Advertisement

Discrete Event Dynamic Systems

, Volume 1, Issue 1, pp 7–35 | Cite as

Algebraic structure of some stochastic discrete event systems, with applications

  • Paul Glasserman
  • David D. Yao
Article

Abstract

Generalized semi-Markov processes (GSMPs) and stochastic Petri nets (SPNs) are generally regarded as performance models (as opposed to logical models) of discrete event systems. Here we take the view that GSMPs and SPNS are essentially automata (generators) driven by input sequences that determine the timing of events. This view combines the deterministic, logical aspects and the stochastic, timed aspects of the two models. We focus on two conditions, (M) and (CX) (which we previously developed to study monotonicity and convexity properties of GSMPs), and the antimatroid and lattice structure they imply for the language generated by a GSMP or SPN. We illustrate applications of these structural properties in the areas of derivative estimation, simulation variance reduction, parallel simulation, and optimal control.

Key Words

antimatroid generalized semi-Markov processes infinitesimal perburtation analysis optimal control stochastic Petri nets 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. F. Baccelli, “Ergodic theory of stochastic decision free Petri nets,” inProc. 28th IEEE Conf. Decision and Control, 1989, pp. 1521—1527. (Extended version to appear inAnn. Probab.)Google Scholar
  2. F. Baccelli and A. M. Makowski, “Queueing models for systems with synchronization constraints,”Proc. IEEE, Vol. 77, pp. 138–161, 1989.Google Scholar
  3. A. Björner “On matroids, groups, and exchange languages,” inMatroid Theory and Its Applications (eds. L. Lovász and A. Rechksi), Colloq. Math. Soc. J. Bolyai 40, North-Holland: Amsterdam, 1985; pp. 25–60.Google Scholar
  4. P. Bratley, B. L. Fox, and L. E. Schrage,A Guide to Simulation, Springer-Verlag: New York, 1983.Google Scholar
  5. C. G. Cassandras and S. G. Strickland, “Sample path properties of timed discrete event systems,”Proc. IEEE, Vol. 77, pp. 59–71, 1989.Google Scholar
  6. K. M. Chong, “Some extensions of a theorem of Hardy, Littlewood and Polya and their applications,”Canad. J. Math., Vol. 26, pp. 1321–1340, 1974.Google Scholar
  7. G. Cohen, P. Moller, J.-P., Quadrat, and M. Viot, “Algebraic tools for the performance analysis of discrete event systems,Proc. IEEE, Vol. 77, pp. 39–58, 1989.Google Scholar
  8. S. Crespi-Reghizzi, “Petri nets and Szilard languages,”Inform. Control, Vol. 33, pp. 177–192, 1977.Google Scholar
  9. S. Crespi-Reghizzi and D. Mandrioli, “A decidability theorem for a class of vector-addition systems,”Inform. Process. Lett., Vol. 3, pp. 78–80, 1975.Google Scholar
  10. B. L. Dietrich, “Matroids and antimatroids—a survey,”Discrete Math., Vol. 78, pp. 223–237, 1989.Google Scholar
  11. P. Glasserman, “Structural conditions for perturbation analysis derivative estimation: Finite-time performance indices, 1989,Operations Research, Forthcoming.Google Scholar
  12. P. Glasserman and D. D. Yao, “Monotonicity in generalized semi-Markov processes,” 1989,Math. of Oper. Res., forthcoming.Google Scholar
  13. P. Glasserman and D. D. Yao, “Generalized semi-Markov processes: Antimatroid structure and second-order properties, 1990,Math. Oper. Res., forthcoming.Google Scholar
  14. P. W. Glynn, “A GSMP formalism for discrete event systems,”Proc. IEEE, Vol. 77, pp. 14–23, 1989.Google Scholar
