T. Since-?, 0 as ? ? 0 by the no generation condition (F2) of SPF, there exists a ? 1 < ? 0 such that µ T (s ? 1 ) = ? by the intermediate value property of continuity. By the bounded stabilization time condition (F5), there are no transitions

M. Akian, S. Gaubert, and C. Walsh, Discrete max-plus spectral theory, Idempotent Mathematics and Mathematical Physics, pp.53-78, 2005.
DOI : 10.1090/conm/377/06984
URL : https://hal.archives-ouvertes.fr/inria-00000968

D. Aldous, Random walks on finite groups and rapidly mixing markov chains, Séminaire de probabilités de Strasbourg, pp.243-297, 1983.
DOI : 10.1214/aop/1176994578

D. Aldous and P. Diaconis, Shuffling Cards and Stopping Times, The American Mathematical Monthly, vol.93, issue.5, pp.333-348, 1986.
DOI : 10.2307/2323590

D. Aldous and P. Diaconis, Strong uniform times and finite random walks, Advances in Applied Mathematics, vol.8, issue.1, pp.69-97, 1987.
DOI : 10.1016/0196-8858(87)90006-6

J. Peter and . Ashenden, The Designers Guide to VHDL, 2008.

B. Awerbuch, Complexity of network synchronization, Journal of the ACM, vol.32, issue.4, pp.804-823, 1985.
DOI : 10.1145/4221.4227

M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani et al., Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proceedings of the National Academy of Sciences, pp.1232-1237, 2008.
DOI : 10.1073/pnas.0711437105

C. Valmir, E. Barbosa, and . Gafni, Concurrency in heavily loaded neighborhoodconstrained systems, ACM Transactions on Programming Languages and Systems, vol.11, issue.4, pp.562-584, 1989.

G. Barnes and U. Feige, Short Random Walks on Graphs, SIAM Journal on Discrete Mathematics, vol.9, issue.1, pp.19-28, 1996.
DOI : 10.1137/S0895480194264988

C. José, B. W. Barros, and . Johnson, Equivalence of the arbiter, the synchronizer, the latch, and the inertial delay, IEEE Transactions on Computers, vol.32, issue.7, pp.603-614, 1983.

M. J. Bellido-díaz, J. Juan-chico, A. J. Acosta, M. Valencia, and J. L. Huertas, Logical modelling of delay degradation effect in static CMOS gates, IEE Proceedings ? Circuits, Devices, and Systems, pp.107-117, 2000.
DOI : 10.1049/ip-cds:20000197

J. Manuel, J. Bellido-díaz, M. Juan-chico, and . Valencia, Logic-Timing Simulation and the Degradation Delay Model, 2006.

D. Vincent, J. M. Blondel, A. Hendrickx, J. N. Olshevsky, and . Tsitsiklis, Convergence in multiagent coordination, consensus, and flocking, Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (CDC-ECC), pp.2996-3000, 2005.

A. Bouillard and B. Gaujal, Coupling time of a (max,plus) matrix, Proceedings of the Workshop on Max-Plus Algebra at the 1st IFAC Symposium on System Structure and Control, 2001.
URL : https://hal.archives-ouvertes.fr/inria-00100573

A. Bouillard and . Thierry, An algorithmic toolbox for network calculus. Discrete Event Dynamic Systems, pp.3-49, 2008.
URL : https://hal.archives-ouvertes.fr/inria-00123643

S. Michael and . Branicky, Universal computation and other capabilities of hybrid and continuous dynamical systems, Theoretical Computer Science, vol.138, issue.1, pp.67-100, 1995.

A. Brauer, On a Problem of Partitions, American Journal of Mathematics, vol.64, issue.1/4, pp.299-312, 1942.
DOI : 10.2307/2371684

A. Richard, H. J. Brualdi, and . Ryser, Combinatorial Matrix Theory, 1991.

A. Janusz, J. C. Brzozowski, and . Ebergen, On the delay-sensitivity of gate networks, IEEE Transactions on Computers, vol.41, issue.11, pp.1349-1360, 1992.

