Deadline Scheduling for Real-Time Systems - EDF and Related Algorithms (The Springer International Series in Engineering and Computer Science)

DEADL~DULING

FOR REAL-TIME SYSTEMS EDF and Related Algoritluns

THE KLUWER INTERNATIONAL SERIES IN ENGINEERING AND COMPUTER SCIENCE REAL-TIME SYSTEMS Consulting Editor John A. Stankovic HARD REAL-TIME COMPUTING SYSTEMS: Predictable Scheduling Algorithms and Applications, by Giorgio C. Buttazzo, ISBN: 0-7923-9994-3 REAL-TIME DATABASE AND INFORJ\iATION RESEARCH ADVANCES, by Azer Bestavros, Victor Wolfe, ISBN: 0-7923-8011-8 REAL-TIME SYSTEMS: Design Principles for Distributed Embedded Applications, by Hennann Kopetz, ISBN: 0-7923-9894-7 REAL-TIME DATABASE SYSTEMS: Issues alld Applications, edited by Azer Bestavros, Kwei-Jay Lin and Sang Hyuk Son, ISBN: 0-7923-9897-1 FAULT-TOLERANT REAL-TIME SYSTEMS: The Problem ofReplica Determinism, by Stefan Poledna, ISBN: 0-7923-9657-X RESPONSIVE COMPUTER SYSTEMS: Steps Toward Fault-Tolerant Real-Time Systems, by Donald Fussell and Miroslaw Malek, ISBN: 0-7923-9563-8 IMPRECISE AND APPROXIMATE COMPUTATION, by Swaminathan Natarajan, ISDN: 0-7923-9579-4 FOUNDATIONS OF DEPENDABLE COMPUTING: System Implementation, edited by Gary M. Koob and Clifford G. Lau, ISBN: 0-7923-9486-0 FOUNDATIONS OF DEPENDABLE COMPUTING: Paradigms for Dependable Applications, edited by Gary M. Koob and Clifford G. Lau, ISBN: 0-7923-9485-2 FOUNDATIONS OF DEPENDABLE COMPUTING: Models and Frameworks for Depelldable Systems, edited by Gary M. Koob and Clifford G. Lau, ISBN: 0-7923-9484-4 THE TESTABILITY OF DISTRIBUTED REAL-TIME SYSTEMS, Werner Schlitz; ISBN: 0-7923-9386-4 A PRACTITIONER'S HANDBOOK FOR REAL-TIME ANALYSIS: Guide to Rate llfollotollic Analysis for Real-Time Systems, Carnegie Mellon University (Mark Klein, Thomas Ralya, Bill Pollak, Ray Obenza, Michale Gonzalez Harbour); ISBN: 0-7923-9361-9 FORMAL TECHNIQUES IN REAL-TIME FAULT-TOLERANT SYSTEMS, J. Vytopil; ISBN: 0-7923-9332-5 SYNCHRONOUS PROGRAMMING OF REACTIVE SYSTEMS, N. Halbwachs; ISBN: 0-7923-9311-2 REAL-TIME SYSTEMS ENGINEERING AND APPLICATIONS, M. Schiebe, S. Pferrer; ISBN: 0-7923-9196-9 SYNCHRONIZATION IN REAL-TIME SYSTEMS: A Priority Inheritance Approach, R. Rajkumar; ISBN: 0-7923-9211-6 CONSTRUCTING PREDICTABLE REAL TIME SYSTEMS, W. A. Halang, A. D. Stoyenko; ISBN: 0-7923-9202-7 FOUND ATIONS OF REAL-TIME COMPUTING: Formal Specificatiolls and Met/lOds, A. M. van Tilborg, G. M. Koob; ISBN: 0-7923-9167-5 FOUNDATIONS OF REAL-TIME COMPUTING: Scheduling alld Resource Managemellt, A. M. van Tilborg, G. M. Koob; ISBN: 0-7923-9166-7 REAL-TIME UNIX SYSTEMS: Design alldApplication Guide, B. Furht, D. Grostick, D. Gluch, G. Rabbat, J. Parker, M. McRoberts, ISBN: 0-7923-9099-7

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EDF and Related Algorithms by ~~! .

