Article

Free access

A decomposition approach to variance reduction

Author:

Barry L. NelsonAuthors Info & Claims

WSC '85: Proceedings of the 17th conference on Winter simulation

Pages 23 - 32

https://doi.org/10.1145/21850.253059

Published: 15 December 1985 Publication History

Abstract

For analyzing stochastic models, simulation trades the tractability problems of analytical techniques for the problem of sampling variability. Variance reduction techniques (VRTs) attack this problem by transforming the simulation experiment in a way that makes it more statistically efficient. Unfortunately, VRTs are infrequently used, even though significant reductions are possible in practical problems. This tutorial introduces some basic concepts of variance reduction, and uses a new taxonomy of VRTs as the basis for an algorithm to select appropriate VRTs for general simulation experiments.

References

[1]

Bratley, P., B.L. Fox and L.E. Schrage (1983), A Guide to Simulation, Springer-Verlag, NY.

Digital Library

[2]

Law, A,M, and W.D. Kelton (1962), Simulation Modeling and Analysis, McGraw-Hill, NY.

Digital Library

[3]

Kleijnen, J.P.C. (1974), Stastical Techniques in Simulation, Vol. I, Marcel Dekker, MY, Chap. 3.

[4]

McGrath, E. and D.C. Irving (1973), "Techniques for Efficient Monte Carlo Simulation, Vol. III, Variance Reduction," ORLC Report SAI-72-509-LJ.

[5]

Nelson, B.L. and B.W. Schmeiser (1985), "Decomposition of Some Well-Known Variance Reduction Techniques," J. Statist, Comput, Simul, forthcoming.

Digital Library

[6]

Wilson, J.R. (1984), "Variance Reduction Techniques for Digital Simulation," Am. J. Math. Mgt. Sci, 4, 3 & 4, 277-312,

[7]

Burt, J.M. and M.B. Garman (1771), "Conditional Monte Carlo: A Simulation Technique for Stochastic Network Analysis," Mgt. SoL, 18, 3, 207-217.

[8]

Burt. J.M. and M.B. German (1971), "Monte Carlo Techniques for Stochastic PERT Network Analysis," INFOR, 9, 3, 248-262.

[9]

German, M.B. (1972), "More on Conditioned Sampling in the Simulation of Stochastic Networks,"Mgt Sci, 19,1, 90-95.

Digital Library

[10]

Grant, F.H. (1983), "A Note on Efficiency of the Antithetic Variate Method for Simulating Stochastic Activity Networks," Mgt, Sci., 29, 3, 381-384.

Digital Library

[11]

Grant, F.H. and J.J. Solberg (1983), "Variance Reduction Techniques in Stochastic Shortest Route Analysis: Application, Procedures and Results," Math. Comput. Simul. XXV, 366-375.

[12]

Fishman, G.S. (1985), "Estimating Network Characteristics in Stochastic Activity Networks," Mgt. Sci, 31, 5, 579-593.

Digital Library

[13]

Fishman, G.S. (1985), "Estimating Critical Path and Arc Probabilities in Stochastic Activity Networks," Naval Res. Log. Quart., 32, 2, 249-261.

[14]

Sigal, C.E., A.A.B. Pritsker and J,J. Solberg (1980), "The Use of Cutsets in Monte Carlo Analysis of Stochastic Networks," Math. Comput Simul, XXI 376-384.

[15]

Sullivan, R., J. Hayya and R. Schual (1982), "Efficiency of the Antithetic Variate Method for Simulating Stochastic Networks," Mgt. Sci, 28, 5, 563-572.

Digital Library

[16]

Bayes, A.J, (1972), "A Minimum Variance Sampling Technique for Simulation Models," J. ACM, 19, 4, 734-741.

Digital Library

[17]

Fishman, G.S. (1983), "Accelerated Accuracy in the Simulation of Markov Chains," Op. Res., 31, 3, 466-487.

[18]

Fishman, G.S.(1983), "Accelerated Convergence in the Simulation of Countably infinite State Markov Chains," Op. Res., 31, 6, 1074-1089.

Digital Library

[19]

Heidelberger, P. (1977), "Variance Reduction Techniques for Simulating Markov Chains," Proceedings of the 1977 Winter Simulation Conference, IEEE, 161-164.

