# Flow Networks

## 1st Edition

### Analysis and Optimization of Repairable Flow Networks, Networks with Disturbed Flows, Static Flow Networks and Reliability Networks

**Authors:**Michael Todinov

**Hardcover ISBN:**9780123983961

**eBook ISBN:**9780123984067

**Imprint:**Elsevier

**Published Date:**1st February 2013

**Page Count:**320

## Description

Dedication

Preface

1. Flow Networks – Existing Analysis Approaches and Limitations

1.1 Repairable Flow Networks and Static Flow Networks

1.2 Repairable Flow Networks and Stochastic Flow Networks

1.3 Networks with Disturbed Flows and Stochastic Flow Networks

1.4 Performance of Repairable Flow Networks

2. Flow Networks and Paths – Basic Concepts, Conventions and Algorithms

2.1 Basic Concepts and Conventions: Data Structures for Representing Flow Networks

2.2 Pseudo-Code Conventions Used in the Algorithms

2.3 Efficient Representation of Flow Networks with Complex Topology

2.4 Paths: Algorithms Related to Paths in Flow Networks

2.5 Determining the Smallest-Cost Paths from the Source

2.6 Topological Sorting of Networks Without Cycles

2.7 Transforming Flow Networks

3. Key Concepts, Results and Algorithms Related to Static Flow Networks

3.1 Path Augmentation in Flow Networks

3.2 Bounding the Maximum Throughput Flow by the Capacity of s–t Cuts

3.3 A Necessary and Sufficient Condition for a Maximum Throughput Flow in a Static Network: The Max-Flow Min-Cut Theorem

3.4 Classical Augmentation Algorithms for Determining the Maximum Throughput Flow in Networks

3.5 General Push-Relabel Algorithm for Maximising the Throughput Flow in a Network

3.6 Applications

3.7 Successive Shortest-Path Algorithm for Determining the Maximum Throughput Flow at a Minimum Cost

4. Maximising the Throughput Flow in Single- and Multi-Commodity Networks: Removing Parasitic Directed Loops of Flow in Networks Optimised by Classical Algorithms

4.1 Eliminating Parasitic Directed Loops of Flow in Networks Optimised by Classical Algorithms

4.2 A Two-Stage Augmentation Algorithm for Determining the Maximum Throughput Flow in a Network

4.3 A New, Efficient Algorithm for Maximising the Throughput Flow of the Useful Commodity in a Multi-Commodity Flow Network

4.4 Network Flow Transformation Along Cyclic Paths

5. Networks with Disturbed Flows Dual Network Theorems for Networks with Disturbed Flows: Reoptimising the Power Flows in Active Power Networks in Real Time

5.1 Reoptimising the Flow in Networks with Disturbed Flows After Edge Failures and After Choking the Edge Flows

5.2 A Fast Augmentation Algorithm for Reoptimising the Flow in a Repairable Network After an Edge Failure

5.3 An Algorithm for Maximising the Throughput Flow of Oil After a Component Failure in Multi-Commodity Oil Production Networks

