Optimization via simulation: A review. (English) Zbl 0833.90089
Summary: We review techniques for optimizing stochastic discrete-event systems via simulation. We discuss both the discrete parameter case and the continuous parameter case, but concentrate on the latter which has dominated most of the recent research in the area. For the distance parameter case, we focus on the techniques for optimization from a finite set: multiple-comparison procedures and ranking-and-selection procedures. For the continuous parameter case, we focus on gradient-based methods, including perturbation analysis, the likelihood ratio method, and frequency domain experimentation. For illustrative purposes, we compare and contrast the implementation of the techniques for some simple discrete-event systems such as the \((s, S)\) inventory system and the \(GI/G/1\) queue.
Finally, we speculate on future directions for the field, particularly in the context of the rapid advances being made in parallel computing.
Finally, we speculate on future directions for the field, particularly in the context of the rapid advances being made in parallel computing.
MSC:
| 90C15 | Stochastic programming |
| 90C31 | Sensitivity, stability, parametric optimization |
| 62J15 | Paired and multiple comparisons; multiple testing |
| 62F07 | Statistical ranking and selection procedures |
| 90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |
Keywords:
response surface methodology; stochastic discrete-event systems; simulation; discrete parameter case; continuous parameter case; multiple- comparison procedures; gradient-based methods; perturbation analysis; parallel computingReferences:
| [1] | V.M. Aleksandrov, V.I. Sysoyev and V.V. Shemeneva, Stochastic optimization, Eng. Cybern. 5(1968)11–16. |
| [2] | A.E. Albert and L.A. Gardner, Jr.,Stochastic Approximation and Nonlinear Regression, Research Monograph No. 42 (M.I.T. Press, Cambridge, MA, 1967). · Zbl 0162.21502 |
| [3] | S. Andradóttir, A new algorithm for stochastic approximation,Proc. 1990 Winter Simulation Conf. (1990) pp. 364–366. |
| [4] | F. Azadivar and J.J. Talavage, Optimization of stochastic simulation models, Math. Comp. Simul. 22(1980)231–241. · Zbl 0452.90053 · doi:10.1016/0378-4754(80)90050-6 |
| [5] | J. Banks and J.S. Carson,Discrete-Event System Simulation (Prentice-Hall, Englewood Cliffs, N.J., 1984). · Zbl 0525.68071 |
| [6] | R.E. Bechhofer, A single-sample multiple decision procedure for ranking means of normal populations with known variances, Ann. Math. Statist. 25(1954)16–39. · Zbl 0055.13003 · doi:10.1214/aoms/1177728845 |
| [7] | A. Benveniste, M. Metivier and P. Priouret,Adaptive Algorithms and Stochastic Approximation (Springer, New York, 1990). · Zbl 0639.93002 |
| [8] | W.E. Biles and J.J. Swain,Optimization and Industrial Experimentation (Wiley-Interscience, New York, 1980). |
| [9] | R.E. Bixby, J.W. Gregory, I.J. Lustig, R.E. Marsten and D.F. Shanno, Very large-scale programming: a case study in combining interior point and simplex methods, Oper. Res. 25(1992)885–897. · Zbl 0758.90056 · doi:10.1287/opre.40.5.885 |
| [10] | G.E.P. Box and N.R. Draper,Empirical Model-Building and Response Surfaces (Wiley, New York, 1987). · Zbl 0614.62104 |
