Mathematical Methods for Construction of Queueing Models

1 Substantive Formulation of the Problem of Queueing Model Construction.- 1.1 Construction of a Model as an Object of Study.- 1.2 The Problem of Identification of Classes and Values of Model Parameters.- 1.3 The Problem of Model Simplification. Approximation of Models.- 1.4 The Stability Problem.- 1.5 A General Schema of Model Construction.- 1.6 Discussion and Review of Literature.- 2 The Concept of Characterization as a General Mathematical Schema for Constructing Queueing Models.- 2.1 Queueing Models. Formalization of Queueing Models.- 2.2 The Problem of Pure Characterization of a Queueing Model.- 2.3 The Direct Characterization Problem and Its Stability.- 2.4 The Inverse Characterization Problem and Its Stability.- 2.5 Discussion and Review of the Literature 44.- 3 Probability Metrics.- 3.1 Introductory Remarks.- 3.2 The Concept of Probability Metric.- 3.3 Examples of Probability Metrics.- 3.4 Classification of Probability Metrics.- 3.5 The Concept of Minimality of Probability Metrics.- 3.6 Dual Relations for Compound and Related Minimal Metrics.- 3.7 Explicit Representations for Minimal Metrics.- 3.8 The Concept of Ideality of Probability Metrics.- 3.9 Topological Properties of Probability Metrics. The Concept of Compactness.- 3.10 Relations Between Metrics.- 3.11 Discussion of Results and Review of Literature.- 4 Characterization of the Components of Queueing Models.- 4.1 Formulation of the Problem.- 4.2 Characterization of a Poisson Flow in Terms of the Aging Property. Evaluation of Stability.- 4.3 Characterization of an Erlang Flow in Terms of the Aging Property.- 4.4 Characterization of a Renewal Flow in Terms of the Aging Property.- 4.5 Characterization of a Poisson Flow in Terms of the Lack-of-Memory Property.- 4.6 Stability of the Characterization of a Poisson Flow as a Stationary, Ordinary Flow Without Memory.- 4.7 Chaxacterization of an Erlang Flow in Terms of the LM-Property.- 4.8 Characterization of Random Flows in Terms of Sample Characteristics.- 4.9 Auxiliary Results from a Comparison of Metrics.- 4.10 Discussion of Results and Review of Literature.- 5 Methods of Analysis of the Continuity of Queueing Models.- 5.1 Formulation of the Problem and Its Practical Import.- 5.2 Continuity on Finite Time Intervals (Output Data as a Random Sequence).- 5.3 Evaluations for Sequences Generated by Piecewise-Linear Transforms.- 5.4 Examples.- 5.5 Continuity on Finite Time Intervals (Output Data as a Stochastic Process).- 5.6 Continuity Uniform in Time.- 5.7 Examples.- 5.8 Discussion of Results and Review of Literature.- 6 Construction of Queueing Models from Observations of Their Inputs - Direct Problems of Characterization.- 6.1 Introductory Remarks.- 6.2 Characterization of Queueing Models in Terms of the Aging Property.Evaluations of Stability.- 6.3 Characterization of Queueing Models in Terms of the Lack-of-Memory Property.- 6.4 Characterization of Queueing Models in Terms of Empirical Distributions of the Input Flow and Service Flow.- 6.5 Evaluations Uniform in Time of the Stability of Characterization.- 6.6 Characterization of the Multiphase-Multichannel Model.- 6.7 Unsolved Problems.- 7 Identification of Individual Queueing Models from Observations of Output Data - Inverse Characterization Problems.- 7.1 Introductory Remarks.- 7.2 Identification of Queueing Models.- 7.3 Identification of a Class of Queueing Models.- 7.4 Remarks and Comments.- 8 Simplification and Approximation of Probability Models.- 8.1 Formulation of the Problem, and an Approach to the Solution.- 8.2 Hyper-Erlang Approximation of Distribution Functions and Estimates of Its Accuracy.- 8.3 Finite Approximation of Countable Markov Chains.- 8.4 Finite Approximation of Noncountable Markov Chains.- 8.5 Approximation of Output Flows.- 8.6 The Problem of Asymptotic Consolidation of States.- 8.7 Discussion and Review of Literature.- Appendix 1 The Prokhorov Criterion and Skorokhod's Theorem of the Weak Convergence of Measures.- Appendix 2 The Duality Theorem of Linear Programming.- Appendix 3 Uniformly Integrable Random Variables.- Appendix 4 The Space D)[0, ?).- Appendix 5 Ancillary Information from the Theory of Markov Processes.- Appendix 6 Estimates of the Coupling Times for Discrete Renewal Processes.- References.- Index of Notation.