Jobn A. Stankovic

(i..i 3}'!; 912-

University of Virginia Charlottesville, Virginia, USA Marco Spuri Merloni Elettrodomestici Spa Fabriano, Italy Kritbi Ramamritbam University ofMassachusetts Amherst, Massachusetts, USA Giorgio C. Buttazzo Scuola Superiore S. Anna Pisa, Italy

KLUWER ACADEMIC PUBLISHERS Boston / Dordrecht / London

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Distributors for North, Central and South America: Kluwer Academic Publishers 101 Philip Drive Assinippi Park Norwell, Massachusetts 02061 USA Telephone (781) 871-6600 Fax (781) 871-6528 E-Mail Distributors for all other countries: Kluwer Academic Publishers Group Distribution Centre Post Office Box 322 3300 AH Dordrecht, THE NETHERLANDS Telephone 31 78 6392 392 Fax 31 786546474 E-Mail

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Library of Congress Cataloging-in-Publication Data Deadline scheduling for real-time systems: EDF and related algorithms / by John A. Stankovic ... [et al.]. p. cm. -- (The Kluwer international series in engineering and computer science; SECS 460. Real-time systems) Includes bibliographical references and index. ISBN 0-7923-8269-2 (alk. paper) I. Real-time data processing. 2. Computer algorithms. 3. Scheduling. I. Stankovic, John. A. II. Series: Kluwer international series in engineering and computer science ; SECS 460. III. Series; Kluwer international series in engineering and computer science. Real-time systems. QA76.54.043 1998 98-39004 004' .33 -- dc21

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Copyright © 1998 by Kluwer Academic Publishers All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any fonn or by any means, mechanical, photocopying, recording, or otherwise, without the prior written pennission of the Assinlppi Park, Norwell, publisher, Kluwer Academic Publishers, 101 Philip Drive, Assimppi Massachusetts 02061 Printed on

acid~free

paper.

Printed in the United States of America The cover of this book was designed by Curtis Buyrn.

CONTENTS

LIST OF FIGURES

ix

LIST OF TABLES

xiii

PREFACE

xv

1

INTRODUCTION 1.1 1.2 1.3 1.4 1.5

Real-Time Systems Common Misconceptions A Typical Example of a Real-Time Application Purpose of this Book FormaL of the Book

1 1 4 4 7 8

REFERENCES

11

2

13 14 19 22

TERMINOLOGY AND ASSUMPTIONS 2.1 2.2 2.3

Task Models, Assumptions and Notation Static versus Dynamic Scheduling Metrics

REFERENCES

25

3

FUNDAMENTALS OF EDF SCHEDULING

27

3.1 3.2 3.3

28 31 61

Optimality on Uni-Processor Systems Feasibility Analysis Summary

DEADLINE SCHEDULING FOR REAL-TIME SYSTEMS

Vl

REFERENCES 4

RESPONSE TIMES UNDER EDF SCHEDULING 4.1 4.2 4.3 4.4

Finding Local Maxima Deadline Busy Periods Algorithm Description Extended Task Modeling

4.5 4.6

Case Study Summary

REFERENCES 5

PLANNING-BASED SCHEDULING 5.1

Preliminaries: Load, Metrics, Value Functions

5.2 5.3 5.4 5.5

Steps in a Dynamic Planning-Based Scheduling Approach Algorithms for Dynamic Planning Timing of the Planning Implementing Planning-Based Scheduling

5.6 5.7

Dispatching Jobs in a Planning-based Schedule Summary

REFERENCES 6

EDF SCHEDULING FOR SHARED RESOURCES 6.1 6.2 6.3 6.4 6.5 6.6 6.7

The Nature of Resources and the Resulting Scheduling Problems The Priority Inversion Problem The Priority Inheritance Protocol The Dynamic Priority Ceiling Protocol The Stack Resource Policy Resource Scheduling in Planning-based Schedulers Summary

