Advertisement

Computing the Fault-Containment Time of Self-Stabilizing Algorithms Using Markov Chains and Lumping

  • Volker Turau
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10616)

Abstract

The analysis of self-stabilizing algorithms is in the vast majority of all cases limited to the worst case stabilization time starting from an arbitrary configuration. Considering the fact that these algorithms are intended to provide fault tolerance in the long run this is not the most relevant metric. From a practical point of view the worst case time to recover in case of a single fault is much more crucial. This paper presents techniques to derive upper bounds for the mean time to recover from a single fault for self-stabilizing algorithms Markov chains in combination with lumping. To illustrate the applicability of the techniques they are applied to a self-stabilizing coloring algorithm.

Notes

Acknowledgments

Research was funded by Deutsche Forschungsgemeinschaft DFG (TU 221/6-1).

References

  1. 1.
    Azar, Y., Kutten, S., Patt-Shamir, B.: Distributed error confinement. ACM Trans. Algorithms 6(3), 48:1–48:23 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barenboim, L., Elkin, M.: Distributed Graph Coloring: Fundamentals and Recent Developments. Morgan & Claypool Publishers, San Rafael (2013)zbMATHGoogle Scholar
  3. 3.
    Beauquier, J., Delaet, S., Haddad, S.: Necessary and sufficient conditions for 1-adaptivity. In: 20th Internatioal Parallel and Distributed Processing Symposium, pp. 10–16 (2006)Google Scholar
  4. 4.
    Crouzen, P., Hahn, E., Hermanns, H., Dhama, A., Theel, O., Wimmer, R., Braitling, B., Becker, B.: Bounded fairness for probabilistic distributed algorithms. In: 11th International Conference Application of Concurrency to System Design, pp. 89–97, June 2011Google Scholar
  5. 5.
    Lee DeVille, R.E., Mitra, S.: Stability of distributed algorithms in the face of incessant faults. In: Guerraoui, R., Petit, F. (eds.) SSS 2009. LNCS, vol. 5873, pp. 224–237. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-05118-0_16CrossRefGoogle Scholar
  6. 6.
    Dolev, S.: Self-Stabilization. MIT Press, Cambridge (2000)zbMATHGoogle Scholar
  7. 7.
    Dolev, S., Herman, T.: Superstabilizing protocols for dynamic distributed systems. Chicago J. Theor. Comput. Sci. 4, 1–40 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dubois, S., Masuzawa, T., Tixeuil, S.: Bounding the impact of unbounded attacks in stabilization. IEEE Trans. Parallel Distrib. Syst. 23(3), 460–466 (2012)CrossRefGoogle Scholar
  9. 9.
    Duflot, M., Fribourg, L., Picaronny, C.: Randomized finite-state distributed algorithms as Markov chains. In: Welch, J. (ed.) DISC 2001. LNCS, vol. 2180, pp. 240–254. Springer, Heidelberg (2001). doi: 10.1007/3-540-45414-4_17CrossRefzbMATHGoogle Scholar
  10. 10.
    Fribourg, L., Messika, S., Picaronny, C.: Coupling and self-stabilization. Distrib. Comput. 18(3), 221–232 (2006)CrossRefGoogle Scholar
  11. 11.
    Gärtner, F.C.: Fundamentals of fault-tolerant distributed computing in asynchronous environments. ACM Comput. Surv. 31(1), 1–26 (1999)CrossRefGoogle Scholar
  12. 12.
    Ghosh, S., Gupta, A.: An exercise in fault-containment: self-stabilizing leader election. Inf. Process. Lett. 59(5), 281–288 (1996)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ghosh, S., Gupta, A., Herman, T., Pemmaraju, S.: Fault-containing self-stabilizing distributed protocols. Distrib. Comput. 20(1), 53–73 (2007)CrossRefGoogle Scholar
  14. 14.
    Ghosh, S., He, X.: Scalable self-stabilization. J. Parallel Distrib. Comput. 62(5), 945–960 (2002)CrossRefGoogle Scholar
  15. 15.
    Gradinariu, M., Tixeuil, S.: Self-stabilizing vertex coloring of arbitrary graphs. In: 4th International Conference on Principles of Distributed Systems, OPODIS 2000, pp. 55–70 (2000)Google Scholar
  16. 16.
    Johansson, Ö.: Simple distributed \(\delta +1\)-coloring of graphs. Inf. Process. Lett. 70(5), 229–232 (1999)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Springer, Heidelberg (1976)zbMATHGoogle Scholar
  18. 18.
    Köhler, S., Turau, V.: Fault-containing self-stabilization in asynchronous systems with constant fault-gap. Distrib. Comput. 25(3), 207–224 (2012)CrossRefGoogle Scholar
  19. 19.
    Kutten, S., Patt-Shamir, B.: Adaptive stabilization of reactive protocols. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 396–407. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-30538-5_33CrossRefzbMATHGoogle Scholar
  20. 20.
    Lenzen, C., Suomela, J., Wattenhofer, R.: Local algorithms: self-stabilization on speed. In: Guerraoui, R., Petit, F. (eds.) SSS 2009. LNCS, vol. 5873, pp. 17–34. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-05118-0_2CrossRefGoogle Scholar
  21. 21.
    Luby, M.: A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15(4), 1036–1055 (1986)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mitton, N., Fleury, E., Guérin-Lassous, I., Séricola, B., Tixeuil, S.: On fast randomized colorings in sensor networks. In: Proceedings of ICPADS, pp. 31–38. IEEE (2006)Google Scholar
  23. 23.
    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM Society for Industrial and Applied Mathematics, Philadelphia (2000)CrossRefGoogle Scholar
  24. 24.
    Turau, V.: Computing the fault-containment time of self-stabilizing algorithms using Markov chains. Technical report, Hamburg University of Techology (2017)Google Scholar
  25. 25.
    Turau, V., Hauck, B.: A fault-containing self-stabilizing (3–2/(delta+1))-approximation algorithm for vertex cover in anonymous networks. Theoret. Comput. Sci. 412(33), 4361–4371 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yamashita, M.: Probabilistic self-stabilization and random walks. In: 2013 International Conference on Computing, Networking and Communications (ICNC), pp. 1–7 (2011)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of TelematicsHamburg University of TechnologyHamburgGermany

Personalised recommendations