Introduction to scheduling

Full of practical examples, Introduction to Scheduling presents the basic concepts and methods, fundamental results, and recent developments of scheduling theory. With contributions from highly respected experts, it provides self-contained, easy-to-follow, yet rigorous presentations of the material.

The book first classifies scheduling problems and their complexity and then presents examples that demonstrate successful techniques for the design of efficient approximation algorithms. It also discusses classical problems, such as the famous makespan minimization problem, as well as more recent advances, such as energy-efficient scheduling algorithms. After focusing on job scheduling problems that encompass independent and possibly parallel jobs, the text moves on to a practical application of cyclic scheduling for the synthesis of embedded systems. It also proves that efficient schedules can be derived in the context of steady-state scheduling. Subsequent chapters discuss scheduling large and computer-intensive applications on parallel resources, illustrate different approaches of multi-objective scheduling, and show how to compare the performance of stochastic task-resource systems. The final chapter assesses the impact of platform models on scheduling techniques.

From the basics to advanced topics and platform models, this volume provides a thorough introduction to the field. It reviews classical methods, explores more contemporary models, and shows how the techniques and algorithms are used in practice.

Author Notes

Yves Robert is a professor in the computer science laboratory at the École Normale Supérieure de Lyon in France. Dr. Robert is also a senior member of the Institut Universitaire de France.

Frédéric Vivien is a researcher at INRIA in France. Dr. Vivien's research interests include scheduling techniques and parallel algorithms for heterogeneous and distributed platforms.

Peter Brucker and Sigrid KnustJean-Claude König and Rodolphe GiroudeauSusanne AlbersUwe SchwiegelshohnClaire HanenOlivier Marchetti and Alix Munier-KordonOlivier Beaumont and Loris MarchalMatthieu Gallet and Yves Robert and Frédéric VivienPieiTe-Francois Duiot and Krzysztof Rzadca and Erik Saule and Denis TrystramBruno Gaujal and Jean-Marc VincentLionel Eyraud-Dubois and Amaud Legrand

