Monte Carlo pp 493-586 | Cite as

Designing and Analyzing Sample Paths

  • George S. Fishman
Part of the Springer Series in Operations Research book series (ORFE)


As in Ch. 5, we take as our objective the estimation of
$$\mu = \mu \left( g \right) = \int_X {g\left( {\text{x}} \right){\text{d}}F\left( {\text{x}} \right)} $$
, where F denotes an m-dimensional d.f. on \(X \subseteq {\mathbb{R}^m}\). Consider a Monte Carlo Markov sampling experiment composed of n independent replications, each of which begins in a state drawn from an initializing nonequilibrium distribution π0. After a warm-up interval of k — 1 steps on each replication, sampling continues for t additional steps and one uses the n independent truncated sample paths or realizations, each of length t, to estimate µ. Whereas Ch. 5 concentrates on sample path generating algorithms and a conceptual understanding of convergence to an equilibrium state, this chapter focuses on sampling plan design and statistical inference. With regard to design, the chapter shows how the choices of k, n, π0, and t affect computational and statistical efficiency. With regard to statistical inference, it describes methods for estimating the warm-up interval k that significantly mitigate the influence of the initial states drawn from the nonequilibrium distribution π0. Also, it describes methods for computing asymptotically valid confidence intervals for µ in expression (1) as n → ∞ for fixed t, as t → ∞ for fixed n and as both n → ∞ and t → ∞. Because confidence intervals inevitably depend on variance estimates, we need to impose a moderately stronger restriction on g. Whereas the assumption \(\int_X {{g^2}\left( {\text{x}} \right){\text{d}}F\left( {\text{x}} \right)} < \infty \) in Ch. 5 guarantees a finite variance for a single observation on a sample path, the assumption \(\int_X {{g^4}\left( {\text{x}} \right){\text{d}}F\left( {\text{x}} \right)} < \infty \) is necessary for us to obtain consistent estimators of that variance and of other variances that play essential roles in the derivation of asymptotically valid confidence intervals for µ.


