feasible 14/10/2015 · 14/10/2015 2 Simplifying assumptions Single processor Homogeneous task sets Fully preemptive tasks Simultaneous activations No precedence constraints No resource

14/10/2015

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Definitions

A task set is said to be feasible, if there exists an

algorithm that generates a feasible schedule for .

Examples of constraints

• Timing constraints: activation, period, deadline, jitter.

• Precedence: order of execution between tasks.• Resources: synchronization for mutual exclusion.

A schedule is said to be feasible if it satisfies a set of

constraints.

A task set is said to be schedulable with an algorithm A,

if A generates a feasible schedule.

3

Feasibility vs. schedulability

Feasible task sets

Space of all

task sets

Task sets schedulable

with alg. ATask sets

schedulablewith alg. B

unfeasible task set

The scheduling problem

Given a set of n tasks, a set P of p processors, and a setR of r resources, find an assignment of P and R to thatproduces a feasible schedule under a set of constraints.

Schedulingalgorithm

R

P feasible

constraints

Complexity

In 1975, Garey and Johnson showed that thegeneral scheduling problem is NP hard .

In practice, it means that the time for finding a feasibleschedule grows exponentially with the number of tasks.

Fortunately, polynomial time algorithms can befound under particular conditions.

Let’s consider an application with n = 30 tasks on a

processor in which the elementary step takes 1 s

Consider 3 algorithms with the following complexity:

A1: O(n) A2: O(n8) A3: O(8

n)

30 s 182 hours 40.000billion years

Why do we care about complexity?

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Definitions







constraints.



3


Feasible task sets

Space of all

task sets




unfeasible task set



Schedulingalgorithm

R

P feasible

constraints

Complexity







A1: O(n) A2: O(n8) A3: O(8

n)



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Definitions







constraints.



3


Feasible task sets

Space of all

task sets




unfeasible task set



Schedulingalgorithm

R

P feasible

constraints

Complexity







A1: O(n) A2: O(n8) A3: O(8

n)



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Definitions







constraints.



3


Feasible task sets

Space of all

task sets




unfeasible task set



Schedulingalgorithm

R

P feasible

constraints

Complexity







A1: O(n) A2: O(n8) A3: O(8

n)



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Definitions







constraints.



3


Feasible task sets

Space of all

task sets




unfeasible task set



Schedulingalgorithm

R

P feasible

constraints

Complexity







A1: O(n) A2: O(n8) A3: O(8

n)



14/10/2015

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Definitions







constraints.



3


Feasible task sets

Space of all

task sets




unfeasible task set



Schedulingalgorithm

R

P feasible

constraints

Complexity







A1: O(n) A2: O(n8) A3: O(8

n)



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Simplifying assumptions

Single processor

Homogeneous task sets

Fully preemptive tasks

Simultaneous activations

No precedence constraints

No resource constraints

Algorithm taxonomy

Preemptive vs. Non Preemptive

Static vs. dynamic

On line vs. Off line

Optimal vs. Heuristic

Static vs. Dynamic

Staticscheduling decisions are taken based onfixed parameters, statically assigned totasks before activation.

Dynamicscheduling decisions are taken based onparameters that can change with time.

Off-line vs. On-line

Off-lineall scheduling decisions are taken beforetask activation: the schedule is stored in atable (table-driven scheduling ).

On-linescheduling decisions are taken at run timeon the set of active tasks.


OptimalThey generate a schedule that minimizesa cost function, defined based on anoptimality criterion.

HeuristicThey generate a schedule according to aheuristic function that tries to satisfy anoptimality criterion, but there is noguarantee of success.

Optimality criteria

Feasibility : Find a feasible schedule if there exists one.