C. Busch and S. Tirthapura, Analysis of Link Reversal Routing Algorithms, SIAM Journal on Computing, vol.35, issue.2, pp.305-326, 2005.
DOI : 10.1137/S0097539704443598

M. Cao, A. S. Morse, and B. D. Anderson, Reaching a Consensus in a Dynamically Changing Environment: A Graphical Approach, SIAM Journal on Control and Optimization, vol.47, issue.2, pp.575-600, 2008.
DOI : 10.1137/060657005

M. Cao, A. S. Morse, and B. D. Anderson, Reaching a Consensus in a Dynamically Changing Environment: Convergence Rates, Measurement Delays, and Asynchronous Events, SIAM Journal on Control and Optimization, vol.47, issue.2, pp.601-623, 2008.
DOI : 10.1137/060657029

K. M. Chandy and J. Misra, The drinking philosophers problem, ACM Transactions on Programming Languages and Systems, vol.6, issue.4, pp.632-646, 1984.
DOI : 10.1145/1780.1804

B. Charron-bost, Orientation and connectivity based criteria for asymptotic consensus, 2013.

B. Charron-bost, M. F. Ugger, J. L. Welch, and J. Widder, Full Reversal Routing as a Linear Dynamical System, Proceedings of the 18th International Colloquium on Structural Information and Communication Complexity (SIROCCO), pp.101-112, 2011.
DOI : 10.1007/BFb0022451

B. Charron-bost, M. F. Ugger, J. L. Welch, and J. Widder, Partial is Full, Proceedings of the 18th International Colloquium on Structural Information and Communication Complexity (SIROCCO), pp.113-124, 2011.
DOI : 10.1007/978-3-642-05434-1_10

B. Charron-bost and A. Schiper, The Heard-Of model: computing in distributed systems with benign faults, Distributed Computing, vol.384, issue.3, pp.49-71, 2009.
DOI : 10.1007/s00446-009-0084-6

B. Chazelle, -Energy of a Multiagent System, SIAM Journal on Control and Optimization, vol.49, issue.4, pp.1680-1706, 2011.
DOI : 10.1137/100791671
URL : https://hal.archives-ouvertes.fr/hal-01179401

F. Chung, Laplacians and the Cheeger Inequality for Directed Graphs, Annals of Combinatorics, vol.9, issue.1, pp.1-19, 2005.
DOI : 10.1007/s00026-005-0237-z

G. Cohen, P. Moller, J. Quadrat, and M. Viot, Algebraic tools for the performance evaluation of discrete event systems, Proceedings of the IEEE, vol.77, issue.1, pp.39-85, 1989.
DOI : 10.1109/5.21069

G. Cohen, D. Dubois, J. Quadrat, and M. Viot, Analyse du comportement périodique de systèmes de production par la théorie des dio¨?desdio¨?des, INRIA Research Report, vol.191, 1983.

G. Cohen, D. Dubois, J. P. Quadrat, and M. Viot, A linear-system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing, IEEE Transactions on Automatic Control, vol.30, issue.3, pp.210-220, 1985.
DOI : 10.1109/TAC.1985.1103925

C. Constantinescu, Trends and challenges in VLSI circuit reliability, IEEE Micro, vol.23, issue.4, pp.14-19, 2003.
DOI : 10.1109/MM.2003.1225959

E. V. Denardo, Periods of Connected Networks and Powers of Nonnegative Matrices, Mathematics of Operations Research, vol.2, issue.1, pp.20-24, 1977.
DOI : 10.1287/moor.2.1.20

P. Diaconis and D. Stroock, Geometric Bounds for Eigenvalues of Markov Chains, The Annals of Applied Probability, vol.1, issue.1, pp.36-61, 1991.
DOI : 10.1214/aoap/1177005980

]. D. Dolev, M. F. ¨-ugger, C. Lenzen, and U. Schmid, Fault-tolerant algorithms for tickgeneration in asynchronous logic: Robust pulse generation, Proceeding of the 13th International Symposium on Stabilization , Safety, and Security of Distributed Systems (SSS), pp.163-177, 2011.