Digital Library

[20]

Ross, S.M. and Z. Schechner (1985), "Using Simulation to Estimate First Passage Distribution," Mgt. Sci., 31, 2, 224-234.

Digital Library

[21]

Ehrhardt, R, (1983), "Variance Reduction Techniques for (s,S)-Type Inventory Simulations," Prooeedings of the Conference on Simulotion in Inventory and Production Control (H.Bekiroglu, ed,) SCS, 12-16.

[22]

Easton, M.C, and C.K. Wong (1980), "Sequential Destruction Method for Monte Carlo Evaluation of System Reliability," IEEE Trans.Rel., R-29, i, 27-32.

[23]

Kumamoto, H, K, Tanaka and K. Inoue (1977), "Efficient Evaluation of System Reliability by Monte Carlo Method," IEEE Trans.Rel., R-26,5,311-315.

[24]

Kumamoto, H., K. Tanaka, X. Inoue and E.J. Henley (1980), "Dagger- Sampling Monte Carlo for System Unavailability," IEEE Trans.Rel., R-29,2, 122-125.

[25]

Kumamoto, H., K. Tanaka, K. Inoue and E.J. Henley (1980), "State- Transition Monte Carlo for Evaluating Large, Repairable Systems," IEEE Trans. Rel.,R-29, 5, 376-380.

[26]

M. Mazumdar (1975), "Importance Sampling in Reliability Estimation," in Reliability and Fault Tree Analysis, SIAM, Philidelphia, 153-163.

[27]

Lavenberg, S.S. and P.D. Welch (1979), "Using Conditional Expectation to Reduce Variance in Discrete Event Simulation," Proceedings of the 1979 Winter Simulation Conference, IEEE, 291-294.

Digital Library

[28]

Lavenberg, S.S., T.L. Moeller and P.D. Welch (1982), "Statistical Results on Control Variables with Application to Queueing Network Simulation" Op. Res., 30,1,182-202.

Digital Library

[29]

Wilson, J,R, and A,A.B. Pritsker (1984), "Variance Reduction in Queueing Simulation Using Generalized Concomitant Variables," J.Statist COmput.Simul, 19, 129-153.

[30]

Wilson, J.R. and A.A.B. Pritsker (1984), "Experimental Evaluation of Variance Reduction Techniques for Queueing Simulation Using Generalized Concomitant Variables," Mgt. Sci, 30, 12, 1459-1472.

Digital Library

[31]

Bratley, Fox and Schrage (1983), pp. 46-57, 273-277.

[32]

Chang, R.C.H. (1981), "The Use of Antithetic Control Variates in Computer Simulations," Proceedings of the 1981 Winter Simulation Conference, IEEE, 313-318.

Digital Library

[33]

Cheng, R.C.H (1982), "The Use of Antithetic Variates in Computer Simulations," J'. Opt. Res. Soc., 33, 229-237.

[34]

Fishman, G.S. and B.D. Huang (1983), "Antithetic Variates Revisited;' Comm, ACM, 26, 964-971.

Digital Library

[35]

Gal, S., R.Y. Rubinstein and A. Ziv (1984), "On the Optimality and Efficiency of Common Random Numbers," Math. Comput. Simul, 26, 6, 502-512.

[36]

Kleijnen (1974), pp. 187-240.

[37]

Law and Kelton (1982), pp. 350-357.

[38]

Mihram, G.A. (1974), "Blocking in Simular Experimental Designs," J.Statist Comput. Simul, 3, 4, 29-32.

[39]

Rubinstein, R.Y., Samorodnitsky, G. and M. Shaked (1985), "Antithetic Variates, Multivariate Dependence and Simulation of Complex Stochastic Systems," Mgt. Sci, 31, 66-77,

Digital Library

[40]

Schruben, L.W. (1979), "Designing Correlation Induction Strategies for Simulation Experiments," in Current Issues in COmputer Simulation (N.R. Adam, A.dogramaci, eds.), Academic Press, NY, 235-256.

[41]

Sohruben, LA0. and B.H. Margolin (1978), "Pseudorandom Number Assignment in Statistically Designed Simulation and Distribution Sampling Experiments," J.ASA, 73, 363, 504-520.

