to the English edition Many processes that describe the operation of engineering, economic, organiza tional, and other systems are represented as sequences of operations performed on material, information, or other types of flows. Typical examples are processes of connection of telephone users, data transmission and processing, calculation at multi user computer centers, and queueing at service centers. The models studied by the theory of service systems, or queueing theory, are used to describe such processes. The more pessimistic term "queueing theory" is used more often in the non-Soviet…mehr
to the English edition Many processes that describe the operation of engineering, economic, organiza tional, and other systems are represented as sequences of operations performed on material, information, or other types of flows. Typical examples are processes of connection of telephone users, data transmission and processing, calculation at multi user computer centers, and queueing at service centers. The models studied by the theory of service systems, or queueing theory, are used to describe such processes. The more pessimistic term "queueing theory" is used more often in the non-Soviet literature. Random arrivals (requests for service), probability distributions defining queueing processes (distributions of service times and acceptable waiting times), and structure parameters (customer priorities, parameters that delimit acceptable queues, parameters that define paths of customers, etc.) are characteristic com ponents of queueing models. Typical output characteristics of queueing models are the probability distributions of queue lengths, waiting times, lengths of busy periods, and so forth.Hinweis: Dieser Artikel kann nur an eine deutsche Lieferadresse ausgeliefert werden.
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.
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.
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