S. 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

D. Dubois and K. E. Stecke, Dynamic analysis of repetitive decision-free discreteevent processes: The algebra of timed marked graphs and algorithmic issues, Annals of Operations Research, vol.15, issue.5, pp.150-193, 1990.
DOI : 10.1007/BF02248590

A. L. Dulmage and N. S. Mendelsohn, Gaps in the exponent set of primitive matrices, Illinois Journal of Mathematics, vol.8, issue.4, pp.642-656, 1964.

S. Even and S. Rajsbaum, The use of a synchronizer yields the maximum computation rate in distributed networks, Theory of Computing Systems, vol.5, issue.9, pp.447-474, 1997.
DOI : 10.1007/BF02679457

G. Fuchs, M. F. Ugger, A. Steininger-matthias, F. Ugger, and U. Schmid, On the threat of metastability in an asynchronous fault-tolerant clock generation scheme Reconciling fault-tolerant distributed computing and systems-on-chip, Proceedings of the 15th IEEE International Symposium on Asynchronous Circuits and Systems (ASYNC) Distributed Computing, pp.127-136, 2009.

M. J. Gadledge, P. H. Eaton, J. M. Benedetto, M. Carts, V. Zhu et al., Digital Device Error Rate Trends in Advanced CMOS Technologies, IEEE Transactions on Nuclear Science, vol.53, issue.6, pp.3466-3471, 2006.
DOI : 10.1109/TNS.2006.886212

E. M. Gafni and D. P. Bertsekas, Asymptotic optimality of shortest path routing algorithms, IEEE Transactions on Information Theory, vol.33, issue.1, pp.83-90, 1987.
DOI : 10.1109/TIT.1987.1057274

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

R. M. Goverde, P. H. Bovy, and G. J. Olsder, The max-plus algebra approach to transportation problems, Proceedings of the 8th World Conference on Transport Research (WCTR), pp.377-390, 1999.

D. A. Gregory, S. J. Kirkland, and N. J. Pullman, A bound on the exponent of a primitive matrix using Boolean rank, Linear Algebra and its Applications, vol.217, pp.101-116, 1995.
DOI : 10.1016/0024-3795(92)00003-5

C. Hanen, A. Edward, G. Coffmann, and J. , Cyclic scheduling on parallel processors: an overview, Scheduling Theory and Its Applications, pp.193-225, 1995.

M. Hartmann and C. Arguelles, Transience Bounds for Long Walks, Mathematics of Operations Research, vol.24, issue.2, pp.414-439, 1999.
DOI : 10.1287/moor.24.2.414

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence models, analysis, and simulation, Journal of Artificial Societies and Social Simulation, vol.5, issue.3, 2002.

R. Hegselmann and U. Krause, Truth and cognitive division of labour: First steps towards a computer aided social epistemology, Journal of Artificial Societies and Social Simulation, vol.9, issue.3, 2006.

B. Heidergott, G. J. Olsder, and J. Van-der-woude, Max Plus at Work, 2006.
DOI : 10.1515/9781400865239

M. Julien, V. D. Hendrickx, and . Blondel, Convergence of linear and non-linear versions of Vicsek's model, CESAME Research Report, vol.57, 2005.

A. Jadbabaie, J. Lin, and A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Transactions on Automatic Control, vol.48, issue.6, pp.988-1001, 2003.
DOI : 10.1109/TAC.2003.812781

K. H. Kim, An extension of the Dulmage-Mendelsohn theorem, Linear Algebra and its Applications, vol.27, pp.187-197, 1979.
DOI : 10.1016/0024-3795(79)90041-7

J. David and . Kinniment, Synchronization and Arbitration in Digital Systems, 2007.

H. J. Landau and A. M. Odlyzko, Bounds for eigenvalues of certain stochastic matrices, Linear Algebra and its Applications, vol.38, pp.5-15, 1981.
DOI : 10.1016/0024-3795(81)90003-3