Minimize the maximum lateness

Minimize the number of deadline miss

Assign a value to each task, then maximizethe cumulative value of the feasible tasks

Pierre

Texte surligné

Pierre

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Single processor






Algorithm taxonomy


Static vs. dynamic



Static vs. Dynamic









Optimality criteria





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Single processor






Algorithm taxonomy


Static vs. dynamic



Static vs. Dynamic









Optimality criteria





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Single processor






Algorithm taxonomy


Static vs. dynamic



Static vs. Dynamic









Optimality criteria





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Single processor






Algorithm taxonomy


Static vs. dynamic



Static vs. Dynamic









Optimality criteria





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Single processor






Algorithm taxonomy


Static vs. dynamic



Static vs. Dynamic









Optimality criteria





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Task set assumptions

We consider algorithms for different types of tasks:

Single-job tasks (one shot)tasks with a single activation (not recurrent)

Periodic tasksrecurrent tasks regularly activated by a timer (eachtask potentially generates infinite jobs)

Aperiodic/Sporadic tasksrecurrent tasks irregularly activated by events (eachtask potentially generates infinite jobs)

Mixed task sets

Graham’s Notation

denotes the number of processors

denotes the constraints on tasks

denotes the optimality criterion

Examples:

1 preem. RavgUniprocessor algorithm for preemptive tasks that minimizes the average response time.

2 sync LmaxDual-core algorithm for synchronous tasks that minimizes the maximum lateness.

4 preem. LmaxQuad-core algorithm for preemptive tasks that minimizes the maximum lateness.

Classical scheduling policies

First Come First Served

Shortest Job First

Priority Scheduling

Round Robin

Not suited for real-time systems


It assigns the CPU to tasks based on their arrival

times (intrinsically non preemptive):

t

Ready queue

CPU1234

a1 a2 a3 a4

1 2 3 4

s2 s3 s4 f4

17

‐ Very unpredictable

response times strongly depend on task arrivals:

0 4 8 12 16 20 24 28 32t

1 2 3

a1 a2 a3R3 = 26R2 = 26R1 = 20

0 4 8 12 16 20 24 28 32t

3 2 1

a3 a2 a1R3 = 2R2 = 8R1 = 26


18

Shortest Job First (SJF)

Static (Ci is a constant parameter)

It can be used on line or off‐line

Can be preemptive or non preemptive

It minimizes the average response time

It selects the ready task with the shortest

computation time.

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Mixed task sets

Graham’s Notation




Examples:






Shortest Job First

Priority Scheduling

Round Robin





t

Ready queue

CPU1234

a1 a2 a3 a4

1 2 3 4

s2 s3 s4 f4

17



0 4 8 12 16 20 24 28 32t

1 2 3

a1 a2 a3R3 = 26R2 = 26R1 = 20

0 4 8 12 16 20 24 28 32t

3 2 1

a3 a2 a1R3 = 2R2 = 8R1 = 26


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computation time.

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Mixed task sets

Graham’s Notation




Examples:






Shortest Job First

Priority Scheduling

Round Robin





t

Ready queue

CPU1234

a1 a2 a3 a4

1 2 3 4

s2 s3 s4 f4

17



0 4 8 12 16 20 24 28 32t

1 2 3

a1 a2 a3R3 = 26R2 = 26R1 = 20

0 4 8 12 16 20 24 28 32t

3 2 1

a3 a2 a1R3 = 2R2 = 8R1 = 26


18







computation time.

Pierre

Note

Because they don't take account of timing constraints.

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Mixed task sets

Graham’s Notation




Examples:






Shortest Job First

Priority Scheduling

Round Robin





t

Ready queue

CPU1234

a1 a2 a3 a4

1 2 3 4

s2 s3 s4 f4

17



0 4 8 12 16 20 24 28 32t

1 2 3

a1 a2 a3R3 = 26R2 = 26R1 = 20

0 4 8 12 16 20 24 28 32t

3 2 1

a3 a2 a1R3 = 2R2 = 8R1 = 26


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computation time.

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Mixed task sets

Graham’s Notation




Examples:






Shortest Job First

Priority Scheduling

Round Robin





t

Ready queue

CPU1234

a1 a2 a3 a4

1 2 3 4

s2 s3 s4 f4

17



0 4 8 12 16 20 24 28 32t

1 2 3

a1 a2 a3R3 = 26R2 = 26R1 = 20

0 4 8 12 16 20 24 28 32t

3 2 1

a3 a2 a1R3 = 2R2 = 8R1 = 26


18







computation time.

