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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This book covers the design and optimization of computer networks applying a rigorous optimization methodology, applicable to any network technology. It is organized into two parts. In Part 1 the reader will learn how to model network problems appearing in computer networks as optimization programs, and use optimization theory to give insights on them. Four problem types are addressed systematically – traffic routing, capacity dimensioning, congestion control and topology design.
Part 2 targets the design of algorithms that solve network problems like the ones modeled in Part 1. Two main approaches are addressed – gradient-like algorithms inspiring distributed network protocols that dynamically adapt to the network, or cross-layer schemes that coordinate the cooperation among protocols; and those focusing on the design of heuristic algorithms for long term static network design and planning problems.
Following a hands-on approach, the reader will have access to a large set of examples in real-life technologies like IP, wireless and optical networks. Implementations of models and algorithms will be available in the open-source Net2Plan tool from which the user will be able to see how the lessons learned take real form in algorithms, and reuse or execute them to obtain numerical solutions.
An accompanying link to the author’s own Net2plan software enables readers to produce numerical solutions to a multitude of real-life problems in computer networks (www.net2plan.com).
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Table of Contents
Title Page
Copyright
Dedication
About the Author
Preface
Reader Requisites and Intended Audience
Acknowledgments
Chapter 1: Introduction
1.1 What is a Communication Network?
1.2 Capturing the Random User Behavior
1.3 Queueing Theory and Optimization Theory
1.4 The Rationale and Organization of this Book
Part One: Modeling
Chapter 2: Definitions and Notation
2.1 Notation for Sets, Vectors and Matrices
2.2 Network Topology
2.3 Installed Capacities
2.4 Traffic Demands
2.5 Traffic Routing
References
Chapter 3: Performance Metrics in Networks
3.1 Introduction
3.2 Delay
3.3 Blocking Probability
3.4 Average Number of Hops
3.5 Network Congestion
3.6 Network Cost
3.7 Network Resilience Metrics
3.8 Network Utility and Fairness in Resource Allocation
3.9 Notes and Sources
3.10 Exercises
References
Chapter 4: Routing Problems
4.1 Introduction
4.2 Flow-Path Formulation
4.3 Flow-Link Formulation
4.4 Destination-Link Formulation
4.5 Convexity Properties of Performance Metrics
4.6 Problem Variants
4.7 Notes and Sources
4.8 Exercises
References
Chapter 5: Capacity Assignment Problems
5.1 Introduction
5.2 Long-Term Capacity Planning Problem Variants
5.3 Fast Capacity Allocation Problem Variants: Wireless Networks
5.4 MAC Design in Hard-Interference Scenarios
5.5 Transmission Power Optimization in Soft Interference Scenarios
5.6 Notes and Sources
5.7 Exercises
References
Chapter 6: Congestion Control Problems
6.1 Introduction
6.2 NUM Model
6.3 Case Study: TCP
6.4 Active Queue Management (AQM)
6.5 Notes and Sources
6.6 Exercises
References
Chapter 7: Topology Design Problems
7.1 Introduction
7.2 Node Location Problems
7.3 Full Topology Design Problems
7.4 Multilayer Network Design
7.5 Notes and Sources
7.6 Exercises
References
Part Two: Algorithms
Chapter 8: Gradient Algorithms in Network Design
8.1 Introduction
8.2 Convergence Rates
8.3 Projected Gradient Methods
8.4 Asynchronous and Distributed Algorithm Implementations
8.5 Non-Smooth Functions
8.6 Stochastic Gradient Methods
8.7 Stopping Criteria
8.8 Algorithm Design Hints
8.9 Notes and Sources
8.10 Exercises
References
Chapter 9: Primal Gradient Algorithms
9.1 Introduction
9.2 Penalty Methods
9.3 Adaptive Bifurcated Routing
9.4 Congestion Control using Barrier Functions
9.5 Persistence Probability Adjustment in MAC Protocols
9.6 Transmission Power Assignment in Wireless Networks
9.7 Notes and Sources
9.8 Exercises
References
Chapter 10: Dual Gradient Algorithms
10.1 Introduction
10.2 Adaptive Routing in Data Networks
10.3 Backpressure (Center-Free) Routing
10.4 Congestion Control
10.5 Decentralized Optimization of CSMA Window Sizes
10.6 Notes and Sources
10.7 Exercises
References
Chapter 11: Decomposition Techniques
11.1 Introduction
11.2 Theoretical Fundamentals
11.3 Cross-Layer Congestion Control and QoS Capacity Allocation
11.4 Cross-Layer Congestion Control and Backpressure Routing
11.5 Cross-Layer Congestion Control and Power Allocation
11.6 Multidomain Routing
11.7 Dual Decomposition in Non-Convex Problems