D. A. Levin, Y. Peres, and E. L. Wilmer, Markov Chains and Mixing Times, 2009.
DOI : 10.1090/mbk/058

N. A. Lynch, Distributed Algorithms, 1996.

Y. Malka, S. Moran, and S. Zaks, A lower bound on the period length of a distributed scheduler, Algorithmica, vol.29, issue.1, pp.383-398, 1993.
DOI : 10.1007/BF01769705

Y. Malka and S. Rajsbaum, Analysis of distributed algorithms based on recurrence relations, Proceedings of the 5th International Workshop on Distributed Algorithms (WDAG), pp.242-253, 1992.
DOI : 10.1007/BFb0022451

R. Leonard and . Marino, The effect of asynchronous inputs on sequential network reliability, IEEE Transactions on Computers, vol.26, issue.11, pp.1082-1090, 1977.

R. Leonard and . Marino, General theory of metastable operation, IEEE Transactions on Computers, vol.30, issue.2, pp.107-115, 1981.

M. , S. Maza, and M. L. Aranda, Analysis of clock distribution networks in the presence of crosstalk and groundbounce, Proceedings of the 8th IEEE International Conference on Electronics, Circuits, and Systems (ICECS), pp.773-776, 2001.

M. Mendler and T. Stroup, Newtonian arbiters cannot be proven correct. Formal Methods in System Design, pp.233-257, 1993.
DOI : 10.1007/bf01384075

M. Mihail, Conductance and convergence of Markov chains-a combinatorial treatment of expanders, 30th Annual Symposium on Foundations of Computer Science, pp.526-531, 1989.
DOI : 10.1109/SFCS.1989.63529

L. Moreau, Stability of multiagent systems with time-dependent communication links, IEEE Transactions on Automatic Control, vol.50, issue.2, pp.169-18287, 1997.
DOI : 10.1109/TAC.2004.841888

A. Olshevsky and J. N. Tsitsiklis, Convergence Speed in Distributed Consensus and Averaging, SIAM Review, vol.53, issue.4, pp.747-772, 2011.
DOI : 10.1137/110837462

J. Rohit and . Parikh, On context-free languages, Journal of the ACM, vol.13, issue.4, pp.570-581, 1966.

J. W. Pitman, Uniform rates of convergence for Markov chain transition probabilities, Zeitschrift f???r Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol.13, issue.3, pp.193-227, 1974.
DOI : 10.1007/BF00536280

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, ACM SIGGRAPH Computer Graphics, vol.21, issue.4, pp.25-34, 1987.
DOI : 10.1145/37402.37406

F. U. Rosenberger, C. E. Molnar, T. J. Chaney, and T. Fang, Q-modules: internally clocked delay-insensitive modules, IEEE Transactions on Computers, vol.37, issue.9, pp.1005-1018, 1988.
DOI : 10.1109/12.2252

H. Schneider and M. H. Schneider, Max-Balancing Weighted Directed Graphs and Matrix Scaling, Mathematics of Operations Research, vol.16, issue.1, pp.208-222, 1991.
DOI : 10.1287/moor.16.1.208

J. Schoissengeier, The discrepancy of (n?) n ?1, Mathematische Annalen, vol.77, issue.1?2, pp.529-545703, 1970.
DOI : 10.1007/BF01445118

S. Schwarz, On the semigroup of binary relations on a finite set, Czechoslovak Mathematical Journal, vol.20, issue.4, pp.632-679, 1970.

E. Seneta, Non-negative Matrices and Markov Chains, 1993.
DOI : 10.1007/0-387-32792-4

A. Sinclair and M. Jerrum, Approximate counting, uniform generation and rapidly mixing Markov chains, Information and Computation, vol.82, issue.1, pp.93-133, 1989.
DOI : 10.1016/0890-5401(89)90067-9

G. Soto and . Koelemeijer, On the Behaviour of Classes of Min-Max-Plus Systems, 2003.

H. G. Tanner, A. Jadbabaie, and G. J. Pappas, Flocking in Fixed and Switching Networks, IEEE Transactions on Automatic Control, vol.52, issue.5, pp.863-868, 2007.
DOI : 10.1109/TAC.2007.895948

B. Touri and A. Nedi´cnedi´c, Product of Random Stochastic Matrices, IEEE Transactions on Automatic Control, vol.59, issue.2, 2011.
DOI : 10.1109/TAC.2013.2283750

N. John and . Tsitsiklis, Problems in Decentralized Decision Making and Computation, 1984.

H. Stephen and . Unger, Asynchronous sequential switching circuits with unrestricted input changes, IEEE Transaction on Computers, vol.20, issue.12, pp.1437-1444, 1971.