| [11] | P. Brémaud and F. Vázquez-Abad, On the pathwise computation of derivatives with respect to the rate of a point process, Queueing Syst. 10(1993)249–270. · Zbl 0747.60046 · doi:10.1007/BF01159209 |
| [12] | M. Caramanis and G. Liberopoulos, Perturbation analysis for the design of flexible manufacturing flow controllers, Oper. Res. 40(1992)1107–1126. · Zbl 0765.90050 · doi:10.1287/opre.40.6.1107 |
| [13] | G. Liberopolous and M. Caramanis, Infiniresimal perturbation analysis for second derivative estimation and design of manufacturing flow controllers, J. Optim. Theory Appl. (1994), to appear. |
| [14] | C.G. Cassandras and S.G. Strickland, On-line sensitivity analysis of Markov chains, IEEE Trans. Auto. Control AC-34(1989)76–86. · Zbl 0667.93094 · doi:10.1109/9.8651 |
| [15] | E.K.P. Chong and P.J. Ramadge, Convergence of recursive optimization algorithms using infinitesimal perturbation analysis, Discr. Event Dyn. Syst. 1(1992)339–372. · Zbl 0761.93075 |
| [16] | E.K.P. Chong and P.J. Ramadge, Optimization of queues using an IPA based stochastic algorithm with general update times, SIAM J. Control Optim. (1993), to appear. · Zbl 0770.60082 |
| [17] | K.L. Chung, On a stochastic approximation method, Ann. Math. Statist. 25(1954)463–483. · Zbl 0059.13203 · doi:10.1214/aoms/1177728716 |
| [18] | W.G. Cochran and G.M. Cox,Experimental Designs, 2nd ed. (Wiley, New York, 1957). · Zbl 0077.13205 |
| [19] | E.J. Dudewicz and S.R. Dalal, Allocation of measurements in ranking and selection with unequal variances, Sankhya B37(1975)28–78. · Zbl 0337.62016 |
| [20] | P. Dupuis and R. Simha, On sampling controlled stochastic approximation, IEEE Trans. Auto. Control AC-36(1991)915–924. · Zbl 0749.93081 · doi:10.1109/9.133185 |
| [21] | W. Farrell, Literature review and bibliography of simulation optimization,Proc. Winter Simulation Conf. (1977) pp. 117–124. |
| [22] | M.C. Fu, Convergence of a stochastic approximation algorithm for theGI/G/1 queue using infinitesimal perturbation analysis, J. Optim. Theory Appl. 65(1990)149–160. · Zbl 0699.93087 · doi:10.1007/BF00941166 |
| [23] | M.C. Fu, Sample path derivatives for (s, S) inventory systems, Oper. Res. (1993), to appear. |
| [24] | M.C. Fu and K. Healy, Simulation optimization of (s, S) inventory systems,Proc. Winter Simulation Conf. (1992) pp. 506–514. |
| [25] | M.C. Fu and Y.C. Ho, Using perturbation analysis for gradient estimation, averaging, and updating in a stochastic approximation algorithm,Proc. Winter Simulation Conf. (1988) pp. 509–517. |
| [26] | M.C. Fu and J.Q. Hu, Extensions and generalizations of smoothed perturbation analysis in a generalized semi-Markov process framework, IEEE Trans. Auto. Control AC-37(1992)1483–1500. · Zbl 0764.60089 · doi:10.1109/9.256367 |
| [27] | M.C. Fu and J.Q. Hu, On choosing the characterization for smoothed perturbation analysis, IEEE Trans. Auto. Control AC-36(1991)1331–1336. · doi:10.1109/9.100949 |
| [28] | M.C. Fu and J.Q. Hu, Second derivative sample path estimators for theGI/G/m queue, Manag. Sci. 39(1993)359–383. · Zbl 0769.90036 · doi:10.1287/mnsc.39.3.359 |
| [29] | M.C. Fu, J.Q. Hu and L. Shi, Likelihood ratio methods with conditional Monte Carlo and splitting,Proc. 1993 Allerton Conf., to appear. |
| [30] | R. Fujimoto, Parallel discrete event simulation, Commun. ACM 33(1990)30–53. · doi:10.1145/84537.84545 |