REFERENCES

63

67

68 71 73

74 82

83

85

87 91

98 101 111 113 114 116 117

121 122 125 127

HO 142 144

147 149

Contents

7

Vll

PRECEDENCE CONSTRAINTS AND SHARED RESOURCES 7.1 7.2 7.3 7.4 7.5

Scheduling Dependent Tasks with EDF The ;-.ration of Quasi-Normality Integration of Shared Resources and Precedence Extended Task Model Summary

REFERENCES 8

151 152 155 158 161 164 167

APERIODIC TASK SCHEDULING

169

8.1 8.2 8.3 8.4 8.5 8.6 8.7

170

Dynamic Priority Exchange server Dynamic Sporadic Server Total Bandwidth Server Earliest Deadline Late server Improved Priority Exchange server Performance Results Summary

175 179 182 187 190 193

REFERENCES

195

9

197 198 201 223 224

DISTRIBUTED SCHEDULING - PART I 9.1 9.2 9.3 9.4

Distributed Systems - An Overview Holistic Scheduling Based on EDF Performance Summary

REFERENCES

225

10 DISTRIBUTED SCHEDULING - PART II

229

10.1 The Spring Complex Task Set Allocation and Scheduling Algorithm 10.2 Focussed Addressing and Bidding 10.3 Summary

REFERENCES

229 246 256 259

viii

DEADLINE SCHEDULING FOR REAL-TIME SYSTEMS

11 SUMMARY AND OPEN QUESTIONS 11.1 Summary 11.2 Open Questions

263 263 265

REFERENCES

267

INDEX

269

LIST OF FIGURES

Chapter 1 1.1

A Fly-By-Wire Flight Control System.

5

Chapter 2 2.1

Example of EDF schedule.

17

Chapter 3 3.1 3.2 3.3

A set of three real-time jobs. EDF schedule (with time overflow) of the job set of Figure 3.1. EDF schedule of two periodic tasks with different initial phas-

33 34

~.

~

3.4 3.5

EDF schedule of two tasks: a sporadic one and a periodic one. EDF schedule of periodic tasks with deadlines different from their periods. 3.6 Pseudo-code of the algorithm for the feasibility analysis of hybrid task sets. 3.7 Worst case arrival paUern for a task set with release jitter. 3.8 Worst arrival pattern of sporadically periodic tasks. 3.9 Non-preemptive non-idling EDF schedule of two jobs. 3.10 Non-preemptive idling schedule of two jobs. 3.11 Busy period preceding an overflow ill a non-preemptive nonidling EDF schedule.

40 44 50 5'2 53 58 58 60

Chapter 4 4.1 4.1

Busy period preceding an instance completion time. Synchronous arrival pattern possibly giving the worst case response time for task Ti.

69

69

x

DEADLINE SCHEDULING FOR REAL-TIME SYSTEMS

4.2 4.3 4.4 4.5

Pseudo-code of the algorithm for the computation of the maximum deadline busy period lengths. EDF schedule with deadline tolerance. Arrival pattern for the evaluation of a task T; Ti'S 's response time. Local response time maxima when non-preemptive EDF scheduling is assumed.

73 75 76 79

Chapter 5

5.1

(a) Feasible schedule v.;ith Earliest Deadline First, in normal load condition. (b) Overload with domino effect due to the arrival of job Jo. 5.2 Utility functions that can be associated with a job to describe its importance. 5.3 No optimal on-line optimal algorithms exist in overload conditions, since the schedule that maximizes r depends on the knowledge of future arrivals. 5.4 Situation in which EDF has an arbitrarily small competitive fador. 5.5 Scheduling schemes for handling overload situations. a. Best Effort. b. Admission Control. c. Robust. 5.6 The RED acceptance test. 5.7 Basic guarantee algorithm.