H. Wielandt, . Unzerlegbare, and . Nicht-negative-matrizen, Unzerlegbare, nicht negative Matrizen, Mathematische Zeitschrift, vol.192, issue.1, pp.642-645, 1950.
DOI : 10.1007/BF02230720

B. Charron-bost, M. F. Ugger, and T. Nowak, New transience bounds for long walks. Preprint, submitted, arXiv:1209
URL : https://hal.archives-ouvertes.fr/hal-00993814

B. Charron-bost, M. F. Ugger, and T. Nowak, New transience bounds for long walks in weighted digraphs, Proceedings of the 7th European Conference on Combinatorics, Graph Theory, and Applications (EuroComb), pp.623-624, 2013.
DOI : 10.1007/978-88-7642-475-5_103
URL : https://hal.archives-ouvertes.fr/hal-00993814

B. Charron-bost, M. F. Ugger, and T. Nowak, Transience Bounds for Distributed Algorithms, Proceeding of the 11th International Conference on Formal Modeling and Analysis of Timed Systems (FORMATS), pp.77-90, 2013.
DOI : 10.1007/978-3-642-40229-6_6
URL : https://hal.archives-ouvertes.fr/hal-00993785

B. Charron-bost, T. Nowak, G. L. Litvinov, V. P. Maslov, A. G. Kushner et al., General transience bounds in tropical linear algebra via Nachtigall decomposition, Proceedings of the Workshop on Tropical and Idempotent Mathematics Institute for Information Transmission Problems of the Russian Academy of Sciences, pp.46-52, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00993902

F. Matthias, A. K. Ugger, T. Oßler, M. Nowak, and . Zeiner, Lecture Notes in Computer Science Brief announcement: The degrading effect of forgetting on a synchronizer, Proceedings of the 9th International Symposium on Algorithms and Experiments for Sensor Systems, Wireless Networks and Distributed Robotics (Algosensors) Proceeding of the 14th International Symposium on Stabilization , Safety, and Security of Distributed Systems (SSS), pp.185-200, 2012.

F. Matthias, T. Ugger, U. Nowak, and . Schmid, Unfaithful glitch propagation in existing binary circuit models, Proceedings of the 19th IEEE International Symposium on Asynchronous Circuits and Systems (ASYNC), pp.191-199

G. Merlet, T. Nowak, H. Schneider, and S. Sergeev, Generalizations of bounds on the index of convergence to weighted digraphs, Discrete Applied Mathematics, p.2014
URL : https://hal.archives-ouvertes.fr/hal-01107293

G. Merlet, T. Nowak, and S. Sergeev, Weak CSR expansions and transience bounds in max-plus algebra, Linear Algebra and its Applications, vol.461, pp.163-199, 2014.
DOI : 10.1016/j.laa.2014.07.027
URL : https://hal.archives-ouvertes.fr/hal-01058031

T. Nowak and B. Charron-bost, An overview of transience bounds in max-plus algebra, Tropical and Idempotent Mathematics and Applications, pp.277-289
DOI : 10.1090/conm/616/12306
URL : https://hal.archives-ouvertes.fr/hal-00993630

T. Nowak, M. F. Ugger, A. , T. Nowak, M. F. Ugger et al., On the performance of a retransmission-based synchronizer On the performance of a retransmission-based synchronizer, Proceedings of the 18th International Colloquium on Structural Information and Communication Complexity (SIROCCO), pp.234-24525, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00993805