| [31] | A. Gaivoronski, Optimization of stochastic discrete event dynamic systems: a survey of some recent results, Lecture Notes in Economics and Mathematical Systems 374,Proc. Workshop on Simulation and Optimization, Laxenburg, Austria, ed. G. Pflug and U. Dieter (1992) pp. 24–44. · Zbl 0785.90076 |
| [32] | P. Glasserman,Gradient Estimation Via Perturbation Analysis (Kluwer Academic, 1991). · Zbl 0746.90024 |
| [33] | P. Glasserman, Structural conditions for perturbation analysis derivative estimation: finite-time performance indices, Oper. Res. 39(1991)724–738. · Zbl 0743.60086 · doi:10.1287/opre.39.5.724 |
| [34] | P. Glasserman and W.B. Gong, Smoothed perturbation analysis for a class of discrete event systems, IEEE Trans. Auto. Control AC-35(1990)1218–1230. · Zbl 0721.93051 · doi:10.1109/9.59807 |
| [35] | P. Glasserman and D.D. Yao, Some guidelines and guarantees for common random numbers, Manag. Sci. 38(1992)884–908. · Zbl 0758.65091 · doi:10.1287/mnsc.38.6.884 |
| [36] | P.W. Glynn, Likelihood ratio gradient estimation: an overview,Proc. Winter Simulation Conf. (1987) pp. 366–374. |
| [37] | P.W. Glynn, Likelihood ratio gradient estimation for stochastic systems, Commun. ACM 33(1990)75–84. · doi:10.1145/84537.84552 |
| [38] | P.W. Glynn, Optimization of stochastic systems,Proc. Winter Simulation Conf. (1986) pp. 52–59. |
| [39] | P.W. Glynn, Optimization of stochastic systems via simulation,Proc. Winter Simulation Conf. (1989) pp. 90–105. |
| [40] | P.W. Glynn, Stochastic approximation for Monte Carlo optimization,Proc. Winter Simulation Conf. (1986) pp. 356–364. |
| [41] | P.W. Glynn and P. Heidelberger, Analysis of parallel replicated simulations under a completion time constraint, ACM Trans. Modeling Comp. Simul. 1(1991)3–23. · Zbl 0842.65099 · doi:10.1145/102810.102811 |
| [42] | D. Goldsman, B.L. Nelson and B. Schmeiser, Methods for selecting the best system,Proc. Winter Simulation Conf. (1991) pp. 177–186. |
| [43] | W.B. Gong and Y.C. Ho, Smoothed perturbation analysis of discrete-event dynamic systems, IEEE Trans. Auto. Control AC-32(1987)858–867. · Zbl 0634.93076 · doi:10.1109/TAC.1987.1104464 |
| [44] | S.S. Gupta, On a decision rule for ranking means, Inst. Statist. Mimeo. Ser. No. 150, University of North Carolina, Chapel Hill, NC (1956). · Zbl 0073.35901 |
| [45] | J. Haddock and G. Bengu, Application of a simulation optimization system for a continuous review inventory model,Proc. Winter Simulation Conf. (1987) pp. 383–390. |
| [46] | K. Healy and L.W. Schruben, Retrospective simulation response optimization,Proc. Winter Simulation Conf. (1991) pp. 901–906. |
| [47] | K. Healy, Retrospective simulation response optimization, Ph.D. Dissertation, Cornell University (1992). |
| [48] | Y.C. Ho, Special issue on dynamics of discrete event systems, Proc. IEEE 77 (1989). |
| [49] | Y.C. Ho, Perturbation analysis: concepts and algorithms,Proc. Winter Simulation Conf. (1992) pp. 231–240. |
| [50] | Y.C. Ho amd X.R. Cao,Discrete Event Dynamic Systems and Perturbation Analysis (Kluwer Academic, 1991). · Zbl 0744.90036 |
| [51] | Y.C. Ho and X.R. Cao, Optimization and perturbation analysis of queueing networks, J. Optim. Theory Appl. 40(1983)559–582. · Zbl 0496.90034 · doi:10.1007/BF00933971 |
| [52] | Y.C. Ho, X.R. Cao and C.G. Cassandras, Infinitesimal and finite perturbation analysis for queueing networks, Automatica 19(1983)439–445. · Zbl 0514.90028 · doi:10.1016/0005-1098(83)90060-2 |
