. Le-jeton-se-déplace-aléatoirement, après trois sauts (4,1 et 2), revient sur le site 2. La condition de déclenchement de la vague est vérifiée (T max ? 4 = jeton.saut 4 ´ etant le nombre de sites du réseau). La vague est donc propagée sur le sous-arbre construit grâce au mot circulant

. Dans, le système atteint une configuration o` u un seul jeton est présent, en temps polynomial (O(n 3 )) Afin d'accélérer la convergence vers une configuration légale, notre algorithme fusionne aussi les mots circulants contenusàcontenusà l'intérieur de chacun des jetons. L'algorithme construit un arbre avec les mots contenus dans les jetons fusionnés. Comme tous ces jetons sont sur le même site, ils sont enracinés sur le même sommet. Il est alors facile de construire l'union de tous ces arbres. Nous présentons ici l'algorithme qui fusionne deux jetons

. Dans-notre-cas, il s'agit alors de simulationsàévénementssimulationsàsimulationsàévénements discrets : chacun des sites consomme et produit desévénementsdesévénements

R. R. Aleliunas, R. Karp, L. Lipton, C. Lovasz, and . Rackoff, Random walks, universal traversal sequences, and the complexity of maze problems, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979), pp.218-223, 1979.
DOI : 10.1109/SFCS.1979.34

A. Andrzejak, Lectures on Proof Verification and Approximation Algorithms Introduction to Randomized Algorithms, LNCS, vol.1367, issue.2, p.29, 1998.

B. Awerbuch, G. Patt-shamir, S. Varghese, . Bernard, . Bui et al., A new method to automatically compute processing times for random walks based distributed algorithm Topological adaptability for the distributed token circulation paradigm in faulty environment Random distributed self-stabilizing structures maintenance, WDAG94 Distributed Algorithms 8th International Workshop Proceedings, Springer LNCS :857 ISPDC 03, IEEE International Symposium on Parallel and Distributed Computing Proceeding ISPA'04, Internationnal Symposium on Parallel and Distributed Processing and Applications ISADS'04, IEEE Internationnal Symposium on Advanced Distributed Systems Proccedings, pp.326-339, 1994.

]. T. Bbfnar, A. Bernard, O. Bui, F. Flauzac, T. Nolot et al., A multiple random walks based self-stabilizing k-exclusion algorithm in ad-hoc networks Gestion de la mobilité dans un réseau orientéorientéà l'aide d'un mot circulant, ISADS06 Fifth IEEE Internationnal Symposium on Advanced Distributed Systems Actes de la ROADEF, groupe Système Complexe et Décision Distribuée, 2006.

T. Bernard, A. Bui, O. Flauzac, C. Rabat, M. Bui et al., Decentralized resources management for grid Random walks in distributed computing, a survey, RDDS'06 International Workshop on Reliability in Decentralized Distributed systems, volume LNCS 4278 I2CS, 4th International Workshop Innovative Internet Community Systems Blin, A. Cournier, and V. Villain. An improved snap-stabilizing pif algorithm SSS'04 Self-Stabilizing Systems, pp.1530-1539, 2004.

M. Bui, S. K. Das, A. Kumar-datta, and D. Nguyen, RANDOMIZED MOBILE AGENT BASED ROUTING IN WIRELESS NETWORKS, International Journal of Foundations of Computer Science, vol.12, issue.03, pp.365-384, 2001.
DOI : 10.1142/S0129054101000539

A. Boukerche, S. K. Das, and A. Fabbri, Analysis of a Randomized Congestion Control Scheme with DSDV Routing in ad Hoc Wireless Networks, Journal of Parallel and Distributed Computing, vol.61, issue.7, pp.61967-995, 2001.
DOI : 10.1006/jpdc.2000.1723

A. Bui, . Datta, V. Petit, A. K. Villain-bein, V. Datta et al., State-optimal snap-stabilizing PIF in tree networks Snap-stabilizing optimal binary search tree, Proceedings of the Fourth Workshop on Self-Stabilizing Systems International Conference on Distributed Computing Systems) SSS'05 Self-Stabilizing Systems, pp.78-85, 1999.