[42]

McKay, M.D., R.J. Beckman and W.J. Conover (1979), "A Comparison of Three Methods tot Selecting Values of Input Variables in the Analysis of Output From a Computer Code," Technpmatrics, 21,2, 239-245.

Digital Library

[43]

Madow, W.G. and L.H. Madow (1944), "On the Theory of Systematic Sampling," Ann.Math.Stat., 14, 1-24.

[44]

Brately,Fox and Scharge (1983), chap.5.

[45]

Cheng, R.C,H, (1983), "Random Samples with Known Sample Statistics: with Application to Variance Reduction," Proceedings of the 1983 Winter Simulation conference, IEEE, 395-400.

Digital Library

[46]

Chang, R.CH. (1985), "Generation of Multivariate Normal Samples with Given Sample Mean and Covariance Matrix," J. Statist, Comput. Simul., 21, 39-49.

[47]

Fishman, G.S. and L.R. Moore (1984), "Sampling From a Discrete Distribution While Preserving Monotonicity," Amer. Statistician, 38, 3, 219-223.

[48]

Franta, W.R. (1975), "A Note on Random Variate Generators and Antithetic Sampling,"INFOR, 13, 1, 112-117.

[49]

Kleijnen, J,P.C. (1974), "A Note on Sampling Two Correlated Variables," Simulation, 22, 2, 45-46.

[50]

Barterly,fox and Scharge (1983) pp. 57-61, 277-280

[51]

Gaver, D.P. and G.S. Shedler (1971), "Control Variable Methods in the Simulation of a Model or a Multiprogrammed Computer System," Naval Res. Log,Quart, 18, 435-450.

[52]

Kleijnen (1974), pp.138-164.

[53]

Law and I{elton (1982), pp. 357-361.

[54]

Lavenberg, S.S. and P.D. Welch (1981), "A Perspective on the Use of Control Variables to Increase the Efficiency of Monte Carlo Simulations," Mgt. Sci, 27, 3, 322-335.

Digital Library

[55]

Nozari, A., S.F. Arnold and C.D. Pegden (1984), "Control Variates for Multipopulation Simulation Experiments," IIE Trans., 16, 159-169.

[56]

Bratley, Fox and Schrage (1983), pp. 66-70, 285.

[57]

Carter, G. and E.J. Ignall (1975), "Virtual Measures: A Variance Reduction Technique for Simulation," Mgt.Sci., 21,6, 607-616.

Digital Library

[58]

Lavenberg, S.S. and P,D. Welch (1979), "Using Conditional Expectation to Reduce Variance in Discrete Event Simulation," Proceedings of the 1979 Winter Simulation Conference, IEEE, 291-294,

Digital Library

[59]

Law and Kelton (i983), pp. 363-366.

[60]

Fox, B.L. and P.W. Glynn (1985), "Discrete-time Conversion for Simulating Semi-Markcv Processes," Tecnical Report, Departement d'informatique et de recherche operationelle, University de Montreal, Montreal, Canada,

[61]

Bratley, Fox and Schrage (1983), pp. 63-66, 282-285.

[62]

Hammersley, J.H. and D.C. Handscomb (1964), Monte Carlo Methods, Chapman and Hall, London.

[63]

Jeruchim, M.C. (1984), "On the Application of Importance Sampling to the Simulation of Digital Satellite and Hultihop Links," IEEE Trans. Comm., 32, 10, 1088-1092.

[64]

Kleijnen (1974), Pp 164-186.

[65]

Kahn, H. (1956), "Use of Different Monte Carlo Sampling Techniques," in Symposium on Monte Carlo Methods (H. Meyer,ed.), Wiley, NY, 146-190.

[66]

McGrath, E. and D.C, Irving (1973), "Techniques for Efficient Monte Carlo Simulation, Vol, III, Variance Reduction," ORLC Report SAI-72-509-LJ.

[67]

Carson, J.S. and A.M. Law (1977), "Conservation Equations and Variance Reduction in Queueing Simulations," Prceedings of the 1977 Winter Simulation Conference, IEEE, 187-187.

