Advertisement

Discrete Event Dynamic Systems

, Volume 6, Issue 2, pp 159–180 | Cite as

Variance reduction algorithms for parallel replicated simulation of uniformized Markov chains

  • Simon Streltsov
  • Pirooz Vakili
Article

Abstract

We discuss the simulation ofM replications of a uniformizable Markov chain simultaneously and in parallel (the so-called parallel replicated approach). Distributed implementation on a number of processors and parallel SIMD implementation on massively parallel computers are described. We investigate various ways of inducing correlation across replications in order to reduce the variance of estimators obtained from theM replications. In particular, we consider the adaptation of Stratified Sampling, Latin Hypercube Sampling, and Rotation Sampling to our setting. These algorithms can be used in conjunction with the Standard Clock simulation of uniformized chains at distinct parameter values and can potentially sharpen multiple comparisons of systems in that setting. Our investigation is primarily motivated by this consideration.

Keywords

Markov Chain System Theory Parallel Computer Discrete Geometry Stratify Sampling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Avramidis, A. 1992. Variance reduction for quantile estimation via correlation induction.Proceedings of the Winter Simulation Conference, 572–576.Google Scholar
  2. Barnhart, C. M., Wieselthier, J. E., and Ephermides, A. 1993. Improvement in simulation efficiency by means of the standard clock: A quantitative study.Proceedings of the IEEE Conference on Decision and Control, 2217–2223.Google Scholar
  3. Bratley, P. B., Fox, B. L., and Schrage, L. E. 1987.A Guide to Simulation. 2d ed., New York: Springer-Verlag.Google Scholar
  4. Cassandras, C. G., and Strickland, S. G. 1989. On-line sensitivity analysis of markov chains.IEEE Trans. on Automatic Control 34: 76–86.Google Scholar
  5. Cassandras, C. G., and Strickland, S. G. 1989. Observable augmented systems for sensitivity analysis of Markov and semi-Markov processes.IEEE Trans. on Automatic Control 34: 1026–1037.Google Scholar
  6. Cassandras, C. G. 1994. Sample-path-based continuous and discrete optimization of discrete event systems: From gradient estimation to “Rapid Learning.”Proceedings of the 11th International Conference on Analysis and Optimization of Systems.Google Scholar
  7. Chen, C. H., and Ho, Y. C. 1995. An approximation approach of the standard clock method for general discrete event simulation.IEEE Trans. on Control Systems Technology 3(3): 309.Google Scholar
  8. Cheng, R. C. H., Davenport, T. 1989. The problem of dimensionality in stratified sampling.Management Science 35: 1278–1296.Google Scholar
  9. Fishman, G. S. 1983a. Accelerated accuracy in the simulation of Markov chains.Oper. Res. 31: 466–487.Google Scholar
  10. Fishman, G. S. 1983b. Accelerated convergence in the simulation of countably infinite state Markov chains.Oper. Res. 31: 1074–1089.Google Scholar
  11. Fishman, G. S., Huang, B. D. 1983. Antithetic variates revisited.Communications of the ACM 26: 964–971.Google Scholar
  12. Fox, B. L. 1990. Generating Markov-chain transitions quickly: I.ORSA Journal on Computing 2: 126–135.Google Scholar
  13. Fox, B. L., Glynn, P. W. 1990. Discrete time conversion for simulating finite-horizon Markov processes.SIAM Journal on Appl. Math. 50: 1457–1473.Google Scholar
  14. Fujimoto, R. M. 1990. Parallel discrete event simulation.Communication of ACM 33: 30–53.Google Scholar
  15. Glasserman, P., Vakili, P. 1994. Comparing uniformized Markov chains simulated in parallel.Probability in the Engineering and Informational Sciences 8: 309–326.Google Scholar
  16. Glynn, P. W., Heidelberger, P. 1991. Analysis of parallel replicated simulations under a completion time constraint.ACM Transactions on Modeling and Computer Simulation 1: 3–23.Google Scholar
  17. Glynn, P. W., Heidelberger, P. 1992. Analysis of initial transient deletion for parallel steady-state simulations.SIAM Journal on Scientific and Statistical Computing 13: 904–922.Google Scholar
  18. Greenberg, A. G., Lubachevsky, B. D., and Mitrani, I. 1991. Algorithms for unboundedly parallel simulations.ACM Trans. Computer Systems 9: 201–221.Google Scholar
  19. Greenberg, A. G., Schlunk, O., and Whitt, W. 1993. Using distributed-event parallel simulation to study departures from many queues in series.Probability in the Engineering and Informational Sciences 7: 159–186.Google Scholar
  20. Heidelberger, P. 1988. Discrete event simulations and parallel processing: Statistical properties.SIAM J. Sci. Stat. Comput. 9: 1114–1132.Google Scholar
  21. Heidelberger, P., Nicol, D. M. 1991. Simultaneous parallel simulations of continuous time Markov chains at multiple parameter settings.Proceedings of the Winter Simulation Conference, 602–607.Google Scholar
  22. Heidelberger, P., Nicol, D. M., 1993. Conservative parallel simulation of continuous time Markov chains using uniformization.IEEE Transactions on Parallel and Distributed Systems 4: 906–921.Google Scholar
  23. Ho, Y. C., Sreenivas, R., and Vakili, P. 1992. Ordinal optimization of discrete event dynamic systems.Journal of Discrete Event Dynamic Systems, 2(1): 61–88.Google Scholar
  24. Hu, J. Q. 1995. Parallel simulation of DEDS via event synchronization.Journal of Discrete Event Dynamic Systems 5(2/3): 167–186.Google Scholar
  25. Lehmann, E. L. 1966. Some concepts of dependence.Annals of Mathematical Statistics 37: 1137–1153.Google Scholar
  26. McKay M. D., Conover, W. J., Beckman, R. J. 1979. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code.Technometrics 21: 239–245.Google Scholar
  27. Nicol, D. M., Heidelberger, P. 1993. Parallel simulation of Markovian queueing networks using adaptive uniformization.Performance Evaluation Review 21: 122-.Google Scholar
  28. Stein, M. 1987. Large sample properties of simulation using latin hypercube sampling.Technometrics 29: 143–151.Google Scholar
  29. Streltsov, S., Vakili, P. 1993. Parallel replicated simulation of Markov chains: implementation and variance reduction.Proceedings of the Winter Simulation Conference, 430–436.Google Scholar
  30. Strickland, S. G., and Phelan, R. G. 1995. Massively parallel SIMD simulation of Markovian DEDS: Events vs. time synchronous methods.Journal of Discrete Event Dynamic Systems 5(2/3): 141–166.Google Scholar
  31. Vakili, P. 1992. Massively parallel and distributed simulation of a class of discrete event systems: a different perspective.TOMACS 2: 214–238.Google Scholar
  32. Wieslethier, J. E., Barnhart, C. M., and Ephremides, A. 1995. Standard clock simulation and ordinal optimization applied to admission control in integrated communication networks.Journal of Discrete Event Dynamic Systems 5(2/3): 243–280.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Simon Streltsov
    • 1
  • Pirooz Vakili
    • 1
  1. 1.Manufacturing Engineering DeptartmentBoston UniversityBoston

Personalised recommendations