rates. Ann Oper Res 1–27Google Scholar
  36. 36.
    Le Boudec JY (2010) The stationary behaviour of fluid limits of reversible processes is concentrated on stationary points. Technical reportGoogle Scholar
  37. 37.
    Benaïm M, Le Boudec JY (2008) A class of mean field interaction models for computer and communication systems. Perform Eval 65(11–12):823–838CrossRefGoogle Scholar
  38. 38.
    Baier C, Haverkort BR, Hermanns H, Katoen JP (2003) Model-checking algorithms for continuous-time Markov chains. IEEE Trans Softw Eng 29(7):524–541CrossRefzbMATHGoogle Scholar
  39. 39.
    Nickovic D, Maler O (2007) AMT: a property-based monitoring tool for analog systems. In: FORMATS, vol 4763. LNCS, Springer, pp 304–319Google Scholar
  40. 40.
    Pnueli A (1977) The temporal logic of programs. In: SFCS, IEEE computer society, pp 46–57Google Scholar
  41. 41.
    Gómez-Marn AM, Hernndez-Ortz JP (2013) Mean field approximation of Langmuir-Hinshelwood CO-surface reactions considering lateral interactions. J Phys Chem 117(30):15716–15727Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Anna Kolesnichenko
    • 1
  • Anne Remke
    • 2
  • Pieter-Tjerk de Boer
    • 3
  • Boudewijn R. Haverkort
    • 3
  1. 1.UL Transaction Security DivisionLeidenThe Netherlands
  2. 2.Department of Computer ScienceUniversity of MünsterMünsterGermany
  3. 3.Department of Computer ScienceUniversity of TwenteEnschedeThe Netherlands

Personalised recommendations