11.8 Notes and Sources
11.9 Exercises
References
Chapter 12: Heuristic Algorithms
12.1 Introduction
12.2 Heuristic Design Keys
12.3 Local Search Algorithms
12.4 Simulated Annealing
12.5 Tabu Search
12.6 Greedy Algorithms
12.7 GRASP
12.8 Ant Colony Optimization
12.9 Evolutionary Algorithms
12.10 Case Study: Greenfield Plan with Recovery Schemes Comparison
12.11 Notes and Sources
12.12 Exercises
References
Appendix A: Convex Sets. Convex Functions
A.1 Convex Sets
A.2 Convex and Concave Functions
A.3 Notes and Sources
Reference
Appendix B: Mathematical Optimization Basics
B.1 Optimization Problems
B.2 A Classification of Optimization Problems
B.3 Duality
B.4 Optimality Conditions
B.5 Sensitivity Analysis
B.6 Notes and Sources
References
Appendix C: Complexity Theory
C.1 Introduction
C.2 Deterministic Machines and Deterministic Algorithms
C.3 Non-Deterministic Machines and Non-Deterministic Algorithms
C.4 and Complexity Classes
C.5 Polynomial Reductions
C.6 -Completeness
C.7 Optimization Problems and Approximation Schemes
C.8 Complexity of Network Design Problems
C.9 Notes and Sources
References
Appendix D: Net2Plan
D.1 Net2Plan
D.2 On the Role of Net2Plan in this Book
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Introduction
Figure 1.1 Two telecommunication systems
Figure 1.2 Communication service built with dedicated telecommunication systems
Figure 1.3 Switching system (node)
Figure 1.4 Communication network example
Chapter 2: Definitions and Notation
Figure 2.1 Example. Topology with nodes, , links . End nodes of are . Outgoing links of node are , and incoming links of are
Figure 2.2 Example topology
Figure 2.3 (a) Multicast tree with origin and destination nodes . (b) Not a multicast tree. Note that has two input links
Figure 2.4 Example. Traffic matrix is created from the list of demands at the left
Figure 2.5 Bifurcated multicast example. A fraction of the traffic is delivered through multicast tree and node forwards two copies of the traffic, while a fraction is delivered through the tree and node is the one that forwards two copies of this traffic
Chapter 3: Performance Metrics in Networks
Figure 3.1 Node and link model
Figure 3.2 Average buffering delay estimation (3.3). bits, Mbps
Figure 3.3 Node and link blocking model
Figure 3.4 Blocking probability (single-class, Erlang-B).
Figure 3.5 Example. Evolution of blocking probability for two traffic classes of connection sizes , traversing a link (Poisson assumption). (a) , . (b) , ,
Figure 3.6 Concave costs example. Normalized wholesale Internet access prices (GigADSL) for different rates in Spain (p. 92, [18])
Figure 3.7 Simple service availability example
Figure 3.8 (a) 1+1 protection (dedicated), (b) 2:3 protection (shared)
Figure 3.9 Example. Three SRGs associated to three ducts that is estimated that can be accidentally cut
Figure 3.10 Example. Bandwidth allocation example. Demands are assigned a rate , respectively. Link capacities are equal to one
Figure 3.11 -Fair allocations in Figure 3.10, for different values
Figure 3.12 -Utility functions ((3.19), ,
Figure 3.13 Figure Exercise 3.10
Chapter 4: Routing Problems
Figure 4.1 Example. Routing not valid in networks implementing destination-based routing
Figure 4.2 Flow conservation constraint example for a demand and node . values are plotted next to incoming and outgoing links of the node. The node is not a source nor destination of the demand: the traffic of the demand entering and leaving the node are the same ()
Figure 4.3 Example. Potential problems when converting a feasible solution into a routing, for a demand with , from node 1 to node 8. The number in each link illustrates its value
Figure 4.4 Example. Flow conservation constraints for destination-link formulation
Figure 4.5 Routing table examples for destination node 4
Figure 4.6 Anycast unicast graph transformation example
Figure 4.7 Example. Routing of demand with traffic units (T.U). (a) integral routing and, (b) not an integral routing, since one of the paths does not carry an integer amount of traffic
Figure 4.8 ECMP splitting rule example. All the links have cost . 80 units of traffic are delivered from to , nodes 1, 4, and 7 split the traffic equally between two links
Figure 4.9 Example topology
Chapter 5: Capacity Assignment Problems
Figure 5.1 Concave cost evolution example, for different values
Figure 5.2 Example. Feasibility set for problems (5.1) and (5.3) in a network of two links, units of traffic in each, , network cost
Figure 5.3
Figure 5.4
Figure 5.5 Example of multi-path propagation