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Mixed task sets

Graham’s Notation




Examples:






Shortest Job First

Priority Scheduling

Round Robin





t

Ready queue

CPU1234

a1 a2 a3 a4

1 2 3 4

s2 s3 s4 f4

17



0 4 8 12 16 20 24 28 32t

1 2 3

a1 a2 a3R3 = 26R2 = 26R1 = 20

0 4 8 12 16 20 24 28 32t

3 2 1

a3 a2 a1R3 = 2R2 = 8R1 = 26


18







computation time.

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19

SJF - Optimality

SL SJF

S L’t

fS’ fL fL’ = fS<r0

fS’ + fL’ fS + fL

)()(1

)'(1

')(11

Rrfn

rfn

Rn

iii

n

iii

20

’ ’’ *. . .

)''(')()( RRR . . . *)( R

* = SJF

)( SJFR is the minimum response timeachievable by any algorithm

SJF - Optimality

21

Is SJF suited for Real-Time?

‐ It is not optimal in the sense of feasibility

0 4 8 12 16 20 24 28 32t

1 2

d1 d2 d3A SJF feasible

3

0 4 8 12 16 20 24 28 32t

3 2 1

d1 d2 d3SJF not feasible

22

Each task has a priority Pi, typically Pi [0, 255]

The task with the highest priority is selected for

execution.

Tasks with the same priority are served FCFS

Priority Scheduling

pi 1/Ci SJF

pi 1/ai FCFS

NOTE:

23

Priority Scheduling

Problem: starvationlow priority tasks may experience long delaysdue to the preemption of high priority tasks.

A possible solution: agingpriority increases with waiting time

24

The ready queue is served with FCFS, but ...

‐ Each task i cannot execute for more than Q

time units (Q = time quantum).

‐ When Q expires, i is put back in the queue.

CPU

READY queue

Q expired

Round Robin

Pierre

Note

Seems correct for 2 tasks. Generalizable for more than 2.

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19

SJF - Optimality

SL SJF

S L’t



)()(1

)'(1

')(11

Rrfn

rfn

Rn

iii

n

iii

20

’ ’’ *. . .

)''(')()( RRR . . . *)( R

* = SJF


SJF - Optimality

21



0 4 8 12 16 20 24 28 32t

1 2


3

0 4 8 12 16 20 24 28 32t

3 2 1


22



execution.


Priority Scheduling

pi 1/Ci SJF

pi 1/ai FCFS

NOTE:

23

Priority Scheduling



24





CPU

READY queue

Q expired

Round Robin

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19

SJF - Optimality

SL SJF

S L’t



)()(1

)'(1

')(11

Rrfn

rfn

Rn

iii

n

iii

20

’ ’’ *. . .

)''(')()( RRR . . . *)( R

* = SJF


SJF - Optimality

21



0 4 8 12 16 20 24 28 32t

1 2


3

0 4 8 12 16 20 24 28 32t

3 2 1


22



execution.


Priority Scheduling

pi 1/Ci SJF

pi 1/ai FCFS

NOTE:

23

Priority Scheduling



24





CPU

READY queue

Q expired

Round Robin

Pierre

Note

do not take account of deadlines!

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19

SJF - Optimality

SL SJF

S L’t



)()(1

)'(1

')(11

Rrfn

rfn

Rn

iii

n

iii

20

’ ’’ *. . .

)''(')()( RRR . . . *)( R

* = SJF


SJF - Optimality

21



0 4 8 12 16 20 24 28 32t

1 2


3

0 4 8 12 16 20 24 28 32t

3 2 1


22



execution.


Priority Scheduling

pi 1/Ci SJF

pi 1/ai FCFS

NOTE:

23

Priority Scheduling



24





CPU

READY queue

Q expired

Round Robin

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19

SJF - Optimality

SL SJF

S L’t



)()(1

)'(1

')(11

Rrfn

rfn

Rn

iii

n

iii

20

’ ’’ *. . .