| [53] | Y.C. Ho and C. Cassandras, A new approach to the analysis of discrete event dynamic systems, Automatica 19(1983)149–167. · Zbl 0523.93045 · doi:10.1016/0005-1098(83)90088-2 |
| [54] | Y.C. Ho, A. Eyler and T.T. Chien, A gradient technique for general buffer storage design in a serial production line, Int. J. Prod. Res. 17(1979)557–580. · Zbl 0427.90047 · doi:10.1080/00207547908919637 |
| [55] | Y.C. Ho, M.A. Eyler and T.T. Chien, A new approach to determine parameter sensitivity of transfer lines, Manag. Sci. 29(1983)700–714. · doi:10.1287/mnsc.29.6.700 |
| [56] | Y.C. Ho and S. Li, Extensions of perturbation analysis of discrete-event dynamic systems, IEEE Trans. Auto. Control AC-33(1988)427–438. · Zbl 0637.90091 · doi:10.1109/9.1221 |
| [57] | Y.C. Ho, S. Li and P. Vakili, On the efficient generation of discrete event sample paths under different system parameters, Math. Comp. Simul. 30(1988)347–370. · Zbl 0645.60110 · doi:10.1016/S0378-4754(98)90005-2 |
| [58] | Y.C. Ho, L. Shi, L. Dai and W.B. Gong, Optimizing discrete event systems via the gradient surface method, Discr. Event Dyn. Syst. 2(1992)99–120. · Zbl 0762.93081 · doi:10.1007/BF01797723 |
| [59] | Y.C. Ho, R. Sreenevas and P. Vakili, Ordinal optimization of DEDS, Discr. Event Dyn. Syst. 2(1992)61–88. · Zbl 0754.60133 · doi:10.1007/BF01797280 |
| [60] | Y. Hochberg and A.C. Tamhane,Multiple Comparison Procedures (Wiley, New York, 1987). |
| [61] | J.C. Hsu, Simultaneous confidence levels for all distributions from the best, Ann. Statist. 9(1981)1026–1034. · Zbl 0474.62031 · doi:10.1214/aos/1176345582 |
| [62] | J.C. Hsu and B.L. Nelson, Optimization over a finite number of system designs with one-stage sampling and multiple comparisons with the best,Proc. Winter Simulation Conf. (1988) pp. 451–457. |
| [63] | J.S. Hunter and T.H. Naylor, Experimental designs for computer simulation experiments, Manag. Sci. 16(1970)422–434. · doi:10.1287/mnsc.16.7.422 |
| [64] | S.H. Jacobson, Optimal mean squared error analysis of the harmonic gradient estimators, J. Optim. Theory Appl. (1994), to appear. · Zbl 0791.93011 |
| [65] | S.H. Jacobson, Oscillation amplitude consideration in frequency domain experiments,Proc. Winter Simulation Conf. (1989) pp. 406–410. |
| [66] | S.H. Jacobson, Variance and bias reduction techniques for harmonic gradient estimators, Appl. Math. Comp. (1993), to appear. · Zbl 0768.93003 |
| [67] | S.H. Jacobson and L.W. Schruben, A review of techniques for simulation optimization, Oper. Res. Lett. 8(1989)1–9. · doi:10.1016/0167-6377(89)90025-4 |
| [68] | S.H. Jacobson and L.W. Schruben, A simulation optimization procedure using harmonic analysis, Oper. Res. (1992), submitted. |
| [69] | S.H. Jacobson, A. Buss and L.W. Schruben, Driving frequency selection for frequency domain simulation experiments, Oper. Res. 39(1991)917–924. · Zbl 0783.65058 · doi:10.1287/opre.39.6.917 |
| [70] | S.H. Jacobson, D. Morrice and L.W. Schruben, The global simulation clock as the frequency domain experiment index,Proc. Winter Simulation Conf. (1988) pp. 558–563. |
| [71] | H. Kesten, Accelerated stochastic approximation, Ann. Math. Statist. 29(1958)41–59. · Zbl 0087.13404 · doi:10.1214/aoms/1177706705 |