88

94

95

97 102 105 109

Chapter 6

6.1 6.2 6.3 6.4 6.5 6.6

Priority inversion under EDF scheduling. Priority inheritance under EDF scheduling. Different relative priorities under EDF scheduling. Example of transitive blocking. Anomalous push-through blocking. Algorithm for the computation of blocking critical sections.

126 127 130 134 135 136

Chapter 7

7.1 7.2 7.3

The schedule of jobs Jk and J j violates EDF*. The new schedule of Jk and Jj EDF* compliant. Example of a not normal schedule produced by EDF and a resource access protocol.

154 154 157

List of Figures

7.4 7.5

A situatio n in which an EDF schedul er without priority inheritance violates quasi-n ormality and precede nce constrai nts. Algorith m of relative deadline modification.

Xl

160 163

Chapte r 8

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9

Dynami c Priority Exchan ge server example . DPE server schedulability. DPE server resource reclaiming. Dynami c Sporadi c Server example . Total Bandwi dth Server example . Availability function under ED L. a) Idle times available at time t = 8 under EDL. b) Schedul e of the aperiod ic request with the EDL server. Improve d Priority Exchang e server example . Perform ance of dynami c server algorith ms.

171 173 175 176 180 183 185 188 192

Chapte r 9

9.1 9.2 9.3

9.4 9.5 9.6 9.7

End-to- end comput ation delay compon ents. Assume d network topology. a) Busy period precedin g an instance complet ion time. b) Arrival pattern possibly giving the worst case respons e time for task i. Local network scenario for the evaluati on of message worst case commu nication delays. Maximu m delayed token visits at host process or p when full Timed Token protoco l is assumed . Maximu m delayed token visits at host process or p when restricted Timed Token protoco l is assumed . Compon ents of the end-to-e nd comput ation delay.

202 204

209 212 216 217 222

Chapte r 10

10.1 Structu re of Two Periodic Tasks. 10.2 Subtask Commu nication Graph. 10.3 Assignm ent and Schedul e for the Two Periodic Tasks. Chapte r 11

232 239 244

LIST OF TABLES

Chapter 1 Chapter 2 Chapter 3

3.1 3.2

Loading factor computation for the job set of Figure 3.1. Computation of the synchronous lJusy period length for the task set of Figure 3.5.

32 48

Chapter 4

4.1 4.2

List of semaphores and locking pattern for the GAP task set. GAP task set parameters.

82 83

Chapter 5 Chapter 6

6.1 6.2

Job parameters for example Structure of accesses to resources.

123 138

Chapter 7 Chapter 8

8.1

Idle times under EDL.

Chapter 9

184

XlV

Chapter 10 Chapter 11

DEADLINE SCHEDULING FOR REAL-TIME SYSTEMS

PREFACE

Many real-time systems rely on static scheduling algorithms. This includes cyclic scheduling, rate monotonic scheduling and fixed schedules created by offline scheduling techniques such as dynamic programming, heuristic search, and simulated annealing. However, for many real-time systems static scheduling algorithms are quite restrictive and inflexible. For example, highly automated agile manufacturing, command, control and communications, and disLrilJuLed real-time multimedia applications all operate over long lifetimes and in highly non-deterministic environments. Dynamic real-time scheduling algorithms are more appropriate for these systems and are used in such systems. Many of these algorithms are based on earliest deadline first (EDF) policies. There exists a wealth of literature on EDF based scheduling with many extensions to deal with sophisticated issues such as precedence constraints, resource requirements, system overload, multi-processors, and distributed systems. This book aims at collecting a significant body of knowledge on EDF scheduling for real-time systems, but it does not try to be all inclusive (the literature is just too extensive). The book primarily presents the algorithms and associated analysis, but guidelines, rules, and implementation considerations are also discussed especially for the more complicated situations where mathematical analysis is difficult. In general, it is very difficult to codify and taxonomize scheduling knowledge because there are many performance metrics, task characteristics, and system configurations. Also, adding to the complexity is the fact that a variety of algorithms have been designed for different combinations of these considerations. In spite of the recent advances there are still gaps in the solution space and there is a need to integrate the available solutions. For example, a list of issues to consider includes:

•

preemptive versus non-preemptive tasks,

•

uni-processors versus multi-processors,

XVI

DEADLINE SCHEDULING FOR REAL- TIME SYSTEMS

•

using EDF at dispatch time versus EDF-based planning,

•

precedence constraints among tasks,

•

resource constraints,

•

periodic versus aperiodic versus sporadic tasks,

•

scheduling during overload,

•

fault tolerance requirements, and

•

providing guarantees and levels of guarantees (meeting quality of service requirements) .

Chapter 1 defines real-time systems and gives a brief example of a real-time system. Chapter 2 contains the terminology and assumptions used throughout the book. The fundamental results of EDF scheduling for independent tasks are first presented in Chapters 3 and 4. These Chapters include results on preemption, non-preemption, uni-processors and multi-processors. The overall approach taken in this book is to consider preemption and non-preemption and uni-processors and multi-processors throughout the book and not as separate Chapters. Using EDF in planning mode is discussed in Chapter 5. Chapter 6 discusses results and algorithms involving tasks having general resource requirements. Chapter 7 presents results on scheduling tasks with precedence constraints and resource requirements. Chapter 8 considers problems where periodic and aperiodic tasks must both be scheduled. Scheduling for distributed systems is presented in Chapters 9 and 10. Chapter 11 summarizes the book and discusses open issues. This book should be of interest to researchers, real-time system designers, and for instructors and students either as a focussed course on deadline based scheduling for real-time systems, or, more likely, as part of a more general course on real-time computing. While reading the Chapters in order from 1 through 11 is recommended, this is not necessary. After reading Chapters 1, 2, 3, and 4, a reader can move to any of the other Chapters in any order, depending on his or her interest. One exception to this rule is that Chapter 6 should be read before Chapter 7. Special thanks go to Sang Son and Worthy Martin for helpful comments on early versions of this book.

1 INTRODUCTION

Over the last 10 years there has been an explosion of real-time scheduling results. Many of these results are based on either the rate monotonic (RM) or earliest deadline first (EDF) scheduling algorithms. The RM results have been collected and presented in an excellent book [3]. To date, a similar effort has been lacking for EDF. Since EDF has many valuable properties, it is important to address EDF in a comprehensive manner. Consequently, this book presents a compendium of results on earliest deadline first (EDF) scheduling for realtime systems. The simplest results presented utilize pure EDF scheduling. As system situations become more complicated, EDF is used as a key ingredient, but is combined with other solutions, e.g., to deal with shared resources. This Chapter defines and discusses real-time systems, discusses several common misconceptions, presents an example of a real-time application, and gives a detailed purpose and outline for the book.

1.1

REAL-TIME SYSTEMS

Real-time systems are defined as those systems in which the correctness of the system depends not only on the logical result of the computation, but also on the time at which the results are produced [6]. Many real-time systems are characterized by the fact that severe consequences will result if timing as well as logical correctness properties of the system are not satisfied. Typically, a real-time system consists of a controlling system and a controlled system. In an automated factory, the controlled system is the factory floor with its robots, assembling stations, and the assembled parts, while the controlling system is the