Preface	p. xi
Acknowledgments	p. xvii
List of.Contributors	p. xix
1 On the Complexity of Scheduling	p. 1
1.1 Introduction	p. 1
1.2 Scheduling Models	p. 2
1.3 Processor (Machine) Scheduling	p. 6
1.4 Easy and Hard Problems	p. 11
1.5 Complexity Classification of Scheduling Problems	p. 18
References	p. 20
2 Approximation Algorithms for Scheduling Problems	p. 23
2.1 Introduction	p. 23
2.1.1 Approximation Algorithms	p. 24
2.1.2 Definitions	p. 25
2.1.3 Absolute Approximation Algorithms (Case pinf = pSUp)	p. 26
2.2 A Fully Polynomial-Time Approximation Scheme	p. 28
2.3 Introduction of Precedence Constraints	p. 31
2.3.1 Unbounded Number of Processors	p. 31
2.3.2 Bounded Number of Processors	p. 31
2.4 Introduction of Communication Delays	p. 32
2.4.1 Introduction	p. 32
2.4.2 Unbounded Number of Processors	p. 33
2.4.3 Limited Number of Processors	p. 37
2.4.4 Introduction of Duplication	p. 42
2.4.5 Large Communication Delays	p. 44
2.5 Conclusion	p. 47
References	p. 48
3 Online Scheduling	p. 51
3.1 Introduction	p. 51
3.2 Classical Scheduling Problems	p. 52
3.2.1 Makespan Minimization	p. 52
3.2.2 Flow Time Objectives	p. 57
3.2.3 Load Balancing	p. 60
3.3 Energy-Efficient Scheduling	p. 62
3.3.1 Power-Down Mechanisms	p. 63
3.3.2 Dynamic Speed Scaling	p. 67
3.4 Conclusion	p. 73
References	p. 73
4 Job Scheduling	p. 79
4.1 Introduction	p. 79
4.2 Single Machine Problems	p. 82
4.3 Makespan Problems on Parallel Machines	p. 86
4.4 Completion Time Problems on Parallel Machines	p. 91
4.5 Conclusion	p. 99
References	p. 100
5 Cyclic Scheduling	p. 103
5.1 Introduction	p. 103
5.2 Cyclic Scheduling and Uniform Constraints	p. 104
5.2.1 Common Features of Cyclic Scheduling Problems	p. 104
5.2.2 Uniform Task Systems	p. 106
5.2.3 Questions	p. 108
5.3 Periodic Schedules of Uniform Task Systems	p. 109
5.3.1 Properties of Periodic Schedules	p. 109
5.3.2 Critical Circuit of a Strongly Connected Graph	p. 112
5.3.3 Computation of an Optimal Periodic Schedule	p. 113
5.4 Earliest Schedule of Uniform Task Systems	p. 116
5.5 Periodic Schedules of Uniform Task Systems with Resource Constraints	p. 117
5.5.1 Which Periodicity?	p. 117
5.5.2 Complexity and Iteration Vectors	p. 118
5.5.3 Patterns and Iteration Vectors	p. 119
5.5.4 Decomposed Software Pipelining: A Generic Approach	p. 121
5.6 Dynamic Schedules	p. 124
5.7 Conclusion	p. 125
References	p. 126
6 Cyclic Scheduling for the Synthesis of Embedded Systems	p. 129
6.1 Introduction	p. 129
6.2 Problem Formulation and Basic-Notations	p. 131
6.2.1 Synchronous Dataflow Graphs	p. 131
6.2.2 Timed Weighted Event Graphs	p. 132
6.2.3 Problem Formulation	p. 133
6.3 Precedence Relations Induced by a Timed Marked WEG	p. 134
6.3.1 Characterization of Precedence Relations	p. 134
6.3.2 Timed Event Graphs	p. 135
6.3.3 Equivalent Places	p. 135
6.4 Unitary WEGs	p. 137
6.4.1 Definitions	p. 138
6.4.2 Normalization of a Unitary WEG	p. 139
6.4.3 Expansion of a Unitary Timed Marked WEG	p. 141
6.4.4 Relationship between Expansion and Normalization	p. 145
6.5 Periodic Schedule of a Normalized Timed Marked WEG	p. 147
6.5.1 Periodic Schedules	p. 148
6.5.2 Properties of Periodic Schedules	p. 148
6.5.3 Existence of Periodic Schedules	p. 150
6.5.4 Optimal Periodic Schedule	p. 152
6.6 Conclusion	p. 154
References	p. 154
7 Steady-State Scheduling	p. 159
7.1 Introduction	p. 159
7.2 Problem Formulation	p. 161
7.2.1 Platform Model	p. 161
7.2.2 Applications	p. 162
7.3 Compact Description of a Schedule	p. 163
7.3.1 Definition of the Allocations	p. 164
7.3.2 Definition of Valid Patterns	p. 166