| [72] | A.I. Khuri and J.A. Cornell,Response Surfaces (Marcel Dekker, 1987). · Zbl 0632.62069 |
| [73] | J. Kiefer and J. Wolfowitz, Stochastic estimation of the maximum of a regression function, Ann. Math. Statist. 23(1952)462–466. · Zbl 0049.36601 · doi:10.1214/aoms/1177729392 |
| [74] | J.P.C. Kleijnen,Statistical Tools for Simulation Practitioners (Marcel Dekker, New York, 1987). · Zbl 0629.62004 |
| [75] | H.J. Kushner and D.C. Clark,Stochastic Approximation Methods for Constrained and Unconstrained Systems (Springer, New York, 1978). · Zbl 0381.60004 |
| [76] | A.M. Law and W.D. Kelton,Simulation Modeling and Analysis, 2nd ed. (McGraw-Hill, New York, 1991). · Zbl 0489.65007 |
| [77] | P. L’Ecuyer, A unified view of the IPA, SA, and LR gradient estimation techniques, Manag. Sci. 36(1990)1364–1383. · Zbl 0731.65130 · doi:10.1287/mnsc.36.11.1364 |
| [78] | P. L’Ecuyer, An overview of derivative estimation,Proc. Winter Simulation Conf. (1991) pp. 207–217. |
| [79] | P. L’Ecuyer, Convergence rates for steady-state derivative estimators, Ann. Oper. Res. 39(1992)121–136. · Zbl 0778.93105 · doi:10.1007/BF02060938 |
| [80] | P. L’Ecuyer and P.W. Glynn, Stochastic optimization by simulation: convergence proofs for theGI/G/1 queue in steady state, Manag. Sci. (1994), to appear. |
| [81] | P. L’Ecuyer, N. Giroux and P.W. Glynn, Stochastic optimization by simulation: numerical experiments with a simple queue in steady state, Manag. Sci. (1994), to appear. · Zbl 0822.90066 |
| [82] | P. L’Ecuyer and G. Perron, On the convergence rates of IPA and FDC derivative estimators for finite-horizon stochastic simulations, Oper. Res. (1993), submitted. |
| [83] | Y.T. Leung and R. Suri, Finite-time behavior of two simulation optimization algorithms,Proc. Winter Simulation Conf. (1990) pp. 372–376. |
| [84] | L. Ljung, Analysis of recursive stochastic algorithms, IEEE Trans. Auto. Control AC-22(1977)551–575. · Zbl 0362.93031 · doi:10.1109/TAC.1977.1101561 |
| [85] | L. Ljung, Strong convergence of a stochastic approximation algorithm, Ann. Statist. 6(1978)680–696. · Zbl 0402.62060 · doi:10.1214/aos/1176344212 |
| [86] | M. Mitra and S.K. Park, Solution to the indexing problem of frequency domain simulation experiments,Proc. Winter Simulation Conf. (1991) pp. 907–915. |
| [87] | R.H. Myers, A.I. Khuri and W.H. Carter, Response surface methodology: 1966–1988, Technometrics 31(1989)137–157. · Zbl 0693.62062 · doi:10.2307/1268813 |
| [88] | R.H. Myers,Response Surface Methodology (Allyn and Bacon, Boston, 1971). |
| [89] | M.B. Nevelson and R. Zalmanovich, Stochastic approximation and recursive estimation, translation of mathematical monographs, Vol. 47 (1973). |
| [90] | G.Ch. Pflug, On-line optimization of simulated Markovian processes, Math. Oper. Res. 15(1990)381–395. · Zbl 0726.90088 · doi:10.1287/moor.15.3.381 |
| [91] | G.Ch. Pflug, Sampling derivatives of probabilities, Computing 42(1989)315–328. · Zbl 0677.65001 · doi:10.1007/BF02243227 |
| [92] | B.T. Polyak, New method of stochastic approximation type, Auto. Remote Control 51(1990)937–946. · Zbl 0737.93080 |
| [93] | M.I. Reiman and A. Weiss, Sensitivity analysis via likelihood ratios,Proc. Winter Simulation Conf. (1986) pp. 285–289. |
| [94] | M.I. Reiman and A. Weiss, Sensitivity analysis for simulations via likelihood ratios, Oper. Res. 37(1989)830–844. · Zbl 0679.62087 · doi:10.1287/opre.37.5.830 |
