1 Jackson Network Models of Manufacturing Systems.- 1.1 Introduction.- 1.2 Jackson Networks.- 1.2.1 The Open Model.- 1.2.2 The Closed Model.- 1.2.3 The Semi-Open Model.- 1.3 The Throughput Function and Computation.- 1.4 Monotonicity of the Throughput Function.- 1.4.1 Equilibrium Rate.- 1.4.2 PF2 Property.- 1.4.3 Likelihood Ratio Ordering.- 1.4.4 Shifted Likelihood Ratio Ordering.- 1.5 Concavity and Convexity.- 1.6 Multiple Servers.- 1.7 Resource Sharing.- 1.7.1 Aggregation of Servers.- 1.7.2 Aggregation of Nodes.- 1.8 Arrangement and Majorization.- 1.9 Conclusions.- 1.10 Notes.- 1.11 References.- 2 Hierarchical Modeling of Stochastic Networks, Part I: Fluid Models.- 2.1 Introduction.- 2.1.1 Macro, Meso and Microscopic Models for an i.i.d. Sequence.- 2.1.2 Strong Approximations - A Unifying Framework.- 2.1.3 Summary.- 2.2 A Flow Network in Discrete Time.- 2.2.1 The Microscopic Model and Its Dynamics.- 2.2.2 Reformulation in Terms of Cumulants and Oblique Reflection.- 2.2.3 Mesoscopic Models and Strong Approximations.- 2.2.4 Macroscopic Models: FSLLN's.- 2.2.5 Deviations Between Micro and Macro Models: FCLT.- 2.3 Flow Networks in Continuous Time.- 2.3.1 Flow Networks with Time Inhomogeneous Dynamics.- 2.3.2 State-Dependent Dynamics.- 2.4 Linear Fluid Network and Bottleneck Analysis.- 2.4.1 Traffic Equations and Bottleneck Definitions.- 2.4.2 Bottleneck Analysis.- 2.5 Functional Strong Law of Large Numbers.- 2.5.1 FSLLN's for Nonlinear Fluid Networks.- 2.5.2 FSLLN's for Nonparametric Jackson Queueing Networks.- 2.5.3 FSLLN's for State-Dependent Networks.- 2.6 Applications and Hints at Prospects of Fluid Models.- 2.6.1 Stochastic Fluid Models for Manufacturing and Communication Systems.- 2.6.2 Heterogeneous Fluid Networks: Bottleneck Analysis and Scheduling Control.- 2.6.3 Transient Analysis of the Mt/Mt/1 Queue.- 2.7 References and Comments.- 2.8 References.- 3 Hierarchical Modeling of Stochastic Networks, Part II: Strong Approximations.- 3.1 Introduction.- 3.2 The Model.- 3.2.1 Primitives and Dynamics.- 3.2.2 Underlying Assumptions and Parameters.- 3.2.3 Nonparametric Jackson Networks.- 3.3 Preliminaries.- 3.3.1 Traffic Equations and Bottlenecks.- 3.3.2 The Oblique Reflection Mapping.- 3.3.3 Reflected Brownian Motion on the Orthant.- 3.4 The Main Results.- 3.4.1 Functional Strong Approximations.- 3.4.2 Functional Laws of the Iterated Logarithm.- 3.4.3 FSLLN's and Fluid Approximations.- 3.4.4 FCLT's and Diffusion Approximations.- 3.5 Fitting Parametes.- 3.5.1 Nonparametric Jackson Networks.- 3.5.2 Product Form and Single Station.- 3.6 Proof of the Main Results.- 3.7 References, Possible Extensions and Future Research.- 3.8 References.- 4 A GSMP Framework for the Analysis of Production Lines.- 4.1 Introduction.- 4.2 GSMP and Its Scheme.- 4.2.1 The Scheme: GSMS.- 4.2.2 Language and Score Space.- 4.3 Structural Properties of the Scheme.- 4.3.1 Some Useful Properties.- 4.3.2 Condition (M).- 4.3.3 Condition (CX).- 4.3.4 Minimal Elements.- 4.3.5 Monotonicity and Convexity.- 4.3.6 Characteristic Function.- 4.3.7 Subschemes.- 4.3.8 Synchronized Schemes.- 4.4 The (a, 6, k) Tandem Queue.- 4.4.1 Production Lines Under Kanban Control.- 4.4.2 Properties with Respect to Service Times.- 4.5 Properties with Respect to (a, b, k).- 4.5.1 Monotonicity with Respect to (a, b, k).- 4.5.2 Concavity with Respect to (a, b, k).- 4.6 Line Reversal.- 4.6.1 Reversibility of Departure Epochs.- 4.6.2 Full Reversibility.- 4.7 Subadditivity and Ergodicity.- 4.7.1 Event-Epoch Vectorization.- 4.7.2 The Subadditive Ergodic Theorem.- 4.7.3 More General Matrices.- 4.8 Cycle Time Limits.- 4.8.1 Existence of the Limits.- 4.8.2 Rate of Convergence.- 4.9 Notes.