)''(')()( RRR . . . *)( R

* = SJF


SJF - Optimality

21



0 4 8 12 16 20 24 28 32t

1 2


3

0 4 8 12 16 20 24 28 32t

3 2 1


22



execution.


Priority Scheduling

pi 1/Ci SJF

pi 1/ai FCFS

NOTE:

23

Priority Scheduling



24





CPU

READY queue

Q expired

Round Robin

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19

SJF - Optimality

SL SJF

S L’t



)()(1

)'(1

')(11

Rrfn

rfn

Rn

iii

n

iii

20

’ ’’ *. . .

)''(')()( RRR . . . *)( R

* = SJF


SJF - Optimality

21



0 4 8 12 16 20 24 28 32t

1 2


3

0 4 8 12 16 20 24 28 32t

3 2 1


22



execution.


Priority Scheduling

pi 1/Ci SJF

pi 1/ai FCFS

NOTE:

23

Priority Scheduling



24





CPU

READY queue

Q expired

Round Robin

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25

n = number of task in the system

t

nQ nQ

Qi

ii nC

Q

CnQR )(

Round Robin

Each task runs as it was executing alone on a virtualprocessor n times slower than the real one.

Time sharing

26

Round Robin

‐ if Q > max(Ci) then RR FCFS

‐ if Q context switch time () then

t

n(Q + )

Q +

n(Q + )

Q

QnC

Q

CQnR i

ii )(

27

CPU

High priority

Medium priority

Low priority

system tasks

interactive tasks

batch tasks

PRIORITY

RR

FCFS

Multi-Level Scheduling

28


CPU

priority

30

Real-Time Algorithms

They can either schedule tasks by

relative deadlines Di (static )

absolute deadlines d i (dynamic )

Di

diri

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25


t

nQ nQ

Qi

ii nC

Q

CnQR )(

Round Robin


Time sharing

26

Round Robin



t

n(Q + )

Q +

n(Q + )

Q

QnC

Q

CQnR i

ii )(

27

CPU

High priority

Medium priority

Low priority

system tasks

interactive tasks

batch tasks

PRIORITY

RR

FCFS


28


CPU

priority

30





Di

diri

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25


t

nQ nQ

Qi

ii nC

Q

CnQR )(

Round Robin


Time sharing

26

Round Robin



t

n(Q + )

Q +

n(Q + )

Q

QnC

Q

CQnR i

ii )(

27

CPU

High priority

Medium priority

Low priority

system tasks

interactive tasks

batch tasks

PRIORITY

RR

FCFS


28


CPU

priority

30





Di

diri

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25


t

nQ nQ

Qi

ii nC

Q

CnQR )(

Round Robin


Time sharing

26

Round Robin



t

n(Q + )

Q +

n(Q + )

Q

QnC

Q

CQnR i

ii )(

27

CPU

High priority

Medium priority

Low priority

system tasks

interactive tasks

batch tasks

PRIORITY

RR

FCFS


28


CPU

priority

30





Di

diri

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25


t

nQ nQ

Qi

ii nC

Q

CnQR )(

Round Robin


Time sharing

26

Round Robin



t

n(Q + )

Q +

n(Q + )

Q

QnC

Q

CQnR i

ii )(

27

CPU

High priority

Medium priority

Low priority

system tasks

interactive tasks

batch tasks

PRIORITY

RR

FCFS


28


CPU

priority

30





Di

diri

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25


t

nQ nQ

Qi

ii nC

Q

CnQR )(

Round Robin


Time sharing

26

Round Robin



t

n(Q + )

Q +

n(Q + )

Q

QnC

Q

CQnR i

ii )(

27

CPU

High priority

Medium priority

Low priority

system tasks

interactive tasks

batch tasks

PRIORITY

RR

FCFS


28


CPU

priority

30





Di

diri

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31

Earliest Due Date

Problem

Algorithm [Jackson 55]

- Order the ready queue by increasing deadline.

