Mathematical Theory of Connecting Networks and Telephone Traffic

Name: Mathematical Theory of Connecting Networks and Telephone Traffic
Brand: Academic Press
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Erschienen am 14. Mai 2014

335 Seiten

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Front Cover
Mathematical Theory of Connecting Networks and Telephone Traffic
Copyright Page
Contents
Preface
Chapter 1. Heuristic Remarks and Mathematical Problems Regarding the Theory of Connecting Systems
1. Introduction
2. Summary of Chapter 1
3. Historical Sketch
4. Critique
5. General Properties of Connecting Systems
6. Performance of Switching Systems
7. Desiderata
8. Mathematical Models
9. Fundamental Difficulties and Questions
10. The Merits of Microscopic States
11. From Details to Structure
12. The Relevance of Combinatorial and Structural Properties: Examples
13. Combinatorial, Probabilistic, and Variational Problems
14. A Packing Problem
15. A Problem of Traffic Circulation in a Telephone Exchange
16. An Optimal Routing Problem
References
Chapter 2. Algebraic and Topological Properties of Connecting Networks
1. Introduction
2. Summary of Chapter 2
3. The Structure and Condition of a Connecting Network
4. Graphical Depiction of Network Structure and Condition
5. Network States
6. The State Diagram
7. Some Numerical Functions
8. Assignments
9. Three Topologies
10. Some Definitions and Problems
11. Rearrangeable Networks
12. Networks Nonblocking in the Wide Sense
13. Networks Nonblocking in the Strict Sense
14. Glossary for Chapter 2
References
Chapter 3. Rearrangeable Networks
1. Introduction
2. Summary of Chapter 3
3. The Slepian-Duguid Theorem
4. The Number of Calls That Must Be Moved: Paull's Theorem
5. Some Formal Preliminaries
6. The Number of Calls That Must Be Moved: New Results
7. Summary of Sections 8-15
8. Stages and Link Patterns
9. Group Theory Formulation
10. The Generation of Complexes by Stages
11. An Example
12. Some Definitions
13. Preliminary Results
14. Generating the Permutation Group
15. Construction of a Class of Rearrangeable Networks
16. Summary of Sections 17-21
17. The Combinatorial Power of a Network
18. Preliminaries
19. Construction of the Basic Partial Ordering
20. Cost Is Nearly Isotone on T(Cn)
21. Principal Results of Optimization
References
Chapter 4. Strictly Nonblocking Networks
1. Introduction
2. Square Array
3. Three-Stage Strictly Nonblocking Connecting Network
4. Principle Involved
5. Five-Stage Network
6. Seven-Stage Network
7. General Multistage Switching Network
8. Most Favorable Size of Input and Output Switches in the Three-Stage Network
9. Most Favorable Switch Sizes in the Five-Stage Network
10. Search for the Smallest N for a Given n for the Three-Stage Network
11. Cases in the Three-Stage Network Where N = r(mod n)
12. Search for the Minimum Number of Crosspoints between N = 23 and N = 160
13. Search for the Minimum Number of Crosspoints for N = 240
14. Rectangular Array
15. N Inputs and M Outputs in the Three-Stage Array
16. Triangular Network
17. One-way Incoming, One-way Outgoing, and Two-way Trunks
18. Comparison with Existing Networks
19. Conclusion
Reference
Chapter 5. A Sufficient Set of Statistics for a Simple Telephone Exchange Model
1. Theoretical Problems and Methods of Traffic Measurement
2. Description of the Mathematical Model
3. Discussion of the Model
4. Summary of Notations for Chapter 5
5. The Average Traffic
6. Maximum Conditional Likelihood Estimators
7. Practical Estimators Suggested by Maximizing the Likelihood L, Defined in Section 11
8. Other Estimators
9. The Joint Distribution of the Sufficient Statistics
10. The Distributions of Z and M
11. Proof that (n, A, H , Z) Is Sufficient
12. Unconditional Maximum Likelihood Estimators
13. The Joint Distribution of xT , n, A , H , and Z
References
Chapter 6. The Covariance Function of a Simple Trunk Group, with Applications to Traffic Measurement
1. Introduction
2. Discussion, Summary, and Conclusions
3. The Covariance Function
4. The Covariance in Terms of the Recovery Function
5. The Variance of the Number of Paths in Service
6. The Variance of Time Averages
7. Derivation of the Covariance
8. Approximation to the Dominant Characteristic Value
References
Chapter 7. A "Thermodynamic" Theory of Traffic
1. Introduction
2. Preliminaries
3. Summary of Chapter 7 and Discussion
4. Equilibrium
5. The Maximum Entropy Principle
6. The Canonical Distribution
7. Properties of the Partition Function
8. A Reversible Markov Process for Which the Canonical Distribution IS Invariant
9. Analogy with Statistical Mechanics
10. Discussion and Critique
11. A Markov Model Based on Terminal-Pair Behavior
12. The Approach to Equilibrium
13. Covariance of Functions of zt
14. Applications to Sampling Error
15. A Generalization
References
Chapter 8. Markov Processes Representing Traffic in Connecting Networks
1. Introduction
2. Preliminary Remarks and Definitions
3. Summary of Chapter 8
4. Probability
5. Operation
6. Transition Rates
7. Markov Processes
8. Probability of Blocking
9. A Basic Formula
10. Solution of the Equations of Statistical Equilibrium
11. Stationary Probability Measures for Ergodic Markov Processes
12. Expansion of the Stationary Vector p in Powers of ?
13. Expansion of the Probability of Blocking in Powers of ?
14. Combinatorial Interpretation and Calculation of the Constants {cm(x, 0), ~ E S , m 2 0)
14.Combinatorial Interpretation and Calculation of the Constants {cm(x, 0), x e S, m=0}
15. Calculation of c,(x, ?)
References
Suggested Reading
Author Index
Subject Index

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