2

CHAPTER 1

computer and human interfaces that manage and coordinate the activities on the factory floor. Thus, the controlled system can be viewed as the environment with which the computer interacts. The controlling system interacts with its environment based on the information available about the environment from various sensors attached to it. It is imperative that the state of the environment, as perceived by the controlling system, be consistent with the actual state of the environment. Otherwise, the effects of the controlling systems' activities may be disastrous. Hence, periodic monitoring of the environment as well as timely processing of the sensed information is necessary. Timing correctness requirernenLs in a real-time system arise because of the physical impact of the controlling systems' activities upon its environment. For example, if the computer controlling a robot does not command it to stop or turn on time, the robot might collide with another object on the factory floor. Needless to say, such a mishap can result in a major catastrophe. Real-time systems span many application areas. In addition to automated factories, applications can be found in control of automobile engines, avionics, undersea exploration, process control, robot and vision systems, military command and control, and space stations. In other words, the complexity of real-time systems spans the gamut from very simple control of laboratory experiments, to process control applications, to very complicated projects such as a space station. Recently, the need to process continuous streams of audio and video data has given rise to exciting new possibilities for real-time applications. Timing constraints for tasks can be arbitrarily complicated, but the most common timing constraints for tasks are either periodic or aperiodic. An aperiodic task has a deadline by which it must finish or start, or it may have a constraint on both start and finish times. In the case of a periodic task, a period might mean 'once per time interval T' or 'exactly T units apart'. Low-level application tasks, such as those that process information obtained from sensors or those that activate elements in the environment, typically have stringent timing constraints dictated by the physical characteristics of the environment. A majority of sensory processing is periodic in nature. A radar that tracks flights and produces data at a fixed rate is one example. A temperature monilor of a nuclear reactor core should be read periodically to detect any changes promptly. Some of these periodic tasks may exist from the point of system initialization and remain permanent, while others may come into existence dynamically. The temperature monitor is an instance of a permanent

Introduction

3

task. An example of a dynamically created task is a (periodic) task that monitors a particular flight; this comes into existence when the aircraft enters an air traffic control region and ceases to exist when the aircraft leaves the region. More complex types of timing constraints also occur. For example, spray painting a car on a moving conveyor must be started after time t j and completed before time t 2 . Aperiodic requirements can arise from dynamic events such as an object falling in front of a moving robot, or a human operator pushing a button on a console. In addition, time related requirements may also be specified in indirect terms. For example, a value may be attached to the completion of each task where the value may increase or decrease with time; or a value may be placed on the quality of an answer whereby an inexact but fast answer might be considered more valuable than a slow but accurate answer'. In other situations, missing X deadlines might be tolerated, but missing X + 1 deadlines can't be tolerated. This raises the question of wll 1 , C

I

=4

I

18

t

-~

I t

C =4 C 2 =4

!

---~_L__~_---'---_ _

____o,,~

12

Figure 3.1

I

I

"

A set of three real-time jobs.

Theorem 3.5 (Spuri) Each set of real-time jobs is feasibly scheduled by EDF if and only if uS; 1. Proof. "If": Assume there is an overflow, that is, a deadline miss, at time t. The overflow must be preceded by a cpu busy period, that is, a period of

continuous processor utilization, in which only jobs with deadlines less than t are executed. Let t2 = t and t 1 be the last instant preceding t such that there are no pending execution requests of jobs released before t 1 and having deadlines less than or equal to t. Both t 1 and t 2 are well defined. See Figure 3.2 for an example. In particular, after t1, which must be the release time of some job, the processor is allocated to jobs released after t 1 and having deadlines less than t 2 . Since there is an overflow at t2, the amount of cpu time demanded in the interval [t 1 , t2) must be greater than the interval itself, that is

L

Ck

> (t 2 - td·

tl~rk,dk$t2

It follows that

hence

u> 1, a contradiction.

34

CHAPTER

3

'I' I

I

_C_I_=_4_bL

~~

-'------'-n_---"------t

,

18

C2 = 4

---__t=======~L___l_-----'

___o_~

12

I

lME OVERFLOW

.----.--.-, ,.-------.-----.,1 , , I o

2

3

Figure 3.2

5

6

7

8

9

'.>

10

II

•"; , Yo 12

13

b,----,---"---.,--.,----.,_,.--,_. .

>-

14

15

16

17

18

19

20

21

22

,

EDF schedule (with time overflow) of the job set of Figure 3.1.