[. Springer, . Beauquier, C. Gradinariu, J. Johnen, C. Beauquier et al., Token-based self-stabilization uniform algorithms k-stabilization of reactive tasks, PODC98 Proceedings of the Seventeenth Annual ACM Symposium on Principles of Distributed Computing, pp.899-921, 1998.

H. Nader, L. Bshouty, J. Higham, D. Warpechowska-grucailan, and . Zernik, Meeting times of random walks on graphs Random leaders and random spanning trees, WDAG89, pp.259-265, 1989.

U. Basile, M. O. Killijian, D. Powell, J. Burns, and . Pachl, A survey of dependability issues in mobile wireless networks Uniform self-stabilizing rings, LAAS ACM Transactions on Programming Languages and Systems, vol.11, issue.2, pp.330-344, 1989.

A. Broder, Generating random spanning trees, 30th Annual Symposium on Foundations of Computer Science, pp.442-447, 1989.
DOI : 10.1109/SFCS.1989.63516

A. Bui, D. Sohier, [. Bulgannawar, N. H. Vaidya, J. Camp et al., Hitting times computation for theoretically studying peer-to-peer distributed systems On time analysis of random walk based token circulation algorithms A distributed k-mutual exclusion algorithm A survey of mobility models for ad hoc network research Randomized dynamic route maintenance for adaptative routing multihop mobile ad hoc networks Devismes, and V. Villain. A snap-stabilizing dfs with a lower space requirement, IPDPS'04 18th International Parallel and Distributed Processing Symposium ISSADS'05 International School and Symposium on Advanced Distributed Systems, volume LNCS 3563 Proceedings of the 15th International Conference on Distributed Computing Systems (ICDCS'95) SSS'05 Self-Stabilizing SystemsCFS96] Don Coppersmith, Uriel Feige, and James Shearer. Random walks on regular and irregular graphsCR79] E.J.H. Chang and R. Roberts. An improved algorithm for decentralized extrema finding in circular arrangement process. Communications of the Association of the Computing MachineryCR83] O.S.F. Carvalho and G. Roucairol. On mutual exclusion in computer networks . Communications of the ACM, pp.179-63, 1979.

K. Ashok, P. Chandra, W. L. Raghavan, R. Ruzzo, P. Smolensky et al., The electrical resistance of a graph captures its commute and cover times Unreliable failure detectors for asynchronous systems, PODC91 Proceedings of the Tenth Annual ACM Symposium on Principles of Distributed Computing, pp.325-340, 1991.

. Th, . Bernard, . Coppersmith, P. Tetali, and . Winkler, Collisions among random walks on a graph, SIAM Journal on Discrete Mathematics, vol.6, issue.3, pp.363-374, 1993.

Y. Chen and J. L. Welch, Self-stabilizing mutual exclusion using tokens in mobile ad hoc networks, Proceedings of the 6th international workshop on Discrete algorithms and methods for mobile computing and communications , DIALM '02, pp.34-42, 2002.
DOI : 10.1145/570810.570815

J. [. Chen and . Welch, Self-stabilizing dynamic mutual exclusion for mobile ad hoc networks, Journal of Parallel and Distributed Computing, vol.65, issue.9, pp.1072-1089, 2005.
DOI : 10.1016/j.jpdc.2005.03.009

]. V. Dav00 and . Davies, Evaluating mobility models within an ad hoc network, Colorado School of Mines, 2000.

C. [. Dolev, L. Dwork, and . Stockmeyer, On the minimal synchronism needed for distributed consensus, Journal of the Association of the Computing Machinery, 1987.

W. [. Dijkstra, A. J. Feigen, and . Van-gasteren, Derivation of a termination detection algorithm for distributed computations, Information Processing Letters, vol.16, issue.5, pp.217-219, 1983.
DOI : 10.1016/0020-0190(83)90092-3

[. Dijkstra, Self stabilizing systems in spite of distributed control. Communications of the Association of the Computing Machinery, pp.643-644, 1974.