5.4 A High-Speed Control of Large and Complex Active Power Distribution Networks

Appendix A

6. The Dual Network Theorem for Static Flow Networks and Its Application for Maximising the Throughput Flow

6.1 Analysis of the Draining Algorithm for Maximising the Throughput Flow in Static Networks

6.2 The Dual Network Theorem for Static Flow Networks

6.3 Improving the Average Running Time of Maximising the Throughput Flow in the Dual Network

6.4 Application of the Dual Network Theorem for Determining the Maximum Throughput Flow in a Static Flow Network

6.5 Area of Application of the Proposed Throughput Flow Maximisation Algorithm

7. Reliability of the Throughput Flow: Algorithms for Determining the Throughput Flow Reliability

7.1 Probability that an Edge Will Be in a Working State on Demand

7.2 Probability of a Source-to-Sink Flow on Demand

7.3 Probability of a Source-to-Sink Flow on Demand, of Specified Magnitude

8. Reliability Networks

8.1 Series and Parallel Arrangement of the Components in a Reliability Network

8.2 Building Reliability Networks: Difference Between a Physical and Logical Arrangement

8.3 Complex Reliability Networks Which Cannot Be Represented As a Combination of Series and Parallel Arrangements

8.4 Evaluating the Reliability of Complex Systems

9. Production Availability of Repairable Flow Networks

9.1 Discrete-Event Solver for Determining the Production Availability of Repairable Flow Networks

9.2 A Fast Algorithm for Determining the Production Availability of Repairable Flow Networks

9.3 Comparing the Performance of Competing Network Topologies

10. Link Between Topology, Size and Performance of Repairable Flow Networks

10.1 A Software Tool for Analysis and Optimisation of Repairable Flow Networks

10.2 A Comparative Method for Improving the Performance of Repairable Flow Networks

10.3 Investigating the Impact of the Network Topology on the Network Performance

10.4 Investigating the Link Between Network Topology and Network Performance by Using Conventional Reliability Analysis

10.5 Degree of Throughput Flow Constraint

11. Topology Optimisation of Repairable Flow Networks and Reliability Networks

11.1 Theoretical Basis of the Proposed Method for Topology Optimisation

11.2 Topology Optimisation Algorithm

11.3 Solved Examples

11.4 Topology Optimisation of Reliability Networks

Appendix A

Appendix B

12. Repairable Networks with Merging Flows

12.1 The Need for Improving the Running Time of Discrete-Event Solvers for Repairable Flow Networks

12.2 An Algorithm with Linear Running Time, for Maximising the Flow in a Network with Merging Flows

12.3 Optimising the Topology of a Repairable Flow Network with Merging Flows to Minimise the Losses from Failures

13. Flow Optimisation in Non-Reconfigurable Repairable Flow Networks

13.1 Lost Flow Caused by Edge Failures

13.2 Resistance of a Path

13.3 Cyclic Paths: Necessary and Sufficient Conditions for Minimising the Lost Flow in Non-Reconfigurable Repairable Flow Networks

13.4 Guaranteeing Throughput Flow Associated with the Smallest Lost Flow Due to Edge Failures

13.5 Determining the Edge Flows which Minimise the Probability of Flow Disruption Caused by Edge Failures

14. Virtual Accelerated Life Testing of Repairable Flow Networks

14.1 Acceleration Stresses and Acceleration Life Models

14.2 Determining the Availability of a Repairable System

References

## Key Features

- Includes theory and practical examples to build a deep understanding of the issues
- Written by the leading scholar and researcher in this emerging field
- Features powerful software tools for analysis, optimization and control of repairable flow networks

## Readership

Students, researchers, and professionals in engineering, computing, and mathematics who work with networks

## Table of Contents

Dedication

Preface

1. Flow Networks – Existing Analysis Approaches and Limitations

1.1 Repairable Flow Networks and Static Flow Networks

1.2 Repairable Flow Networks and Stochastic Flow Networks

1.3 Networks with Disturbed Flows and Stochastic Flow Networks

1.4 Performance of Repairable Flow Networks

2. Flow Networks and Paths – Basic Concepts, Conventions and Algorithms

2.1 Basic Concepts and Conventions: Data Structures for Representing Flow Networks

2.2 Pseudo-Code Conventions Used in the Algorithms

2.3 Efficient Representation of Flow Networks with Complex Topology

2.4 Paths: Algorithms Related to Paths in Flow Networks

2.5 Determining the Smallest-Cost Paths from the Source

2.6 Topological Sorting of Networks Without Cycles

2.7 Transforming Flow Networks

3. Key Concepts, Results and Algorithms Related to Static Flow Networks

3.1 Path Augmentation in Flow Networks

3.2 Bounding the Maximum Throughput Flow by the Capacity of s–t Cuts

3.3 A Necessary and Sufficient Condition for a Maximum Throughput Flow in a Static Network: The Max-Flow Min-Cut Theorem

3.4 Classical Augmentation Algorithms for Determining the Maximum Throughput Flow in Networks

3.5 General Push-Relabel Algorithm for Maximising the Throughput Flow in a Network

3.6 Applications

3.7 Successive Shortest-Path Algorithm for Determining the Maximum Throughput Flow at a Minimum Cost

4. Maximising the Throughput Flow in Single- and Multi-Commodity Networks: Removing Parasitic Directed Loops of Flow in Networks Optimised by Classical Algorithms

4.1 Eliminating Parasitic Directed Loops of Flow in Networks Optimised by Classical Algorithms

4.2 A Two-Stage Augmentation Algorithm for Determining the Maximum Throughput Flow in a Network

4.3 A New, Efficient Algorithm for Maximising the Throughput Flow of the Useful Commodity in a Multi-Commodity Flow Network

4.4 Network Flow Transformation Along Cyclic Paths

5. Networks with Disturbed Flows Dual Network Theorems for Networks with Disturbed Flows: Reoptimising the Power Flows in Active Power Networks in Real Time

5.1 Reoptimising the Flow in Networks with Disturbed Flows After Edge Failures and After Choking the Edge Flows

5.2 A Fast Augmentation Algorithm for Reoptimising the Flow in a Repairable Network After an Edge Failure

5.3 An Algorithm for Maximising the Throughput Flow of Oil After a Component Failure in Multi-Commodity Oil Production Networks