| [95] | H. Robbins and S. Monro, A stochastic approximation method, Ann. Math. Statist. 22(1951)400–407. · Zbl 0054.05901 · doi:10.1214/aoms/1177729586 |
| [96] | K.W. Ross and J. Wang, Solving product form stochastic networks with Monte Carlo summation,Proc. Winter Simulation Conf. (1990) pp. 270–275. |
| [97] | R.Y. Rubinstein, How to optimize discrete-event systems from a single sample path by the score function method, Ann. Oper. Res. 27(1991)175–212. · Zbl 0714.93063 · doi:10.1007/BF02055195 |
| [98] | R.Y. Rubinstein, The push-out method for sensitivity analysis of discrete event systems, Ann. Oper. Res. (1993), to appear. |
| [99] | R.Y. Rubinstein, Sensitivity analysis and performance extrapolation for computer simulation models, Oper. Res. 37(1989)72–81. · doi:10.1287/opre.37.1.72 |
| [100] | R.Y. Rubinstein,Monte Carlo Optimization: Simulation and Sensitivity of Queueing Networks (Wiley, New York, 1986). · Zbl 0664.65059 |
| [101] | R.Y. Rubinstein and A. Shapiro,Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method (Wiley, 1992). · Zbl 0805.93002 |
| [102] | D. Ruppert, Almost sure approximations to the Robbins-Monro and Kiefer-Wolfowitz processes with dependent noise, Ann. Statist. 16(1982)178–187. · Zbl 0485.62083 |
| [103] | A. Ruszczynski and W. Syski, Stochastic approximation method with gradient averaging for unconstrained problems, IEEE Trans. Auto. Control AC-28(1983)1097–1105. · Zbl 0533.62076 · doi:10.1109/TAC.1983.1103184 |
| [104] | J. Sacks, Asymptotic distribution of stochastic approximation procedures, Ann. Math. Statist. 29(1958)373–405. · Zbl 0229.62010 · doi:10.1214/aoms/1177706619 |
| [105] | M.H. Safizadeh, Optimization in simulation: current issues and the future outlook, Naval Res. Logist. Quart. 37(1990)807–825. · Zbl 0711.90091 · doi:10.1002/1520-6750(199012)37:6<807::AID-NAV3220370602>3.0.CO;2-F |
| [106] | M.H. Safizadeh and B.M. Thornton, Optimization in simulation experiments using response surface methodology, Comp. Ind. Eng. 8(1984)11–27. · doi:10.1016/0360-8352(84)90018-4 |
| [107] | J. Santner and A.C. Tamhane,Design of Experiments: Ranking and Selection (Marcel Dekker, 1984). · Zbl 0542.00012 |
| [108] | R.G. Sargent, Research issues in metamodeling,Proc. Winter Simulation Conf. (1991) pp. 888–893. |
| [109] | R.G. Sargent and T.K. Som, Current issues in frequency domain experimentation, Manag. Sci. 38(1992)667–687. · Zbl 0825.90408 · doi:10.1287/mnsc.38.5.667 |
| [110] | L.W. Schruben, Simulation optimization using frequency domain methods,Proc. Winter Simulation Conf. (1986) pp. 366–369. |
| [111] | L.W. Schruben and V.J. Cogliano, An experimental procedure for simulation response surface model identification, Commun. ACM 30(1987)716–730. · doi:10.1145/27651.27656 |
| [112] | L.W. Schruben and V.J. Cogliano, Simulation sensitivity analysis: a frequency domain approach,Proc. Winter Simulation Conf. (1981) pp. 455–459. |
| [113] | B. Schmeiser, Simulation output data analysis, Ann. Oper. Res. (1994), this volume. |
| [114] | L.Y. Shi, Discontinuous perturbation analysis of discrete event dynamic systems, IEEE Trans. Auto. Control (1993), submitted. |
| [115] | D.E. Smith, An empirical investigation of optimum-seeking in the computer simulation situation, Oper. Res. 21(1973)475–497. · doi:10.1287/opre.21.2.475 |
