Abstract
This paper addresses the performance evaluation and marking optimization of stochastic timed event graphs. The transition firing times are generated by random variables with general distribution. The marking optimization problem consists in obtaining a given cycle time while minimizing a p-invariant criterion (or S-invariant). Some important properties have been established. In particular, the average cycle time is shown to be non-increasing with respect to the initial marking while the p-invariant criterion is non-decreasing. We further prove that the criterion value of the optimal solutions is non-increasing in transition firing times in stochastic ordcring's sense. Lower bounds and upper bounds of the average cycle time are proposed. Based on these bounds, we show that the p-invariant oriterion reaches its minimum when the firing times become deterministic. A sufficient condition under which an optimal solution for the deterministic case remains optimal for the stochastic case is given. These new bounds are also used to provide simple proof of the reachability conditions of a given cycle time. Thanks to the tightness of the bounds for the normal distribution case, two algorithms which leads to near-optimal solutions have been proposed to solve the stochastic marking optimization problem with normal distributed firing times.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
F. BACCELLI, “Ergodic Theory of Stochastic Petri Networks”, Annals of Probability, Vol. 20, No. 1, pp. 350–374, 1992.
F. BACCELLI and Z. LIU, “Comparison Properties of Stochastic Decision Free Petri Nets”, to appear in IEEE Transactions on Automatic Control, 1992.
J. CAMPOS, G. CHIOLA and M. SILVA, “Properties and Performance Bounds for Closed Free Choice Synchronized Monoclass Queueing Networks”, IEEE Transactions on Automatic Control, Vol. 36, No. 12, pp. 1368–1382, December 1991.
P. CHRETIENNE, “Les réseaux de Petri temporisés”, Université Paris VI, Paris, France, Thése d'Etat, 1983.
F. COMMONER, A. HOLT, S. EVEN and A. PNUELI, “Marked Directed Graphs”, Journal of Computer and System Science, Vol. 5, No. 5, pp. 511–523, 1971.
P.J. HAAS and G.S. SHEDLER, “Stochastic Petri Nets: Modeling Power and Limit Theorem”, Probability in Engineering and Informational Sciences, Vol. 5, pp. 477–498, 1991.
H.P. HILLION and J.M. PROTH, “Performance Evaluation of Job-Shop Systems Using Timed Event-Graphs”, IEEE Transactions on Automatic Control, Vol. 34, No. 1, pp. 3–9, January 1989.
S. LAFTIT, J.M. PROTH and X.L. XIE, “Marking Optimization in Timed Event Graphs”, Proc. of 12th Int. Conf. on Applications and Theory of Petri Nets, Denmark, June 1991.
S. LAFTIT, J.M. PROTH and X.L. XIE, “Optimization of Invariant Criteria for Event Graphs”, IEEE Transactions on Automatic Control, Vol. 37, No. 5, pp. 547–555, May 1992.
T. MURATA, “Petri Nets: Properties, Analysis and Applications,” Proceedings of the IEEE, vol. 77, N∘ 4, pp. 541–580, April 1989.
J.M. PROTH, N. SAUER and X.L. XIE, “Stochastic Timed Event Graphs: Bounds, Cycle Time Reachability and Marking Optimization”, to appear in the Proc. of the 12th IFAC World Congress, 1993.
J.M. PROTH and X.L. XIE, “Performance Evaluation and Optimization of Stochastic Timed Event Graphs”, submitted for publication in IEEE Transactions on Automatic Control, June 1992.
C.V. RAMAMOORTHY and G.S. HO, “Performance Evaluation of Asynchronous Concurrent Systems using Petri Nets”, IEEE Trans. Software Eng., Vol. SE-6, No. 5, pp. 440–449, 1980.
J.A.C. RESING, R.E. de VRIES, G. HOOGHIEMSTRA, M.S. KEANE and G.J. OLSDER, “Asymptotic Behaviour of Random Discrete Event Systems”, Stochastic Processes and their Applications, Vol. 36, pp. 195–216, 1990.
M. SILVA, “Petri Nets and Flexible Manufacturing,” in Advances in Petri Nets 1989, G. Rozenberg (ed.), Lecture Notes of Computer Science, Springer Verlag, pp. 374–417, 1989.
D. STOYAN, “Comparison Methods for Queues and Other Stochastic Models”, English translation (D.j. Daley editor), J. Wiley and Sons, New York, 1984.
X.L. XIE, “Superposition Properties and Performance Bounds of Stochastic Timed Event Graphs”, Submitted to IEEE Trans. on Automatic Control, 1992
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sauer, N., Xie, X. (1993). Marking optimization of stochastic timed event graphs. In: Ajmone Marsan, M. (eds) Application and Theory of Petri Nets 1993. ICATPN 1993. Lecture Notes in Computer Science, vol 691. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56863-8_56
Download citation
DOI: https://doi.org/10.1007/3-540-56863-8_56
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56863-6
Online ISBN: 978-3-540-47759-4
eBook Packages: Springer Book Archive