Digital Library

[68]

Carson, J.S. and A.M. Law (1977), "Conservation Equations and Variance Reduction in Queueing Simulations," Op. Res., 28, 3, 535-546.

[69]

Law, A.M. (1975), "Efficient Estimators for Simulated Queueing Systems," Mgt. Sci, 22, 1, 30-41.

[70]

Law and Kelton (1982), pp. 361-363.

[71]

Minh, D.L. and R.M. Sorli (1983), "Simulating the GI/G/1 Queue in Heavy Traffic," Op. Res., 31, 5, 966-971.

Digital Library

[72]

Bickel, P.J. and K.A. Doksum (1977), Mathematical Statistics, Basic Ideas and Selected Topicas, Holden-Day, San Francisco.

[73]

Cochran, W.G. (1977), Sampling Techniques, Wiley, NY, pp. 134-135.

[74]

Kleijnen, J.P.C. (1974), pp. 116-121.

[75]

Nelson, B.L. and B.W. Schmeiser (1985), "Decomposition of Some Well-Known Variance Reduction Techniques," J. Statist. Comput. Simul., forthcoming.

Digital Library

[76]

Bratley, Fox and Schrage (1983), pp. 61-63

[77]

Cochran, W.G. (1977), Sampling Techniques, Wiley, NY, Chaps. 5, 5A.

[78]

Kleijnen (1974), pp. 110-133.

[79]

Hopmans, A.C.M. and J.P.C. Kleijnen (1979), "Importance Sampling in Systems Simulation: A Practical Failure?", Math. Comput,. Simul XXI, 2, 209-220.

[80]

Kahn, H. (1956), "Use of Different Monte Carlo Sampling Techniques," in Symposium on Monte Carlo Methods (H. Meyer, ed.), Wiley, NY, 146-190.

[81]

Kioussis, L.C. and D.R. Miller (1983), "An Importance Sampling Scheme for Simulating the Degradation and Failure of Complex Systems During Finite Missions," Proceedings of the 1983 Winter Simulation Conference, IEEE, 631-639.

Digital Library

[82]

HcGrath, E. and D.C. Irving (1973), "Techniques for Efficient Monte Carlo Simulation, Vol. III, Variance Reduction," ORLC Report SAI-72-509-LJ.

[83]

Cochran, W.G., Sampling Techniqe, Wiley, NY, 1977.

[84]

Hammersley, J.M. and D.C. Handscomb, Monte Carlo Methods. Chapman and Hall, London, 1964.

[85]

Nelson, B.L. and B. W. Schmeiser, "A Mathematical-Statistical Framework for Variance Reduction, Part I: Simulation Experiments," Research Memorandum No. 84-4, School of Industrial Engineering, Purdue University, West Lafayette. IN, 1984.

[86]

Nelson, B.L. and B. W. Schmeiser, "A Mathematical-Statistical Framework for Variance Reduction, Part II: Classes of Transformations," Research Memorandum No. 84-5, School of Industrial Engineering, Purdue University, West Lafayette, IN, 1984.

[87]

Bickel, B.J. and K.A. Doksum, Methematical Statistics, Basic Ideas and Selected Topics, Holden-Day, San Francisco, 1977.

[88]

Nelson, B.L. and B.W. Schmeiser, "Decomposition of Some Well- Known Variance Reduction Techniques," J. Statist, Comput. Simul., forthcoming.

Digital Library

[89]

Nelson, B.L., "A Perspective on Variance Reduction in Simulation Experiments," Working Paper Series No. 1986-011, Dept. of Industrial and Systems Engineering, The Ohio State University, Columbus, OH, 1985.

[90]

Bratley, P., B.L, Fox and L.E. Schrage, A Guide to Simulation, Springer-Verlag, NY, 1983.

Digital Library

[91]

Nelson, B.L., "On Control Variate Estimators," Norking Paper" Series No, 1985-006, Dept. of Industrial and Systems Engineering, The Ohio State University, Columbus, OH, 1985.