"Only If": Since the schedule is feasible, the amount of time demanded in each interval of time must be less than or equal to the length of the interval, that is

It follows that

hence uS1.

o Note that the result just shown also confirms the optimality of the EDF algorithm for uni-processor systems. From this point of view the theorem is equivalent to that by Dertouzos, previously described in Section 3.1. The main outcome of the theorem, however, is that the problem of assessing the feasibility of a task set is equivalent to the problem of computing the loading factor of the same task set. This fact is used in the following sections to show several other resul ts.

35

Fundamentals oj EDF Scheduling

3.2.2

Synchronous and Asynchronous Periodic Tasks

As already mentioned, the first researchers to address the feasibility analysis of a task set under EDF scheduling were Liu and Layland [21]. In their remarkable work, the reference model consists of synchronous independent periodic tasks (i.e., all tasks share the same start time, which, without loss of generality, can be assumed to be 0), and with relative deadlines equal to their respective periods (i.e., Vi, D i = T;). With this hypothesis, a necessary and sufficient condition for the feasibility of a task set is that the processor utilization is not greater than 1. This result can now be easily seen as a consequence of Theorem 3.5.

Corollary 3.1 (Liu and Layland) Any set of n synchronous periodic tasks with processor utilization U = 2::'=1 ~; is feasibly scheduled by EDF if and only if

Proof. It is sufficient to show that the loading 'factor 1.£ of the task set (i. e., the loading factor of the instances generated by the task set) is equal to U. By Theorem 3.5 the thesis then follows. For any interval [t 1 , t 2 ):

that is, U[tl,t,) ::;

Now, let t 1

U.

= 0 and t 2 = Icm(T1 , ••• , Tn): 2: n !:2C U[tl,!o)

=

i=~2 T;

,

= U.

It follows 1.£=

U.

o In the previous corollary it is assumed that all the periodic tasks have a null initial phasing, that is, for each task Ti, the first instance is released at time

36

CHAPTER

'2

=0

10

n

10 II 12

n

16

19

3

22

C 2 =3 T 2 =5

o

Figure 3.3

I

3

4

5

6

7

,

9

14 15 16 17 18 192021 1123 2425

t

EDF schedule of two periodic tasks with different initial phasing.

t = O. Liu and Lay land's result is very simple and efficient to use, however, in

actual systems it may not be practical to start all periodic tasks at the same time. A first rela..xation to their reference model is thus to let each task have its own start time. Namely, each task Ti is allowed to have a start time Si, the time at which the first job of the task is released. In this way, the k th job of Ti, Ji,k> Ji,k, has release time ri,k = Si + (k - l)Ti and deadline di,k =

Si

+ (k -

l)Ti

+ Di ,

where di is the deadline of task i. Again, such task sets where the first instance has a non-zero start time are termed asynchronous. It is still assumed that Vi, D i = T i . See Figure 3.3 for an example of EDF scheduling with different initial phasing. ColTman [6] showed that the condition of having a processor utilization not exceeding 1 is still sufficient for the feasibility under EDF of asynchronous periodic task sets. His result, too, can be seen as a consequence of Theorem 3.5. Corollary 3.2 (Coffman) Any set of n asynchronous periodic tasks with processor utilization U = L:~l ~:' is feasibly scheduled by EDF if and only if

Proof. Once again, it is sufficient to show that u

= U.

For any interval [t 1 , tz):

I

',J Iv; l'

Fundamentals of EDF Scheduling

37

that is,

Consider now the interval [0, s + mH), where m is an integer greater than 0, s = max{ SI,' .. , sn}, and H = lcm(T), .. . , Tn): n

h[O,s+mH)

2':

mH

LT i=l

Gi

n

Gi = mH

LT i=l

l

= mHU.

1

It follows mH

U[O,s+mH)

U

2': s + mH '

and for arbitrary m U

2': U,

from which it can be finally concluded that follows.

U

= U.

By Theorem 3.5, the thesis 0

The importance of this result is that the feasibility condition introduced by Liu and Layland can be efficiently used in asynchronous systems. That is, as long~~..~t~(l,.QliJ~§.!!re.equ,al~