Stochastic-Process Limits

Transcription

1 Ward Whitt Stochastic-Process Limits An Introduction to Stochastic-Process Limits and Their Application to Queues With 68 Illustrations Springer

2 Contents Preface vii 1 Experiencing Statistical Regularity A Simple Game of Chance Plotting Random Walks When the Game is Fair The Final Position Making an Interesting Game Stochastic-Process Limits A Probability Model Classical Probability Limits Identifying the Limit Process Limits for the Plots Invariance Principles The Range of Brownian Motion Relaxing the IID Conditions Different Step Distributions The Exception Makes the Rule Explaining the Irregularity The Centered Random Walk with p = 3/ Back to the Uncentered Random Walk with p = 1/ Summary 45

3 xviii Contents 2 Random Walks in Applications Stock Prices The Kolmogorov-Smirnov Statistic A Queueing Model for a Buffer in a Switch Deriving the Proper Scaling Simulation Examples Engineering Significance Buffer Sizing Scheduling Service for Multiple Sources 68 3 The Framework for Stochastic-Process Limits Introduction The Space V The Space D The Continuous-Mapping Approach Useful Functions Organization of the Book 89 4 A Panorama of Stochastic-Process Limits Introduction Self-Similar Processes General CLT's and FCLT's Self-Similarity The Noah and Joseph Effects Donsker's Theorem The Basic Theorems Multidimensional Versions Brownian Limits with Weak Dependence The Noah Effect: Heavy Tails Stable Laws Ill Convergence to Stable Laws Convergence to Stable Levy Motion Extreme-Value Limits The Joseph Effect: Strong Dependence Strong Positive Dependence Additional Structure Convergence to Fractional Brownian Motion Heavy Tails Plus Dependence Additional Structure Convergence to Stable Levy Motion Linear Fractional Stable Motion Summary 136

4 Contents xix Heavy-Traffic Limits for Fluid Queues Introduction A General Fluid-Queue Model Input and Available-Processing Processes Infinite Capacity Finite Capacity Unstable Queues Fluid Limits for Fluid Queues Stochastic Refinements Heavy-Traffic Limits for Stable Queues Heavy-Traffic Scaling The Impact of Scaling Upon Performance Identifying Appropriate Scaling Functions Limits as the System Size Increases Brownian Approximations The Brownian Limit The Steady-State Distribution The Overflow Process One-Sided Reflection First-Passage Times Planning Queueing Simulations The Standard Statistical Procedure Invoking the Brownian Approximation Heavy-Traffic Limits for Other Processes The Departure Process The Processing Time Priorities A Heirarchical Approach Processing Times 190 Unmatched Jumps in the Limit Process Introduction Linearly Interpolated Random Walks Asymptotic Equivalence with M\ Simulation Examples Heavy-Tailed Renewal Processes Inverse Processes The Special Case with m = A Queue with Heavy-Tailed Distributions The Standard Single-Server Queue Heavy-Traffic Limits Simulation Examples Rare Long Service Interruptions Time-Dependent Arrival Rates 220

5 xx Contents 7 More Stochastic-Process Limits Introduction Central Limit Theorem for Processes Hahn's Theorem A Second Limit Counting Processes CLT Equivalence FCLT Equivalence Renewal-Reward Processes Fluid Queues with On-Off Sources Introduction A Fluid Queue Fed by On-Off Sources The On-Off Source Model Simulation Examples Heavy-Traffic Limits for the On-Off Sources A Single Source Multiple Sources M/G/oo Sources Brownian Approximations The Brownian Limit Model Simplification Stable-Levy Approximations The RSLM Heavy-Traffic Limit The Steady-State Distribution Numerical Comparisons Second Stochastic-Process Limits M/G/l/K Approximations Limits for Limit Processes Reflected Fractional Brownian Motion An Increasing Number of Sources Gaussian Input Reflected Gaussian Processes Single-Server Queues Introduction The Standard Single-Server Queue Heavy-Traffic Limits The Scaled Processes Discrete-Time Processes Continuous-Time Processes Superposition Arrival Processes Split Processes Brownian Approximations Variability Parameters 307

6 Contents xxi Models with More Structure Very Heavy Tails Heavy-Traffic Limits First Passage to High Levels An Increasing Number of Arrival Processes Iterated and Double Limits Separation of Time Scales Approximations for Queueing Networks Parametric-Decomposition Approximations Approximately Characterizing Arrival Processes A Network Calculus Exogenous Arrival Processes Concluding Remarks Multiserver Queues Introduction Queues with Multiple Servers A Queue with Autonomous Service The Standard m-server Model Infinitely Many Servers Heavy-Traffic Limits Gaussian Approximations An Increasing Number of Servers Infinite-Server Approximations Heavy-Traffic Limits for Delay Models Heavy-Traffic Limits for Loss Models Planning Simulations of Loss Models More on the Mathematical Framework Introduction Topologies Definitions Separability and Completeness The Space V Probability Spaces Characterizing Weak Convergence Random Elements Product Spaces The Space D J 2 and M 2 Metrics The Four Skorohod Topologies Measurability Issues The Compactness Approach 386

7 xxii Contents 12 The Space D Introduction Regularity Properties of D Strong and Weak M 1 Topologies Definitions Metric Properties Properties of Parametric Representations Local Uniform Convergence at Continuity Points Alternative Characterizations of M\ Convergence SMi Convergence WM X Convergence Strengthening the Mode of Convergence Characterizing Convergence with Mappings Topological Completeness Noncompact Domains Strong and Weak M 2 Topologies Alternative Characterizations of M 2 Convergence M2 Parametric Representations SM 2 Convergence WM 2 Convergence Additional Properties of M 2 Convergence Compactness Useful Functions Introduction Composition Composition with Centering Supremum One-Dimensional Reflection Inverse The Standard Topologies The M[ Topology First Passage Times Inverse with Centering Counting Functions Queueing Networks Introduction The Multidimensional Reflection Map A Special Case Definition and Characterization Continuity and Lipschitz Properties The Instantaneous Reflection Map Definition and Characterization Implications for the Reflection Map 480

8 Contents xxiii 14.4 Reflections of Parametric Representations Mi Continuity Results and Counterexamples M x Continuity Results Counterexamples Limits for Stochastic Fluid Networks Model Continuity Heavy-Traffic Limits Queueing Networks with Service Interruptions Model Definition Heavy-Traffic Limits The Two-Sided Regulator Definition and Basic Properties With the M x Topologies Related Literature The Spaces E and F Introduction Three Time Scales More Complicated Oscillations The Space E Characterizations of M 2 Convergence in E Convergence to Extremal Processes The Space F Queueing Applications 535 References 541 Appendix A Regular Variation 569 Appendix B Contents of the Internet Supplement 573 Notation Index 577 Author Index 579 Subject Index 585