Abstract
This paper deals with the asymptotic behavior of the stochastic dynamics of discrete event systems. In this paper we focus on a wide class of models arising in several fields and particularly in computer science. This class of models may be characterized by stochastic recurrence equations in ℝK of the form T(n+1) = φ n+1(T(n)) where φ n is a random operator monotone and 1—linear. We establish that the behaviour of the extremas of the process T(n) are linear. The results are an application of the sub-additive ergodic theorem of Kingman. We also give some stability properties of such sequences and a simple method of estimating the limit points.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baccelli, B. Ergodic Theory of Stochastic Petri networks. The Annals of Probability, 20(1):375–396, 1992.
Baccelli, B. and Liu, Z. On a class of stochastic evolution equations. The Annals of Probability, 21(1):350–374, 1992.
Baccelli, F., Cohen, G., Olsder, G. and Quadrat, J.-P. Synchronization and Linearity. John Wiley and Sons, 1992.
Baccelli, F. and Foss, S. On the saturation rule for the stability of queues. J. Appl. Prob., 32(2):494–507, 1995.
Baccelli, F. and Mairesse, J. Ergodic Theory of Stochastic Operators and Discrete Event Networks. Technical Report RR-2641, INRIA, 1995. To appear in Idempotency, Publications of the Isaac Newton Institute, Cambridge University Press, in 1996.
Borovkov, A.A. Stochastic processes in queuing theory. Springer-Verlag, 1976.
Brams, G.W. Réseaux de Pétri: théorie et pratique. Masson, 1983.
Doob, J.L. Stochastic Processes. John Wiley & Sons, New-York, 1950.
Durrett, R. Probability: Theory and Examples. Wadsworth & Brooks/Cole, California, 1991.
Furstenberg, N. and Kesten, H. Products of Random Matrices. Ann. Math. Statist., 31:457–469, 1972.
Glasserman, P. and Yao, D. Stochastic Vector Difference Equations with Stationary Coefficients. Journal of Applied Probability, 32(4):851–866, 1995.
Glasserman, P. and Yao, D. Subadditivity and Stability of a Class of Discrete-Event Systems. IEEE Trans. on Automatic Control, 40(9):1514–1527, September 1995.
Ho, Y-C., editor. Special Issue on Dynamics of Discrete Event Systems, volume 77. Proceedings of the IEEE, January 1989.
Kesten, H. Percolation Theory and first-passage percolation. The Annals of Probability, 15(4):1231–1271, 1987.
Kingman, J.F.C. The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser., 30(3):499–510, 1968.
Kingman, J.F.C. Subadditive ergodic theory. The annals of probability, 1(6):883–909, 1973.
Lazowska, E.D., Zahorjan, J., Graham, G.S. and Sevcik, K.S. Quantitative System Performance: Computer System Analysis Using Queuing Networks Theory. Prentice-Hall, 1984.
Liggett, T.M. An improved subadditive Ergodic theorem. The Annals of Probability, 13(4):1279–1285, 1985.
Loynes, R.M. The stability of queues with non independent inter-arrival and service times. Proc. Cambridge Ph. Soc., 58:497–520, 1962.
Madala, S. and Sinclair, J.B. Performance of Synchronous Parallel Algorithms with Regular Structures. IEEE Trans. on Parallel and Distributed Systems, 2(1):105–116, 1991.
Plateau, B and Atif, K. Stochastic Automata Networks for Modelling Parallel Systems. IEEE Transactions on Software Engineering, 17(10):1093–1108, October 1991.
Schürger, Klaus. Almost subadditive extensions of Kingman's theorem. The Annals of Probability, 19(4):1575–1586, 1991.
Steele, J. Michael. Kingman's subadditive ergodic theorem. Ann. Inst. Poincaré, 25(1):93–98, 1989.
Vincent, J-M. Stability condition of a service system with precedence constraints between tasks. Performance Evaluation, 12(1):61–66, 1991.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Vincent, JM. Some Ergodic Results on Stochastic Iterative Discrete Events Systems. Discrete Event Dynamic Systems 7, 209–232 (1997). https://doi.org/10.1023/A:1008276017457
Issue Date:
DOI: https://doi.org/10.1023/A:1008276017457