Figure 5.6 Example. Three node network and its capacity region , when a node cannot receive simultaneously traffic from two nodes (black) and when this constraint does not exist (black and gray)
Figure 5.7 Example of interference map
Figure 5.8
Figure 5.9 Capacity region of a network composed of two links with a common receiver node and , , and transmission powers from 0 to 100 mW
Chapter 6: Congestion Control Problems
Figure 6.1 Examples of utility functions, (a) inelastic source and (b) elastic source
Figure 6.2 Window-based congestion control. The average rate of a connection is limited by
Figure 6.3 TCP Reno slot window evolution
Figure 6.4 Duplicated ACKs example
Figure 6.5 Simplified AIMD evolution
Figure 6.6 Example. Connection (), RTT of 100 ms, connection (), RTT of 200 ms. Losses only occur in the (shared) bottleneck link. According to (6.10),
Figure 6.7 RED marking probability example. is the slope of the line
Figure 6.8
Chapter 7: Topology Design Problems
Figure 7.1 Example: node location plots
Figure 7.2 Node location trends: number of core nodes for different values of maximum node connectivity (
K
) and core node costs (
C
)
Figure 7.3 Example of topologies in an Abilene node set. , ,
Figure 7.4 Topology trends: number of links in the topology for different values of link cost per km (
C
)
Chapter 8: Gradient Algorithms in Network Design
Figure 8.1
Figure 8.2 Basic gradient algorithm iterations for , with , . Lipschitz constant , . Condition number of is . The method needs 290 iterations to arrive to a distance of units of the optimum
Figure 8.3
Figure 8.4
Figure 8.5 Application of the heavy-ball method with , to same example as Fig, 8.2 (same step also). The method needs 130 iterations (instead of 290) to arrive at a distance of units of the optimum
Chapter 9: Primal Gradient Algorithms
Figure 9.1 Example: Unstable routing. Demands , two potential routings each through links, or , of the same capacity. If the routing is non-bifurcated, the minimum delay routing is unstable for some initial conditions if and always unstable if
Figure 9.2 Test network. Link capacities
Figure 9.3 function for a link with
Figure 9.4 Evolution of Algorithm 4 in asynchronous case with signaling losses, . The upper graph plots the evolution for each path. The lower graph plots the cost evolution (9.7a). Optimum solutions are marked with squares on the right hand-side
Figure 9.5 Evolution of Algorithm 4 in the asynchronous case with signaling losses, , but limiting the routing changes to . The upper graph plots the evolution for each path. The lower graph plots the cost evolution (9.7a). Optimum solutions are marked with squares on the right hand-side
Figure 9.6 Evolution of Algorithm 4 in the asynchronous case with signaling losses, , , random signaling delay in the interval . The upper graph plots the evolution for each path. The lower graph plots the cost evolution (9.7a). Optimum solutions are marked with squares on the right hand-side
Figure 9.7 Evolution of Algorithm 5 in the asynchronous case with signaling losses, , . The upper graph plots the evolution for each path. The lower graph plots the network utility evolution (9.14a). Optimum solutions are marked with squares on the right-handside
Figure 9.8 Same example as Figure 9.7,
Figure 9.9 Evolution of Algorithm 5 in asynchronous case with signaling losses, diagonal scaling, and ,
Figure 9.10 Example figure
Figure 9.11 Evolution of Algorithm 6 in the asynchronous case with signaling losses, , . The upper graphs plot the and evolution for each link. The lower graph plots the network utility evolution (9.20a). Optimum solutions are marked with squares on the right-hand side
Figure 9.12 Same example as Figure 9.11, but capacity observations are subject to a noise uniformly chosen in the range of the true link capacity
Figure 9.13 Example network: uplink channels in a cell
Figure 9.14 Evolution of Algorithm 7 in the asynchronous case with signaling losses, . The upper graphs plot the and evolution for each link. The lower graph plots the network utility evolution (9.24). Optimum solutions are marked with squares on the right-hand side
Figure 9.15 Same example as Figure 9.14, with a heavy-ball variation (9.28) with
Chapter 10: Dual Gradient Algorithms
Figure 10.1
Figure 10.2 Evolution of Algorithm 9 in the asynchronous case with signaling losses, , . The upper graph plots the evolution for each path, the graph shows the traffic in each link, graph is the link weight evolution, and the lower graph plots the regularized cost . Optimum solutions are marked with squares on the right-hand side. Regularized costs are lower than the optimal occuring since in the first phase, some links are oversubscribed, and the solution is unfeasible
