Stochastic Modeling and Analysis of Manufacturing Systems

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.