Abstract
The notions introduced in Braker and Resing (1992), concerning periodicity of 2×2 matrices in a generalized setup, are extended to the case of general square matrices. The asymptotic behaviour of a series of matrices is related to a particular type of circuit, called a generalized critical circuit, which is a circuit that is critical with respect to all the nodes it contains. It is shown that the asymptotic behaviour is determined only by the lengths of some generalized critical circuits, the weights of arcs in these circuits and some critical paths. This formulation is an appealing extension of the concepts of periodicity and critical circuit in the ‘usual’ max algebra. The term ‘asymptotic’ may give the impression that the described behaviour is only reached after a long or even infinite transient. However, the periodic regular part may be reached in only a few steps, after which the generalized critical circuits determine the future behaviour. As a corollary we obtain that each aperiodic graph contains at least one generalized critical circuit.
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References
F. Baccelli, G. Cohen, G. J. Olsder, and J. P. Quadrat.Synchronization and Linearity: An Algebra for Discrete Event Systems. Wiley, New York, 1992.
J. G. Braker and J. A. C. Resing. On a generalized asymptoticity problem in max algebra. In S. Balemi, P. Kozak, and R. Smedinga, editors,Discrete Event Systems: Modeling and Control, pp. 125–139. Birkhäuser, 1992.
R. A. Brualdi and H. J. Ryser,Combinatorial Matrix Theory. Cambridge University Press, Cambridge, 1991.
G. Cohen, D. Dubois, J. P. Quadrat, and M. Viot. Analyse du comportement périodique de systèmes de production par la théorie des dioïdes. Technical Report INRIA, Le Chesnay, France, no. 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 Ac 30:210–220, 1985.
K. L. Cooke and E. Halsey: The shortest route through a network with time-dependent internodal transit times.J. Math. Anal. and Appl., 14:493–498, 1966.
R. A. Cuninghame-Green.Minimax Algebra: Springer-Verlag, New York, 1979.
E. V. Denardo:Dynamic Programming, Models and Applications. Prentice-Hall, New Jersey, 1982.
S. E. Dreyfus. An appraisal of some shortest path algorithms.Operations Research, 17:395–412, 1969.
M. Gondran and M. Minoux:Graphs and Algorithms. Wiley, New York, 1984.
R. A. Howard.Dynamic Programming and Markov Processes. MIT Press, Cambridge, MA, 1960.
V. Maslov and S. Samborskiį, editors.idempotent Analysis, Vol. 13 ofAdv. in Sov. Math. AMS, Rhode Island, 1992.
M. Minoux. Structures algébriques généralisées des problèmes de cheminement dans les graphes.RAIRO Recherche Opérationnelle, 10:33–62, 1976.
J. A. C. Resing.Asymptotic Results in Feedback Systems. Ph.D. thesis, Delft University of Technology, 1990.
U. Zimmermann.Linear and Combinatorial Optimization in Ordered Algebraic Structures, Annals of Discrete Mathematics, 10. North-Holland, Amsterdam, 1981.
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Braker, J.G., Resing, J.A.C. Periodicity and critical circuits in a generalized max-algebra setting. Discrete Event Dyn Syst 4, 293–314 (1994). https://doi.org/10.1007/BF01438711
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DOI: https://doi.org/10.1007/BF01438711