Advances in Stochastic Simulation Methods

Advances in Stochastic Simulation Methods

N. Balakrishnan V.B. Melas S. Ermakov Editors

Birkhäuser Boston • Basel • Berlin

Contents

xiii xv xxi xxv

Preface Contributors List of Tables List of Figures PART I: SIMULATION MODELS

1 Solving the Nonlinear Algebraic Equations with Monte Carlo Method S. Ermakov and I. Kaloshin Introduction 4 1.1 Neumann-Ulam Scheme 4 1.2 Simples Nonlinear Problems References 14

7

2 Monte Carlo Algorithms For Neumann Boundary Value Problem Using Fredholm Representation Y. N. Kashtanov and I. N. Kuchkova 2.1 2.2 2.3 2.4 2.5

Introduction 17 Integral Representation 18 Monte Carlo Estimators 19 The Two-Dimensional Case 23 An Application to Navier-Stokes Equations References 28

3.1 Introduction 29 3.2 Strong Weakly-Continuous Derivatives 3.3 General Results 34

33

17

26

3 Estimation Errors for Functionals on Measure Spaces N. Golyandina and V. Nekrutkin

v

3

29

3.4 Stratification 40 3.4.1 General stratification scheme 3.4.2 Examples 41 References 45

40

4 The Multilevel Method of Dependent Tests Stefan Heinrich 4.1 4.2 4.3 4.4

Introduction 47 The Standard Method of Dependent Tests The Multilevel Approach 50 Integrals Depending on a Parameter 52 References 60

48

5 Algebraic Modelling and Performance Evaluation of Acyclic Fork-Join Queueing Networks Nikolai K. Krivulin 5.1 5.2 5.3 5.4

5.5 5.6 5.7

5.8

Introduction 63 Preliminary Algebraic Definitions and Results 65 Further Algebraic Results 67 An Algebraic Model of Queueing Networks 68 5.4.1 Fork-Join queueing networks 69 5.4.2 Examples of network modeis 71 A Monotonicity Property 72 Bounds on the Service Cycle Completion Time 74 Stochastic Extension of the Network Model 75 5.7.1 Some properties of expectation 76 5.7.2 Existence of the cycle time 77 5.7.3 Calculating bounds on the cycle time 78 Discussion and Examples 78 References 81

PART IL EXPERIMENTAL DESIGNS

6 Analytical Theory of ^-Optimal Designs for Polynomial Regression V. B. Melas 6.1 6.2 6.3 6.4 6.5 6.6

Introduction 85 Statement of the Problem 86 Duality Theorem 86 The Number of Design Points 87 Tchebysheff Designs 90 Boundary Equation 91 An Extremal Property of Positive Polynomial Representations 93

Contents 6.7 6.8 6.9 6.10

vii Differential Equation 95 Limiting Design 103 Taylor Expansion 109 Particular Cases 110 6.10.1 Boundary equation 110 6.10.2 Matrices Ji, n , and vectors JJI (°)

Z

111

(0)

6.10.3 Tables of coefficients 112 6.10.4 Studying of convergence radius References 114

113

7 Bias Constrained Minimax Robust Designs for Misspecified Regression Models Douglas P. Wiens

117

7.1 Introduction 117 7.2 General Theory 118 7.3 Fitting a Second Order Response in Several Regressors 122 7.3.1 S an ellipsoid 122 7.3.2