K. Ajoy, C. Datta, F. Johnen, V. Petit, and . Villain, Selfstabilizing depth-first token circulation in arbitrary rooted networks, Distributed Compututing, vol.13, issue.4, pp.207-218, 2000.

M. [. Dolev, M. Klawe, and . Rodeth, An O(n log n) unidirectional distributed algorithm for extrema finding in a circle, Journal of Algorithms, vol.3, issue.3, pp.245-260, 1982.
DOI : 10.1016/0196-6774(82)90023-2

[. Dolev, Self Stabilization, Journal of Aerospace Computing, Information, and Communication, vol.1, issue.6, 2000.
DOI : 10.2514/1.10141
URL : https://hal.archives-ouvertes.fr/inria-00627780

C. [. Dijkstra and . Scholten, Termination detection for diffusing computations, Information Processing Letters, vol.11, issue.1, pp.1-4, 1980.
DOI : 10.1016/0020-0190(80)90021-6

G. Peter, J. L. Doyle, and . Snell, Random Walks and Electric Networks, 2000.

S. Dolev, E. Schiller, and J. L. Welch, Random walk for selfstabilizing group communication in ad-hoc networks, Proc. 21st Symposium on Reliable Distributed Systems, 2002.

[. Feige, A tight lower bound for the cover time of random walks on graphs. Random structures and algorithms, pp.433-438, 1995.

[. Feige, A tight upper bound for the cover time of random walks on graphs. Random structures and algorithms, pp.51-54, 1995.

C. [. Foster and . Kesselman, The Grid : Blueprint for a Future Computing Infrastructure, 1999.

U. Flauzac, Conception d'algorithmes distribués de routagetoì erants aux fautes, 2000.

[. Flauzac, Random circulating word information management for tree construction and shortest path routing tables computation, On Principle Of DIstributed Systems, pp.17-32, 2001.

N. [. Fischer, J. Lynch, A. Burns, and . Borodin, Distributed fifo allocation of identical ressources using small shared space. Transaction on Programmning Languages and systems, pp.90-114, 1989.

N. [. Fischer, M. S. Lynch, and . Paterson, Impossibility of distributed consensus with one faulty process, Journal of the ACM, vol.32, issue.2, pp.374-382, 1985.
DOI : 10.1145/3149.214121

[. Flauzac and V. Villain, An implementable dynamic automatic selfstabilizing protocol. In I-SPAN'97, Third International Symposium on Parallel Architectures, Algorithms and Networks Proceedings, pp.91-97, 1997.
DOI : 10.1109/ispan.1997.645076

[. Goff, B. Nael, D. S. Abu-ghazaleh, R. Phatak, and . Kahvecioglu, Preemptive routing in ad hoc networks, MobiCom '01 : Proceedings of the 7th annual international conference on Mobile computing and networking, pp.43-52, 2001.

R. Michael, D. S. Garey, and . Johnson, Computers and Intractability : A Guide to the Theory of NP-Completeness, 1979.

[. Gouda and N. Multari, Stabilizing communication protocols, IEEE Transactions on Computers, vol.40, issue.4, pp.448-458, 1991.
DOI : 10.1109/12.88464

]. M. Gou95 and . Gouda, The triumph and tribulation in system stabilization, WDAG95, 9th Internationnal Workshop on Distributed Algorithm, volume LNCS :972, pp.1-18, 1995.

[. Hong, M. Gerla, G. Pei, and C. Chiang, A group mobility model for ad hoc wireless networks, Proceedings of the 2nd ACM international workshop on Modeling, analysis and simulation of wireless and mobile systems , MSWiM '99, pp.53-60, 1999.
DOI : 10.1145/313237.313248

[. Howell, M. Nesterenko, and . Mizuno, Finite-state self-stabilizing protocols in message-passing systems, Proceedings of the Fourth Workshop on Self-Stabilizing Systems (published in association with ICDCS99 The 19th IEEE International Conference on Distributed Computing Systems), pp.62-69, 1999.