Figure 10.3 Same example as Figure 10.2, effect of reducing to
Figure 10.4 Same example as Figure 10.2, effect of signaling delay
Figure 10.5 Same example as Figure 10.2, effect of measurement error in idle link capacity of of link capacity
Figure 10.6 Evolution of Algorithm 10 in asynchronous case with signaling losses, , . The upper graph plots the resulting route evolution, the medium graph the queue sizes in the average number of packets, and the lower graph the average number of hops
Figure 10.8 Same example as Figure 10.6,
Figure 10.8 Same example as Figure 10.6,
Figure 10.9 Evolution of Algorithm 11 in the asynchronous case with signaling losses, . The upper graph plots the resulting injected traffics , medium graph the link multipliers and the lower graph the network utility to maximize
Figure 10.10 Evolution of Algorithm 12 in the asynchronous case with measurement noise, . The upper graph plots the resulting link capacities , the medium graph the TAs (link multipliers) and the lower graph network utility to maximize. Optimum solutions are marked with squares on the right-hand side
Figure 10.11 Same example as Figure 10.10,
Chapter 11: Decomposition Techniques
Figure 11.2 Evolution of Algorithm 16 in the asynchronous case with signaling losses, . Plots from upper to lower are the demand rates, routes evolution, queue sizes in average number of packets, and network utility. Optimum values are plotted as squares on the right-hand side
Figure 11.1 Evolution of Algorithm 15, (congestion control), (QoS split). Plots from upper to lower are the capacity splits, demand rates, and network utility. Optimum values are plotted as squares on the right-hand side
Figure 11.3 Evolution of Algorithm 17. Optimum values are plotted as squares in the right-hand side
Figure 11.4 Topology example. Cluster 1, nodes , Cluster 2, nodes , Cluster 3, nodes . Link capacity , offered traffic in file abilene_N12_E30_withTrafficAndClusters3.n2p
Figure 11.5 Evolution of Algorithm 18, . Plots from upper to lower are the traffic in the links, values, and total bandwidth consumed in the links (target to minimize). Optimum values are plotted as squares on the right-hand side
Figure 11.6 Evolution of Algorithm 19, decreasing step . Plots from upper to lower are link multipliers and primal and dual costs. Optimum cost is 210
Chapter 12: Heuristic Algorithms
Figure 12.1 NSFNET topology. The network has 21 bidirectional links, all with the same capacity ( Gbps). Total offered traffic is as in NSFNet_N14_E42.n2p file, normalized to sum 4 Tbps
Figure 12.2
Figure 12.3
Figure 12.4
Figure 12.5
Figure 12.6 Results from the ACO algorithm for a OSPF weight setting problem in a 5 minute run (270 ACO iterations), , , , , initial pheromone values . Best solution found has a congestion metric (0.51 is a lower bound)
Figure 12.7 Crossover examples in evolutionary algorithms
Figure 12.8 Results from an Evolutionary Algorithm for OSPF weight setting problem in a 5 minutes run (356 generations), , , . Best solution found has a congestion metric (0.51 is a lower bound)
Appendix A: Convex Sets. Convex Functions
Figure A.1 Convex and non-convex set examples in . Dark points belong to the set, white do not. (a) and (d) are convex sets. The rest are not, since it is possible to find segments with the end points in the set, but that have points not belonging to the set
Figure A.2 Convex hull examples in . Dark points belong to the set , gray points and black points are in
Figure A.3 (a) Set , (b) set , (c) set (not convex), (d) (convex)
Figure A.4 Convex function . For any two points , the segment is above the function graph . The function is not strictly convex, since for some points, for example , the segment between them is not strictly above the graph
Figure A.5 Concave function . For any two points , the segment is below the function graph . The function is not strictly concave, since for some points, for example , the segment between them is not strictly below the graph
Figure A.6 Convex and concave differentiable functions
Figure A.7 Subgradients in convex and concave functions
Figure A.8 Pointwise maximum and minimum properties for convex and concave functions
Figure A.9 Sub-level sets. (a) is convex and its sub-level set for is the interval . (b) is non-convex, and its sub-level set is the set , that is not convex
Figure A.10 Epigraph of a convex function
Appendix B: Mathematical Optimization Basics
Figure B.1 Example. . Local maximums are . Local minimums are . Global maximum is and global minimum
Figure B.2 Linear program example
Figure B.3 Convex program examples