5.4 A High-Speed Control of Large and Complex Active Power Distribution Networks

Appendix A

6. The Dual Network Theorem for Static Flow Networks and Its Application for Maximising the Throughput Flow

6.1 Analysis of the Draining Algorithm for Maximising the Throughput Flow in Static Networks

6.2 The Dual Network Theorem for Static Flow Networks

6.3 Improving the Average Running Time of Maximising the Throughput Flow in the Dual Network

6.4 Application of the Dual Network Theorem for Determining the Maximum Throughput Flow in a Static Flow Network

6.5 Area of Application of the Proposed Throughput Flow Maximisation Algorithm

7. Reliability of the Throughput Flow: Algorithms for Determining the Throughput Flow Reliability

7.1 Probability that an Edge Will Be in a Working State on Demand

7.2 Probability of a Source-to-Sink Flow on Demand

7.3 Probability of a Source-to-Sink Flow on Demand, of Specified Magnitude

8. Reliability Networks

8.1 Series and Parallel Arrangement of the Components in a Reliability Network

8.2 Building Reliability Networks: Difference Between a Physical and Logical Arrangement

8.3 Complex Reliability Networks Which Cannot Be Represented As a Combination of Series and Parallel Arrangements

8.4 Evaluating the Reliability of Complex Systems

9. Production Availability of Repairable Flow Networks

9.1 Discrete-Event Solver for Determining the Production Availability of Repairable Flow Networks

9.2 A Fast Algorithm for Determining the Production Availability of Repairable Flow Networks

9.3 Comparing the Performance of Competing Network Topologies

10. Link Between Topology, Size and Performance of Repairable Flow Networks

10.1 A Software Tool for Analysis and Optimisation of Repairable Flow Networks

10.2 A Comparative Method for Improving the Performance of Repairable Flow Networks

10.3 Investigating the Impact of the Network Topology on the Network Performance

10.4 Investigating the Link Between Network Topology and Network Performance by Using Conventional Reliability Analysis

10.5 Degree of Throughput Flow Constraint

11. Topology Optimisation of Repairable Flow Networks and Reliability Networks

11.1 Theoretical Basis of the Proposed Method for Topology Optimisation

11.2 Topology Optimisation Algorithm

11.3 Solved Examples

11.4 Topology Optimisation of Reliability Networks

Appendix A

Appendix B

12. Repairable Networks with Merging Flows

12.1 The Need for Improving the Running Time of Discrete-Event Solvers for Repairable Flow Networks

12.2 An Algorithm with Linear Running Time, for Maximising the Flow in a Network with Merging Flows

12.3 Optimising the Topology of a Repairable Flow Network with Merging Flows to Minimise the Losses from Failures

13. Flow Optimisation in Non-Reconfigurable Repairable Flow Networks

13.1 Lost Flow Caused by Edge Failures

13.2 Resistance of a Path

13.3 Cyclic Paths: Necessary and Sufficient Conditions for Minimising the Lost Flow in Non-Reconfigurable Repairable Flow Networks

13.4 Guaranteeing Throughput Flow Associated with the Smallest Lost Flow Due to Edge Failures

13.5 Determining the Edge Flows which Minimise the Probability of Flow Disruption Caused by Edge Failures

14. Virtual Accelerated Life Testing of Repairable Flow Networks

14.1 Acceleration Stresses and Acceleration Life Models

14.2 Determining the Availability of a Repairable System

References

## Details

- No. of pages:
- 320

- Language:
- English

- Copyright:
- © Elsevier 2013

- Published:
- 1st February 2013

- Imprint:
- Elsevier

- eBook ISBN:
- 9780123984067

- Hardcover ISBN:
- 9780123983961

## About the Author

### Michael Todinov

Prof. Todinov’s background is Engineering, Mathematics and Computer Science. He holds a PhD and a higher doctorate (DEng) from the University of Birmingham. His name is associated with key results in the areas: Reliability and Risk, Flow networks, Probability, Statistics of inhomogeneous media, Theory of phase transformations, Residual stresses and Probabilistic fatigue and fracture.

M.Todinov pioneered research on: the theory of repairable flow networks and networks with disturbed flows, risk-based reliability analysis - driven by the cost of system failure, fracture initiated by flaws in components with complex shape, reliability dependent on the relative configurations of random variables and optimal allocation of a fixed budget to achieve a maximal risk reduction.

A sample of M.Todinov’s results include: introducing the hazard stress function for modelling the probability of failure of materials and deriving the correct alternative of the Weibull model; stating a theorem regarding the exact upper bound of properties from multiple sources and a theorem regarding variance of a distribution mixture; the formulation and proof of the necessary and sufficient conditions of the Palmgren-Miner rule and Scheil’s additivity rule; deriving the correct alternative of the Johnson-Mehl-Avrami-Kolmogorov equation and stating the dual network theorems for static flows networks and networks with disturbed flows.

### Affiliations and Expertise

Department of Mechanical Engineering and Mathematical Sciences, Oxford Brookes University, Oxford, UK

## Reviews

"...a solid publication targeted at graduate students and academia, as well as industry researchers...useful mainly due to the presentation of models and algorithms applicable to real network problems." **--IEEE Communications Magazine, December 2014**