Sample Path Batch Size Percent Confidence Interval Independent Replication Autocovariance Function 
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  1. Aldous, D. (1987). On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing, Prob. in Eng. and Infor. Sci.,1 33–46.Google Scholar
  2. Blomqvist, N. (1967). The covariance function of the M/G/1 queueing system, Skandinavisk Aktuarietidskrift, 50, 157–174.Google Scholar
  3. Brillinger, D.R. (1973). Estimation of the mean of a stationary time series by sampling, J. Appl. Prob., 10, 419–431.CrossRefGoogle Scholar
  4. Brockwell, P.J. and R.A. Davis (1991). Time Series: Theory and Methods, 2nd ed., Springer-Verlag, New York.CrossRefGoogle Scholar
  5. Chien, Chia-Hon (1989). Small sample theory for steady state confidence intervals, Tech. Rep. 37, Department of Operations Research, Stanford University, Stanford, CA.Google Scholar
  6. Crane, M.A. and D.L. Iglehart (1974a). Simulating stable stochastic systems I: general multiserver queues, J. ACM, 21, 103–113.CrossRefGoogle Scholar
  7. Crane, M.A. and D.L. Iglehart (1974b). Simulating stable stochastic systems II: Markov chains, J. ACM, 21, 114–123.CrossRefGoogle Scholar
  8. Damerdji, H. (1994). Strong consistency of the variance estimator in steady-state simulation output analysis, Math. Oper. Res., 19, 494–512.CrossRefGoogle Scholar
  9. Diaconis, P. and B. Sturmfels (1993). Algebraic algorithms for sampling from conditional distributions, Tech. Rep. 430, Department of Statistics, Stanford University, Stanford, CA.Google Scholar
  10. Fishman, G.S. (1973a). Statistical analysis for queueing simulations, Management Science, 20, 363–369.CrossRefGoogle Scholar
  11. Fishman, G.S. (1973b). Concepts and Methods in Discrete Event Simulation, Wiley, New York.Google Scholar
  12. Fishman, G.S. (1974). Estimation in multiserver queueing simulations, Operations Research, 22, 72–78.CrossRefGoogle Scholar
  13. Fishman, G.S. (1978). Principles ofDiscrete Event Simulation, Wiley, New York.Google Scholar
  14. Fishman, G.S. (1994). Choosing sample path length and number of sample paths when starting in the steady state, Oper. Res. Letters, 16, 209–219.CrossRefGoogle Scholar
  15. Fishman, G.S. and V.G. Kulkarni (1992). Improving Monte Carlo efficiency by increasing variance, Man. Sci., 38, 1432–1444.CrossRefGoogle Scholar
  16. Fishman, G.S. and P.J. Kiviat (1967). The analysis of simulation generated time series, Man. Sci., 13, 525–557.CrossRefGoogle Scholar
  17. Fishman, G.S. and L.S. Yarberry (1990). RAPIDS: Routing algorithm performance investigation and design simulation, UNC/OR/TR/90–12, Department of Operations Research, University of North Carolina at Chapel Hill.Google Scholar
  18. Fishman, G.S. and L.S. Yarberry (1994). An implementation of the batch means method, UNC/OR/TR/93–1, Department of Operations Research, University of North Carolina at Chapel Hill.Google Scholar
  19. Fox, B.L., D. Goldsman and J.J. Swain (1991). Spaced batch means, Oper. Res. Letters, 10, 255–266.CrossRefGoogle Scholar
  20. Gelman, A. and D.B. Rubin (1992). Inference from iterative simulation using multiple sequences, Statistical Sciences, 7, 457–511.CrossRefGoogle Scholar
  21. Glynn, P. (1987) Limit theorems for the method of replication, Stochastic Models, 3, 343–355.CrossRefGoogle Scholar
  22. Glynn, P. and D. Iglehart (1988). A new class of strongly consistent variance estimators for steady-state simulations, Stochastic Processes and Their Applications, 28, 71–80.CrossRefGoogle Scholar
  23. Glynn, P. and D. Iglehart (1990). Simulation output analysis using standardized time series, Math. Opns. Res., 15, 1–16.CrossRefGoogle Scholar
  24. Gross, D. and C. Harris (1985). Fundamentals of Queueing Theory, 2nd ed., Wiley, New York.Google Scholar
  25. Iosifescu, M. (1968). La loi du logarithme itéré pour une classe de variables aléatoires dépendantes, Teorija Veroj, 13, 315–325.Google Scholar
  26. Iosifescu, M. (1970). Addendum to La du logarithme itéré pour une classe de variables aléatoires dépendantes, Teorija Veroj, 15, 170–171.Google Scholar
  27. Johnson, N.L. and S. Kotz (1970). Continuous Univariate Distributions, Houghton Mifflin. Johnson, N.L. and B.L. Welch (1939). On the calculation of the cumulants of the x-distribution, Biometrika, 31, 216–218.Google Scholar
  28. Komlbs, J., P. Major and G. Tusnâdy (1975). An approximation of partial sums of independent r.v.’s and the sample d.f. I, Z. Wahrsch. Verw. Geb., 32, 111–131.CrossRefGoogle Scholar
  29. Komlôs, J., P. Major and G. Tusnâdy (1976). An approximation of partial sums of independent r.v.’s and the sample d.f. II, Z. Wahrsch. Verw. Geb., 34, 33–58.CrossRefGoogle Scholar
  30. Major, P. (1976). The approximation of partial sums in independent r.v.’s, Z. Wahrsch. Verw. Geb., 35, 213–220.CrossRefGoogle Scholar
  31. Meketon, M.S. and P. Heidelberger (1982). A renewal theoretic approach to bias reduction in regenerative simulation, Man. Sci., 28, 173–181.CrossRefGoogle Scholar
  32. Meketon, M.S. and B.W. Schmeiser (1984). Overlapping batch means: something for nothing? Proc. Winter Sim. Conf., 227–230.Google Scholar
  33. Mykland, P. L. Tierney and B. Yu (1992). Regeneration in Markov chain samplers, Tech. Rep. 585, School of Statistics, University of Minnesota.Google Scholar
  34. Nummelin, E. (1984). General Irreducible Markov Chains and Non-negative Operators, Cambridge University Press, Cambridge, England.Google Scholar
  35. Parzen, E. (1962). Stochastic Processes,Holden Day.Google Scholar
  36. Peskun, P.H. (1973). Optimum Monte-Carlo sampling using Markov chains, Biometrika, 60, 607–612.CrossRefGoogle Scholar
  37. Philipp, W. (1969). The law of the iterated logarithm for mixing stochastic processes, Ann. Math. Statist., 40, 1985–1991.CrossRefGoogle Scholar
  38. Philipp, W. and W. Stout (1975). Almost sure invariance principle for partial sums of weakly dependent random variables, Memoirs of the American Mathematical Society, 161.Google Scholar
  39. Reznik, M.Kh. (1968). The law of the iterated logarithm for some classes of stationary processes, Theory, Probability Appl., 8, 606–621.CrossRefGoogle Scholar
  40. Schmeiser, B.W. (1982). Batch size effects in the analysis of simulation output, Oper. Res., 30, 556–568.CrossRefGoogle Scholar
  41. Schmeiser, B.W. and W.T. Song (1987). Correlation among estimators of the variance of the sample mean, Proc. Winter Sim. Conf., 309–317.Google Scholar
  42. von Neumann, J. (1941). Distribution of the ratio of the mean square successive difference and the variance, Ann. Math. Stat., 12, 367–395.CrossRefGoogle Scholar
  43. Yaglom, A. (1962). An Introduction to the Theory of Stationary Random Functions, translated by R.A. Silverman, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
  44. Yarberry, L.S. (1993). Incorporating a dynamic batch size selection mechanism in a fixedsample-size batch means procedure, unpublished Ph.D. thesis, Dept. of Operations Research, University of North Carolina, Chapel Hill.Google Scholar
  45. Young, L.C. (1941). Randomness in ordered sequences, Ann. Math. Statist, 12, 293–300.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • George S. Fishman
    • 1
  1. 1.Department of Operations ResearchUniversity of North CarolinaChapel HillUSA

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