Cited By

Barton RNakayama MSchruben L(2017)History of improving statistical efficiencyProceedings of the 2017 Winter Simulation Conference10.5555/3242181.3242193(1-23)Online publication date: 3-Dec-2017
https://dl.acm.org/doi/10.5555/3242181.3242193
Barton RNakayama MSchruben L(2017)History of improving statistical efficiency2017 Winter Simulation Conference (WSC)10.1109/WSC.2017.8247788(158-180)Online publication date: Dec-2017
https://doi.org/10.1109/WSC.2017.8247788
Guthke PBárdossy A(2012)Reducing the number of MC runs with antithetic and common random fieldsAdvances in Water Resources10.1016/j.advwatres.2012.03.01443(1-13)Online publication date: Jul-2012
https://doi.org/10.1016/j.advwatres.2012.03.014
Show More Cited By

A decomposition approach to variance reduction
1. Computing methodologies
  1. Modeling and simulation
    1. Model development and analysis
2. Mathematics of computing
  1. Probability and statistics
    1. Probabilistic reasoning algorithms

Recommendations

Variance Reduction Techniques for Estimating Value-at-Risk

This paper describes, analyzes and evaluates an algorithm for estimating portfolio loss probabilities using Monte Carlo simulation.Obtaining accurate estimates of such loss probabilities is essential to calculating value-at-risk, which is a quantile of ...
L2 Model reduction and variance reduction

In this contribution we examine certain variance properties of model reduction. The focus is on L"2 model reduction, but some general results are also presented. These general results can be used to analyze various other model reduction schemes. The ...
Asymptotically optimal allocation of stratified sampling with adaptive variance reduction by strata

To enhance efficiency in Monte Carlo simulations, we develop an adaptive stratified sampling algorithm for allocation of sampling effort within each stratum, in which an adaptive variance reduction technique is applied. Given the number of replications ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

WSC '85: Proceedings of the 17th conference on Winter simulation

December 1985

620 pages

ISBN:0911801073

DOI:10.1145/21850

Editors:
Donald T. Gantz
George Mason Univ., Fairfax, VA
,
Gerard C. Blais
Federal Computer Performance Evaluation and Simulation Center
,
Susan L. Solomon
California State Univ., Sacramento

Copyright © 1985 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

SIGSIM: ACM Special Interest Group on Simulation and Modeling

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 15 December 1985

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Acceptance Rates

Overall Acceptance Rate 3,413 of 5,075 submissions, 67%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

7
Total Citations
View Citations
442
Total Downloads

Downloads (Last 12 months)30
Downloads (Last 6 weeks)6

Reflects downloads up to 26 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Barton RNakayama MSchruben L(2017)History of improving statistical efficiencyProceedings of the 2017 Winter Simulation Conference10.5555/3242181.3242193(1-23)Online publication date: 3-Dec-2017
https://dl.acm.org/doi/10.5555/3242181.3242193
Barton RNakayama MSchruben L(2017)History of improving statistical efficiency2017 Winter Simulation Conference (WSC)10.1109/WSC.2017.8247788(158-180)Online publication date: Dec-2017
https://doi.org/10.1109/WSC.2017.8247788
Guthke PBárdossy A(2012)Reducing the number of MC runs with antithetic and common random fieldsAdvances in Water Resources10.1016/j.advwatres.2012.03.01443(1-13)Online publication date: Jul-2012
https://doi.org/10.1016/j.advwatres.2012.03.014
Frost VLaRue WShanmugan K(2006)Efficient techniques for the simulation of computer communications networksIEEE Journal on Selected Areas in Communications10.1109/49.1927386:1(146-157)Online publication date: 1-Sep-2006
https://dl.acm.org/doi/10.1109/49.192738
Glazier JStolfo S(1993)Dynamic neighborhood bounding for Monte Carlo simulationProceedings of the 25th conference on Winter simulation10.1145/256563.256689(466-473)Online publication date: 1-Dec-1993
https://dl.acm.org/doi/10.1145/256563.256689
Nelson B(1987)Variance reduction for simulation practitionersProceedings of the 19th conference on Winter simulation10.1145/318371.318382(43-51)Online publication date: 1-Dec-1987
https://dl.acm.org/doi/10.1145/318371.318382
Nelson B(1987)A perspective on variance reduction in dynamic simulation experimentsCommunications in Statistics - Simulation and Computation10.1080/0361091870881259716:2(385-426)Online publication date: Jan-1987
https://doi.org/10.1080/03610918708812597

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Table of Contents