Figure B.4 Non-convex program example
Figure B.5 Example of nonlinear problem (B.5) with two local minimum , with cost 4 and , with cost 1
Figure B.6 Integer program example
Figure B.7
Figure B.8 Optimum point in the interior of the feasibility set
Figure B.9 Point in the boundary of the feasibility set with one active constraint
Figure B.10 Point in the boundary of the feasibility set with two active constraints
Figure B.11
Figure B.12
Appendix C: Complexity Theory
Figure C.1 (a) Execution path in a deterministic machine. (b) Tree of execution paths created by a non-deterministic machine
Figure C.2 Polynomial reduction, reduces to ()
Figure C.3 Example equivalence between independent sets and cliques. Gray nodes highlight (a) an independent set of , (b) a clique of
Figure C.4 The dilemma. (a) , the widely accepted conjecture, (b) , then -complete problems are all
Figure C.5 Example of reduction of SAT to ISet. Representation of the problem: . Links among the nodes of the same clause are not shown. In gray, an independent set that reflects a SAT solution: ,
Figure C.6 PTAS reduction of optimization problem into ,
List of Tables
Chapter 3: Performance Metrics in Networks
Table 3.1 Typical MTBF MTTR and availability values [19]
Chapter 4: Routing Problems
Table 4.1 Convexity of some functions and constraints with respect to routing variables
Chapter 5: Capacity Assignment Problems
Table 5.1
Table 5.2 Discrete capacities available in different layer 1/2 technologies
Chapter 9: Primal Gradient Algorithms
Table 9.1 Case studies in Chapter 9
Chapter 10: Dual Gradient Algorithms
Table 10.1 Case studies in Chapter 9
Chapter 11: Decomposition Techniques
Table 11.1 Case studies in Chapter 11
Chapter 12: Heuristic Algorithms
Table 12.1 Case study results
Appendix A: Convex Sets. Convex Functions
Table A.1 Some basic convex functions
Table A.2 Convexity of the composite function
Appendix B: Mathematical Optimization Basics
Table B.1 Brute force enumeration for problem (B.7)
Appendix C: Complexity Theory
Table C.1 Case studies in Chapter 9
Table C.2 Complexity of some optimization problems of interest in network design
OPTIMIZATION OF COMPUTER NETWORKS - MODELING AND ALGORITHMS
A HANDS-ON APPROACH
Pablo Pavón Mariño
This edition first published 2016
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Library of Congress Cataloging-in-Publication Data
Names: Marino, Pablo Pavon, author.
Title: Optimization of computer networks : modeling and algorithms : a hands-on approach / Pablo Pavon Marino.
Description: Chichester, West Sussex, United Kingdom : John Wiley & Sons, Inc., [2016] | Includes bibliographical references and index.
Identifiers: LCCN 2015044522 (print) | LCCN 2016000694 (ebook) | ISBN 9781119013358 (cloth) | ISBN 9781119013334 (ePub) | ISBN 9781119013341 (Adobe PDF)
Subjects: LCSH: Network performance (Telecommunication)–Mathematical models. | Computer networks–Mathematical models. | Computer algorithms.
Classification: LCC TK5102.83 .M37 2016 (print) | LCC TK5102.83 (ebook) | DDC 004.601–dc23
LC record available at http://lccn.loc.gov/2015044522
A catalogue record for this book is available from the British Library.
To my sons, Pablo and Guille, and to my wife Victoria, the smiles of my life.
About the Author
Pablo Pavón Mariño is Associate Professor at the Universidad Politécnica de Cartagena (Spain) and Head of GIRTEL research group, MSc and Ph.D in Telecommunications, and MSc in Mathematics, with specialization in operations research. His research interests in the last 15 years are in optimization, planning, and performance evaluation of computer networks. He has more than a decade track as a lecturer in network optimization courses. He is author or co-author of more than 100 research papers in the field, published in top journals and international conferences, as well as several patents. He leads the Net2Plan open-source initiative, which includes the Net2Plan tool and its associated public repository of algorithms and network optimization resources (www.net2plan.com). Pablo Pavón has served as chair in international conferences like IEEE HPSR 2011, ICTON 2013 or ONDM 2016. He is Technical Editor of the Optical Switching and Networking journal, and has participated as Guest Editor in other journals such as Computer Networks, Photonic Network Communications, and IEEE/OSA Journal of Optical Communications and Networking.
Preface
Lesen Sie weiter in der vollständigen Ausgabe!
Lesen Sie weiter in der vollständigen Ausgabe!
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Lesen Sie weiter in der vollständigen Ausgabe!
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