7.4 From Allocations and Valid Patterns to Schedules	p. 167
7.4.1 Conditions and Weak Periodic Schedules	p. 167
7.4.2 Weak Periodic Schedules and Cyclic Scheduling	p. 169
7.5 Problem Solving in the General Case	p. 172
7.5.1 Existence of a Compact Solution	p. 173
7.5.2 Resolution with the Ellipsoid Method	p. 175
7.5.3 Separation in the Dual Linear Program	p. 176
7.6 Toward Compact Linear Programs	p. 178
7.6.1 Introduction	p. 178
7.6.2 Efficient Computation of Valid Patterns under the Bidirectional One-Port Model	p. 179
7.6.3 Efficient Computation of Allocations	p. 182
7.7 Conclusion	p. 184
References	p. 185
8 Divisible Load Scheduling	p. 187
8.1 Introduction	p. 187
8.1.1 Motivating Example	p. 188
8.1.2 Classical Approach	p. 188
8.2 Divisible Load Approach	p. 191
8.2.1 Bus-Shaped Network	p. 192
8.2.2 Star-Shaped Network	p. 195
8.3 Extensions of the Divisible Load Model	p. 201
8.3.1 Introducing Latencies	p. 201
8.3.2 Multi-Round Strategies	p. 204
8.3.3 Return Messages	p. 214
8.4 Conclusion	p. 216
References	p. 217
9 Multi-Objective Scheduling	p. 219
9.1 Motivation	p. 220
9.1.1 Once Upon a Time	p. 220
9.1.2 Diversity of Objectives	p. 221
9.1.3 Motivating Problems	p. 222
9.1.4 Summary of Results on Single Objective Problems	p. 223
9.1.5 Beyond the Scope of This, This Chapter	p. 223
9.1.6 Chapter Organization	p. 224
9.2 What Is Multi-Objective Optimization?	p. 225
9.3 Overview of the Various Existing Approaches	p. 228
9.3.1 Algorithms Building One Trade-off Solution	p. 228
9.3.2 Complexity Issues	p. 230
9.4 Zenith Approximation on MaxAndSum	p. 233
9.5 Pareto Set Approximation on EfficientReliable	p. 235
9.5.1 Motivation	p. 235
9.5.2 Definition of Pareto Set Approximation	p. 236
9.5.3 The Thresholding Approach	p. 237
9.6 Fairness as Multi-Objective Optimization	p. 241
9.6.1 The Meaning of Fairness	p. 241
9.6.2 Axiomatic Theory of Fairness	p. 242
9.6.3 Application to Two Agent MinSum	p. 243
9.6.4 Problems with Different Objective Functions	p. 245
9.6.5 Aggregative Fairness	p. 246
9.7 Conclusion	p. 247
References	p. 248
10 Comparisons of Stochastic Task-Resource Systems	p. 253
10.1 Motivation	p. 253
10.2 Task-Resource Models	p. 255
10.2.1 Static Systems	p. 255
10.2.2 Dynamic Systems	p. 256
10.3 Stochastic Orders	p. 257
10.3.1 Orders for Real Random Variables	p. 258
10.3.2 Orders for Multidimensional Random Variables	p. 263
10.3.3 Association	p. 264
10.4 Applications to Static Problems	p. 265
10.4.1 The 1 ¿Ci Problem, Revisited	p. 266
10.4.2 PERT Graphs	p. 267
10.5 Applications to Dynamic Systems	p. 268
10.5.1 Single Queues	p. 269
10.5.2 Networks of Queues	p. 272
10.5.3 Stochastic Comparisons and Simulation Issues	p. 275
References	p. 279
11 The Influence of Platform Models on Scheduling Techniques	p. 281
11.1 Introduction	p. 281
11.2 Platform Modeling	p. 282
11.2.1 Modeling the Topology	p. 282
1l.2.2 Modeling Point-to-Point Communication Time	p. 284
11.2.3 Heterogeneity	p. 288
11.2.4 Modeling Concurrent Communications	p. 289
11.2.5 Interaction between Communication and Computation	p. 291
11.3 Scheduling Divisible Load	p. 292
11.3.1 Single Round	p. 293
11.3.2 Multi-Round	p. 296
11.4 Iterative Algorithms on a Virtual Ring	p. 298
11.4.1 Problem Statement	p. 299
11.4.2 Complete Homogeneous Platform	p. 299
11.4.3 Complete Heterogeneous Platform	p. 300
11.4.4 Arbitrary Heterogeneous Platform	p. 301
11.5 Data Redistribution	p. 302
11.5.1 The Matching Approach	p. 304
11.5.2 The Elastic Flows Approach	p. 306
11.6 Conclusion	p. 307
References	p. 307
Index	p. 311

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