Markov Chains Competing for Transitions: Application to Large-Scale Distributed Systems

  • Emmanuelle Anceaume
  • François Castella
  • Romaric Ludinard
  • Bruno Sericola


We consider the behavior of a stochastic system composed of several identically distributed, but non independent, discrete-time absorbing Markov chains competing at each instant for a transition. The competition consists in determining at each instant, using a given probability distribution, the only Markov chain allowed to make a transition. We analyze the first time at which one of the Markov chains reaches its absorbing state. We obtain its distribution and its expectation and we propose an algorithm to compute these quantities. We also exhibit the asymptotic behavior of the system when the number of Markov chains goes to infinity. Actually, this problem comes from the analysis of large-scale distributed systems and we show how our results apply to this domain.


Asymptotic analysis Competing Markov chains Large-scale distributed systems Markov chains 

AMS 2000 Subject Classifications

60J10 65C40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anceaume E, Brasiliero F, Ludinard R, Ravoaja A (2008) Peercube: an hypercube-based p2p overlay robust against collusion and churn. In: IEEE proceedings of the international conference on self autonomous and self organizing systems (SASO)Google Scholar
  2. Awerbuch B, Scheideler C (2004) Group spreading: a protocol for provably secure distributed name service. In: Procs of the international colloquium on automata, languages and programming (ICALP)Google Scholar
  3. Beaugendre P (2005) Un opérateur d’extension linéaire explicite. Revue de la Filière Mathématique, vol 116(1)Google Scholar
  4. Fiat A, Saia J, Young M (2005) Making chord robust to byzantine attacks. In: Proceedings of the annual European symposium on algorithms (ESA)Google Scholar
  5. Fourneau J-M (2008) Discrete time Markov chains competing over resources: product form steady-state distribution. In: Proceedings of the 5th international conference on the quantitative evaluation of systems (QEST’08), Saint-Malo, France, pp 147–156Google Scholar
  6. Harvey N, Jones MB, Saroui S, Theimer M, Wolman A (2003) Skipnet: a scalable overlay network with practical locality properties. In: Proceedings of the 4th USENIX symposium on internet technologies and systems (USITS)Google Scholar
  7. Karlin S, Taylor HM (1981) A second course in stochastic processes. Academic, New YorkzbMATHGoogle Scholar
  8. Locher T, Schmid S, Wattenhofer R (2006) eQuus: a provably robust and locality-aware peer-to-peer system. In: Proceedings of the international conference on peer-to-peer computing (P2P)Google Scholar
  9. Malkhi D, Naor M, Ratajzcak D (2003) Viceroy: scalable emulation of butterfly networks for distributed hash tables. In: Proceedings of the annual symposium on principles of distributed computing (PODC)Google Scholar
  10. Manku GS, Bawa M, Raghavan P (2003) Symphony: distributed hashing in a small world. In: Proceedings of the 4th USENIX symposium on internet technologies and systems (USITS)Google Scholar
  11. Meyer CD (2000) Matrix analysis and applied linear algebra. SIAM, PhiladelphiazbMATHCrossRefGoogle Scholar
  12. Neuts MF (1981) Matrix-geometric solutions in stochastic models: an algorithmic approach. Johns Hopkins University Press, BaltimorezbMATHGoogle Scholar
  13. Ratnasamy S, Francis P, Handley M, Karp R, Shenker S (2001) A scalable content-addressable network. In: Proceedings of the ACM SIGCOMMGoogle Scholar
  14. Riordan J (1967) An introduction to combinatorial analysis. Wiley, New YorkGoogle Scholar
  15. Rowstron A, Druschel P (2001) Pastry: scalable, distributed object location and routing for large-scale peer-to-peer systems. In: Proceedings of the international conference on distributed systems platforms (Middleware)Google Scholar
  16. Stoica I, Liben-Nowell D, Morris R, Karger D, Dabek F, Kaashoek MF, Balakrishnan H (2001) Chord: a scalable peer-to-peer lookup service for internet applications. In: Proceedings of the ACM SIGCOMMGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Emmanuelle Anceaume
    • 1
  • François Castella
    • 2
  • Romaric Ludinard
    • 3
  • Bruno Sericola
    • 3
  1. 1.IRISA - CNRSRennes cedexFrance
  2. 2.Université de Rennes 1Rennes cedexFrance
  3. 3.INRIA Rennes - Bretagne AtlantiqueRennes cedexFrance

Personalised recommendations