- 4.10 References.- 5 Stochastic Convexity and Stochastic Majorization.- 5.1 Introduction.- 5.2 Stochastic Order Relations: Functional Characterizations.- 5.3 Second-Order Stochastic Properties.- 5.3.1 Stochastic Convexity.- 5.3.2 Stochastic Supermodularity and Submodularity.- 5.3.3 Markov Chain Applications.- 5.3.4 A Joint Setup Problem.- 5.3.5 Production with Trial Runs.- 5.4 Arrangement and Likelihood Ratio Orderings.- 5.4.1 The Connection.- 5.4.2 Queueing Network Applications.- 5.5 Stochastic Rearrangement and Majorization.- 5.5.1 The Deterministic Theory.- 5.5.2 The Stochastic Counterpart.- 5.5.3 Connections to Stochastic Convexity and Stochastic Supermodularity.- 5.6 Notes.- 5.7 References.- 6 Perturbation Analysis of Production Networks.- 6.1 Introduction.- 6.2 Overview Through the Single-Machine Model.- 6.3 Differentiation.- 6.3.1 Classes of Random Functions.- 6.3.2 Differentiability of Inputs.- 6.3.3 Differentiability of Recursions.- 6.4 Analysis of the Single-Machine Model.- 6.5 Production Networks.- 6.5.1 The Production Line.- 6.5.2 Finite Buffers.- 6.5.3 Implementation.- 6.5.4 A Kanban System.- 6.5.5 Systems with Rework and Scrap.- 6.5.6 A System with Alternative Sourcing.- 6.5.7 A System with Subassemblies.- 6.6 Steady-State Derivative Estimation.- 6.6.1 Discrete Time.- 6.6.2 Continuous Time.- 6.7 Concluding Remarks.- 6.8 Notes.- 6.9 References.- 7 Scheduling Networks of Queues: Heavy Traffic Analysis of a Bi-Criteria Problem.- 7.1 Introduction.- 7.2 A Single Server Queue.- 7.2.1 The Scheduling Problem.- 7.2.2 The Limiting Control Problem.- 7.2.3 The Workload Formulation.- 7.2.4 Solution to the Workload Formulation.- 7.2.5 Interpreting the Solution to the Workload Formulation.- 7.3 A Closed Network.- 7.3.1 The Workload Formulation.- 7.3.2 The c ? ? Case.- 7.3.3 The c = 0 Case.- 7.3.4 The Bi-Criteria Case.- 7.4 A Network with Controllable Inputs.- 7.4.1 The Workload Formulation.- 7.4.2 Solution to the Workload Formulation.- 7.5 An Example.- 7.6 A Review of Related Results.- 7.6.1 Open Networks.- 7.6.2 Closed Networks.- 7.6.3 Networks with Controllable Inputs.- 7.6.4 Networks with Discretionary Routing.- 7.6.5 Production/Inventory Systems.- 7.6.6 Weak Convergence Results.- 7.7 References.- 8 Scheduling Manufacturing Systems of Re-Entrant Lines.- 8.1 Introduction.- 8.2 Re-Entrant Lines: The Models.- 8.3 Fluctuation Smoothing Scheduling Policies to Reduce Variance of Lateness, Variance of Cycle-Time, and Mean Cycle-Time.- 8.4 Stability of LBFS, SRPTS, EA, EDD and All Least Slack Scheduling Policies.- 8.5 Dynamic Scheduling of a Single Machine with Set-Up Times: A Push Model.- 8.6 Clear-A-Praction Policies.- 8.7 A Lower Bound on Optimal Cost.- 8.8 A Good CAF Policy.- 8.9 Non-Acyclic Manufacturing Systems with Set-Up Times.- 8.10 Concluding Remarks.- 8.11 Notes.- 8.12 References.

Manufacturing systems have become increasingly complex over recent years. This volume presents a collection of chapters which reflect the recent developments of probabilistic models and methodologies that have either been motivated by manufacturing systems research or been demonstrated to have significant potential in such research.
The editor has invited a number of leading experts to present detailed expositions of specific topics. These include: Jackson networks, fluid models, diffusion and strong approximations, the GSMP framework, stochastic convexity and majorization, perturbation analysis, scheduling via Brownian models, and re-entrant lines and dynamic scheduling. Each chapter has been written with graduate students in mind, and several have been used in graduate courses that teach the modeling and analysis of manufacturing systems.