| [116] | D.E. Smith, Automatic optimum-seeking program for digital simulation, Simulation 27(1976)27–32. · Zbl 0326.65040 · doi:10.1177/003754977602700103 |
| [117] | D.E. Smith, Requirements of an optimizer for computer simulation, Naval Res Logist. Quart. 20(1973)161–179. · doi:10.1002/nav.3800200114 |
| [118] | D.W. Sullivan and J.R. Wilson, Restricted subset selection procedures for simulation, Oper. Res. 37(1989)52–71. · Zbl 0663.62034 · doi:10.1287/opre.37.1.52 |
| [119] | R. Suri, Perturbation analysis: the state of the art and research issues explained via theGI/G/1 queue, Proc. IEEE 77(1989)114–137. · doi:10.1109/5.21075 |
| [120] | R. Suri and Y.T. Leung, Single run optimization of discrete event simulations – an empirical study using theM/M/1 queue, IIE Trans. 21(1991)35–49. · doi:10.1080/07408178908966205 |
| [121] | R. Suri and M. Zazanis, Perturbation analysis gives strongly consistent sensitivity estimates for theM/G/1 queue, Manag. Sci. 34(1988)39–64. · Zbl 0636.60102 · doi:10.1287/mnsc.34.1.39 |
| [122] | P. Vakili, Using a standard clock technique for efficient simulation, Oper. Res. Lett. (1991) 445–452. · Zbl 0745.65095 |
| [123] | P. Vakili, Massively parallel and distributed simulation of a class of discrete event systems: a different perspective, ACM Trans. Modeling Comp. Simul. (1993), to appear. · Zbl 0842.68108 |
| [124] | F. Vázquez-Abad and P. L’Ecuyer, Comparing alternative methods for derivative estimation when IPA does not apply directly,Proc. Winter Simulation Conf. (1991) pp. 1004–1011. |
| [125] | Y. Wardi, Simulation-based stochastic algorithm for optimizingGI/G/1 queues, Manuscript, Department of Industrial Engineering, Ben Gurion University (1988). · Zbl 0619.90069 |
| [126] | M.T. Wasan,Stochastic Approximation (Cambridge University Press, 1969). · Zbl 0293.62026 |
| [127] | J.R. Wilson, Future directions in response surface methodology for simulation,Proc. Winter Simulation Conf. (1987) pp. 378–381. |
| [128] | S. Yakowitz, A globally convergent stochastic approximation, SIAM J. Control Optim. (1993), to appear. · Zbl 0769.62060 |
| [129] | S. Yakowitz, T. Jayawadena and S. Li, Theory for automatic learning under partially observed Markov-dependent noise, IEEE Trans. Auto. Control AC-37(1992)1316–1324. · Zbl 0755.60090 · doi:10.1109/9.159569 |
| [130] | W.N. Yang and B.L. Nelson, Using common random numbers and control variates in multiple-comparison procedures, Oper. Res. 39(1991)583–591. · Zbl 0737.65117 · doi:10.1287/opre.39.4.583 |
| [131] | G. Yin, Stochastic approximation via averaging: the Polyak approach revisited, Lecture Notes in Economics and Mathematical Systems Vol. 374,Proc. Workshop on Simulation and Optimization, Laxenburg, Austria, ed. G. Pflug and U. Dieter (1990) pp. 119–134. · Zbl 0768.65030 |
| [132] | M. Zazanis and R. Suri, Estimating first and second derivatives of response time forG/G/1 queues from a single sample path, Technical Report, Northwestern University, Evanston, IL; also Queueing Syst. (1986). |
| [133] | B. Zhang and Y.C. Ho, Performance gradient estimation for very large finite Markov chains, IEEE Trans. Auto. Control AC-36(1991)1218–1227. · Zbl 0736.60069 · doi:10.1109/9.100931 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
