As Soon as Probable: Optimal Scheduling under Stochastic Uncertainty

  • Jean-François Kempf
  • Marius Bozga
  • Oded Maler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7795)


In this paper we continue our investigation of stochastic (and hence dynamic) variants of classical scheduling problems. Such problems can be modeled as duration probabilistic automata (DPA), a well-structured class of acyclic timed automata where temporal uncertainty is interpreted as a bounded uniform distribution of task durations [18]. In [12] we have developed a framework for computing the expected performance of a given scheduling policy. In the present paper we move from analysis to controller synthesis and develop a dynamic-programming style procedure for automatically synthesizing (or approximating) expected time optimal schedulers, using an iterative computation of a stochastic time-to-go function over the state and clock space of the automaton.


Pareto Front Optimal Schedule Discrete Event System Time Density Controller Synthesis 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abdeddaïm, Y., Asarin, E., Maler, O.: Scheduling with timed automata. Theoretical Computer Science 354(2), 272–300 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alur, R., Bernadsky, M.: Bounded Model Checking for GSMP Models of Stochastic Real-Time Systems. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 19–33. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Asarin, E., Degorre, A.: Volume and Entropy of Regular Timed Languages: Analytic Approach. In: Ouaknine, J., Vaandrager, F.W. (eds.) FORMATS 2009. LNCS, vol. 5813, pp. 13–27. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Asarin, E., Maler, O.: As Soon as Possible: Time Optimal Control for Timed Automata. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 19–30. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Asarin, E., Maler, O., Pnueli, A., Sifakis, J.: Controller synthesis for timed automata. In: Proc. IFAC Symposium on System Structure and Control, pp. 469–474 (1998)Google Scholar
  6. 6.
    Carnevali, L., Grassi, L., Vicario, E.: State-density functions over DBM domains in the analysis of non-Markovian models. IEEE Trans. Software Eng. 35(2), 178–194 (2009)CrossRefGoogle Scholar
  7. 7.
    Cassandras, C.G., Lafortune, S.: Introduction to Discrete Event Systems. Springer (2008)Google Scholar
  8. 8.
    David, A., Larsen, K.G., Legay, A., Mikučionis, M., Wang, Z.: Time for Statistical Model Checking of Real-Time Systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 349–355. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Daws, C., Olivero, A., Tripakis, S., Yovine, S.: The Tool KRONOS. In: Alur, R., Sontag, E.D., Henzinger, T.A. (eds.) HS 1995. LNCS, vol. 1066, pp. 208–219. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  10. 10.
    German, R.: Non-Markovian Analysis. In: Brinksma, E., Hermanns, H., Katoen, J.-P. (eds.) FMPA 2000. LNCS, vol. 2090, pp. 156–182. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    Glynn, P.W.: A GSMP formalism for discrete event systems. Proceedings of the IEEE 77(1), 14–23 (1989)CrossRefGoogle Scholar
  12. 12.
    Kempf, J.-F., Bozga, M., Maler, O.: Performance Evaluation of Schedulers in a Probabilistic Setting. In: Fahrenberg, U., Tripakis, S. (eds.) FORMATS 2011. LNCS, vol. 6919, pp. 1–17. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. 13.
    Kempf, J.-F.: On Computer-Aided Design-Space Exploration for Multi-Cores. PhD thesis, University of Grenoble (October 2012)Google Scholar
  14. 14.
    Larsen, K.G., Behrmann, G., Brinksma, E., Fehnker, A., Hune, T., Pettersson, P., Romijn, J.: As Cheap as Possible: Efficient Cost-Optimal Reachability for Priced Timed Automata. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 493–505. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  15. 15.
    Larsen, K.G., Pettersson, P., Yi, W.: UPPAAL in a nutshell. International Journal on Software Tools for Technology Transfer (STTT) 1(1), 134–152 (1997)zbMATHCrossRefGoogle Scholar
  16. 16.
    Legriel, J., Le Guernic, C., Cotton, S., Maler, O.: Approximating the Pareto Front of Multi-criteria Optimization Problems. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 69–83. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  17. 17.
    Maler, O.: On optimal and reasonable control in the presence of adversaries. Annual Reviews in Control 31(1), 1–15 (2007)CrossRefGoogle Scholar
  18. 18.
    Maler, O., Larsen, K.G., Krogh, B.H.: On zone-based analysis of duration probabilistic automata. In: INFINITY, pp. 33–46 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Jean-François Kempf
    • 1
  • Marius Bozga
    • 1
  • Oded Maler
    • 1
  1. 1.CNRS-VERIMAGUniversity of GrenobleFrance

Personalised recommendations