  15. A. G. Greenberg, G. D. Lubachevsky, and I. Mitrani, “Unboundedly parallel simulations via recurrence relations,”Sigmetrics '90, Boulder, CO, 1990.ACM Trans. Comput. Syst., forthcoming.Google Scholar
  16. P. J. Haas and G. S. Schedler, “Modeling power of stochastic Petri nets,”Probab. Eng. Inform. Sci., Vol. 2, pp. 435–459, 1988.Google Scholar
  17. Y. C. Ho, “Performance evaluation and perturbation analysis of discrete event dynamic systems: Perspectives and open problems,”IEEE Trans. Automat. Control, Vol. AC-32, pp. 563–572, 1987.Google Scholar
  18. J. Q. Hu, “Convexity of sample path performances and strong consistency of infinitesimal perturbation analysis estimates,”IEEE Trans. Automat. Control, July, 1991, forthcoming.Google Scholar
  19. M. R. Karp and R. E. Miller, “Parallel program schemata,”J. Comput. Syst. Sci., Vol. 3, pp. 147–195; 1969.Google Scholar
  20. R. E. Ladner and M. J. Fischer, “Parallel prefix computation,”J. ACM, Vol. 27, pp. 831–838, 1980.Google Scholar
  21. F. Lin and D. D. Yao, “Generalized semi-Markov processes: A view through supervisory control,Proc. 28th IEEE Conf. Decision and Control, 1989, pp. 1075–1076.Google Scholar
  22. G. G. Lorentz, “An inequality for rearrangements,”Amer. Math. Monthly, Vol. 60, pp. 176–179, 1953.Google Scholar
  23. R. Parikh, “On context-free languages,”J. ACM, Vol. 13, pp. 570–581, 1966.Google Scholar
  24. J. L. Peterson, “Computation sequence sets,”J. Comput. Syst. Sci., Vol. 13, pp. 1–24, 1976.Google Scholar
  25. J. L. Peterson,Petri Net Theory and the Modeling of Systems, Prentice-Hall: Englewood Cliffs, NJ, 1981.Google Scholar
  26. P. J. Ramadge and W. M. Wonham, “Supervisory control of a class of discrete-event processes,”SIAM J. Control Optim., Vol. 25, pp. 206–230, 1987.Google Scholar
  27. C. V. Ramamoorthy and G. S. Ho, “Performance evaluation of asynchronous concurrent systems using Petri nets,”IEEE Trans. Software Eng., Vol. SE-6, pp. 440–449, 1980.Google Scholar
  28. S. M. Ross,Stochastic Processes, Wiley: New York, 1983.Google Scholar
  29. H. L. Royden,Real Analysis, Macmillan: New York, 1968.Google Scholar
  30. L. Rüschendorf, “Solution of a statistical optimization problem by rearrangement methods,”Metrika, Vol. 30, pp. 55–61, 1983.Google Scholar
  31. R. Schassberger, “On the equilibrium distribution of a class of finite-state generalized semi-Markov processes,”Math. Oper. Res., Vol. 1, pp. 395–406, 1976.Google Scholar
  32. R. Schassberger, “Insensitivity of steady-state distributions of generalized semi-Markov processes,”Ann. Probab., Vol. 5, pp. 87–99, 1978.Google Scholar
  33. J. G. Shanthikumar and D. D. Yao, “Second-order stochastic properties in queueing systems,Proc. IEEE, Vol. 77, pp. 162–170.Google Scholar
  34. P. W. Shor, A. Björner, and L. Lovász, “Chip-firing games on graphs,”Euro. J. Combin., 1988, forthcoming.Google Scholar
  35. R. Suri, “Perturbation analysis: The state of the art and research issues explained via the GI/G/1 queue,”Proc. IEEE, Vol. 77, pp. 114–137.Google Scholar
  36. P. Tsoucas and J. Walrand, “Monotonicity of throughput in non-Markovian networks,”J. Appl. Probab., Vol. 26, pp. 134–141.Google Scholar
  37. W. Whitt, “Continuity of generalized semi-Markov processes,”Math. Oper. Res., Vol. 5, pp. 494–501.Google Scholar

Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • Paul Glasserman
    • 1
  • David D. Yao
    • 2
  1. 1.Graduate School of BusinessColumbia UniversityNew York
  2. 2.IE/OR DepartmentColumbia UniversityNew York

Personalised recommendations