V. [. Hadid and . Villain, A New Efficient Tool for the Design of Self-Stabilizing ???-Exclusion Algorithms: The Controller, WSS'01, LNCS 2194. Sringer, 2001.
DOI : 10.1007/3-540-45438-1_10

[. Hassanein and A. Zhou, Routing with load balancing in wireless Ad hoc networks, Proceedings of the 4th ACM international workshop on Modeling, analysis and simulation of wireless and mobile systems , MSWIM '01
DOI : 10.1145/381591.381614

A. Israeli and M. Jalfon, Token management schemes and random walks yield self-stabilizing mutual exclusion, Proceedings of the ninth annual ACM symposium on Principles of distributed computing , PODC '90, pp.119-131, 1990.
DOI : 10.1145/93385.93409

I. [. Ikeda, N. Kubo, M. Okumoto, and . Yamashita, Fair circulation of a token, IEEE Transactions on Parallel and Distributed Systems, vol.13, issue.4, pp.367-372, 2002.
DOI : 10.1109/71.995817

M. [. Itai and . Rodeh, Symmetry breaking in distributive networks, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981), pp.150-158, 1981.
DOI : 10.1109/SFCS.1981.41

J. [. Kim, L. Lovasz, and V. H. Vu, The cover time, the blanket time, and the matthews bound, IEEE Symposium on Foundations of Computer Science, pp.467-475, 2000.

B. David, . Johnson, A. David, and . Maltz, Dynamic source routing in ad hoc wireless networks, Mobile Computing, 1996.

]. D. Jm96b, D. A. Johnson, and . Maltz, Mobile computing, chapter 5 : The dynamic source routing in ad hoc wireless networks, 1996.

S. [. Korach, S. Kutten, and . Moran, A modular technique for the design of efficient distributed leader finding algorithms, ACM Transactions on Programming Languages and Systems, vol.12, issue.1, pp.84-101, 1990.
DOI : 10.1145/77606.77610

[. Katz and K. Perry, Self-stabilizing extensions for message-passing systems, Proceedings of the ninth annual ACM symposium on Principles of distributed computing , PODC '90, pp.17-26, 1993.
DOI : 10.1145/93385.93405

[. Karlin and P. Raghavan, Random Walks and Undirected Graph Connectivity: A Survey, IMA's, discret probability and algorithm workshop, 1993.
DOI : 10.1007/978-1-4612-0801-3_7

J. [. Kemeny and . Snell, Finite Markov Chains, 1976.

[. Kakugawa and M. Yamashita, Uniform and self-stabilizing token rings allowing unfair daemon, IEEE Transactions on Parallel and Distributed Systems, vol.8, issue.2, pp.154-162, 1997.
DOI : 10.1109/71.577257

M. [. Kakugawa and . Yamashita, A dynamic reconfiguration tolerant selfstabilizing circulation algorithm in ad-hoc network, OPODIS'04, On Principles Of DIstributed Systems, 2004.

[. Lamport, Solved problems, unsolved problems and non-problems in concurrency, invited address, PODC84 Proceedings of the Third Annual ACM Symposium on Principles of Distributed Computing, pp.1-11, 1984.

I. Lavallée, ContributionàContributionà l'algorithmiqueparalì ele et distribuée, applicationà applicationà l'optimisation combinatoire, thèse d'´ etat, 1986.

]. G. Lel77 and . Lelann, Towards a formal approach, Information Processing '77, pp.155-160, 1977.

B. Liang and Z. J. Haas, Predictive distance-based mobility management for multidimensional pcs networks, IEEE/ACM Transactions on Networking, vol.11, issue.5, pp.718-732, 2003.
DOI : 10.1109/TNET.2003.815301

A. J. Martinmr95-]-r, P. Motwani, . Raghavan, A. Merlin, . [. Segall et al., Randomized Algorithm A fail safe distributed routing protocol Extended impossibility results for asynchronous complete networks Markov Chains Ad hoc networking An o(n log n) unidirectionnal algorithm for the circular extrema problem A lower bound for broadcasting in mobile ad hoc networks A space-efficient and self-stabilizing depth-first token circulation protocol for asynchronous message-passing systems, Sci. Comput. Program Euro-Par'97 Parallel Processing, Proceedings LNCS :1300, pp.265-276, 1979.

. Springer-verlag-[-ra81-]-g, A. K. Ricart, ]. M. Agrawalarab05, and . Rakotoarisoa, An optimal algorithm for mutual exclusion in computer networks Simulateur dasor : http ://cosy.univ-reims.fr/?crabat/dasor Simulation distribuée de réseaux de files d'attente On theory of resonance, Collected scientific papers, pp.9-17, 1981.

M. Raynal, [. Royer, P. M. Melliar-smith, and L. Moser, A simple taxonomy for distributed mutual exclusion algorithms, ICC'O1, IEEE International Conference on Communications Self-stabilization. ACM Computing Surveys, pp.47-5045, 1983.
DOI : 10.1145/122120.122123
URL : https://hal.archives-ouvertes.fr/inria-00075198

M. Sanchez and P. Manzoni, ANEJOS: a Java based simulator for ad hoc networks, Future Generation Computer Systems, vol.17, issue.5, pp.573-583, 2001.
DOI : 10.1016/S0167-739X(00)00040-6

D. Sohier, Marches aléatoires dans les systèmes complexes : exemples du contrôle aérien et des algorithmes distribuésdistribuésà bases de marches aléatoires, 2005.

[. Tajibnapis, A correctness proof of a topology information maintenance protocol for a distributed computer network, Communications of the Association of the Computing Machinery, pp.477-485, 1977.
DOI : 10.1145/359636.359701

]. G. Tar95 and . Tarry, Leprobì eme des labyrinthes, Nouvelles annales de Mathématique, vol.14, p.1895

G. Tel, Introduction to distributed algorithms, 1994.
DOI : 10.1017/CBO9781139168724

[. Tetali, Random walks and the effective resistance of networks, Journal of Theoretical Probability, vol.2, issue.1, 1991.
DOI : 10.1007/BF01046996

]. V. Tol99 and . Tolety, Load reduction in ad hoc networks using mobile servers. Master's thesis, Colorado School of Mines, 1999.

[. Toueg, I. T. Watson-research, and . Center, An all-pairs shortest-path distributed algorithm, Performance Evaluation, vol.1, issue.1, 1980.
DOI : 10.1016/0166-5316(81)90048-1

P. [. Tetali and . Winkler, On a random walk problem arising in selfstabilizing token management, 10th ACM Symposium on Principles Of Distributed Computing, pp.273-280, 1991.

P. [. Tetali and . Winkler, Simultaneous reversible markov chains, Combinatorics : Paul Erdos is Eighty, pp.422-452, 1993.

G. Varghese, Self-stabilization by counter flushing, Proceedings of the thirteenth annual ACM symposium on Principles of distributed computing, pp.244-253, 1994.

. M. Vt95-]-a, G. Verweij, and . Tel, A monte carlo algorithm for election, Proceedings of the 2nd Colloquium on Structural Information and Communication Complexity, pp.77-88, 1995.

G. [. Walter, M. Cao, and . Mohanty, A k-mutual exclusion algorithm for ad hoc wireless networks, POMC01, Proceedings of the first annual Workshop on Principles of Mobile Computing, 2001.

B. [. Wang and . Li, Group mobility and partition prediction in wireless adhoc networks, ICC'02. IEEE International Conference on Communications, pp.1017-1021, 2002.

J. E. Walter, J. L. Welch, and N. H. Vaidya, A mutual exclusion algorithm for ad hoc mobile networks, Wireless Networks, vol.7, issue.6, pp.585-600, 2001.
DOI : 10.1023/A:1012363200403