Assumptions- All the tasks are simultaneously activated

- Preemption is not needed.- Static (Di is fixed)

Property- It minimizes the maximum lateness (Lmax)

1 sync Lmax

32

Lateness

L i > 0

di f iri

di

L i < 0

f iri

L i = fi di

33

Maximum Lateness

Lmax = maxi (L i)

if (Lmax < 0) then

no task exceeds its deadline

34

EDD - Optimality

Lmax = La = fa da

La’ = fa’ da < fa da

Lb’ = fb’ db < fa da

L’ max < Lmax

AB EDD

A B’t

fa’ fb fb’ = fa<r0 da db

35

EDD - Optimality

’ ’’ *. . .

)''(')()( maxmaxmax LLL . . . *)(max L

* = EDD

)(max EDDL is the minimum valueachievable by any algorithm

36

EDD guarantee test (off line)

tf1 f2 f3 f4

1 2 3 4

A task set is feasible iff ii dfi

i

kki Cf

1i

i

kk DCi

1

Pierre

Note

1 Proc, concurrent, Makespan

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31

Earliest Due Date

Problem






1 sync Lmax

32

Lateness

L i > 0

di f iri

di

L i < 0

f iri

L i = fi di

33

Maximum Lateness

Lmax = maxi (L i)

if (Lmax < 0) then


34

EDD - Optimality

Lmax = La = fa da



L’ max < Lmax

AB EDD

A B’t


35

EDD - Optimality

’ ’’ *. . .


* = EDD


36


tf1 f2 f3 f4

1 2 3 4


i

kki Cf

1i

i

kk DCi

1

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31

Earliest Due Date

Problem






1 sync Lmax

32

Lateness

L i > 0

di f iri

di

L i < 0

f iri

L i = fi di

33

Maximum Lateness

Lmax = maxi (L i)

if (Lmax < 0) then


34

EDD - Optimality

Lmax = La = fa da



L’ max < Lmax

AB EDD

A B’t


35

EDD - Optimality

’ ’’ *. . .


* = EDD


36


tf1 f2 f3 f4

1 2 3 4


i

kki Cf

1i

i

kk DCi

1

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31

Earliest Due Date

Problem






1 sync Lmax

32

Lateness

L i > 0

di f iri

di

L i < 0

f iri

L i = fi di

33

Maximum Lateness

Lmax = maxi (L i)

if (Lmax < 0) then


34

EDD - Optimality

Lmax = La = fa da



L’ max < Lmax

AB EDD

A B’t


35

EDD - Optimality

’ ’’ *. . .


* = EDD


36


tf1 f2 f3 f4

1 2 3 4


i

kki Cf

1i

i

kk DCi

1

Pierre

Note

TO Check!

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31

Earliest Due Date

Problem






1 sync Lmax

32

Lateness

L i > 0

di f iri

di

L i < 0

f iri

L i = fi di

33

Maximum Lateness

Lmax = maxi (L i)

if (Lmax < 0) then


34

EDD - Optimality

Lmax = La = fa da



L’ max < Lmax

AB EDD

A B’t


35

EDD - Optimality

’ ’’ *. . .


* = EDD


36


tf1 f2 f3 f4

1 2 3 4


i

kki Cf

1i

i

kk DCi

1

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31

Earliest Due Date

Problem






1 sync Lmax

32

Lateness

L i > 0

di f iri

di

L i < 0

f iri

L i = fi di

33

Maximum Lateness

Lmax = maxi (L i)

if (Lmax < 0) then


34

EDD - Optimality

Lmax = La = fa da



L’ max < Lmax

AB EDD

A B’t


35

EDD - Optimality

’ ’’ *. . .


* = EDD


36


tf1 f2 f3 f4

1 2 3 4


i

kki Cf

1i

i

kk DCi

1

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37

Problem

Algorithm [Horn 74]

- Order the ready queue by increasing absolute deadline

Assumptions

- Tasks can be activated dynamically

- Dynamic algorithm (di depends on ai)

- Tasks can be preempted at any time


1 preem. Lmax

Earliest Deadline First

38

EDF Example

39

EDF Guarantee test (on line)

t

c1(t)

c2(t)

c3(t)

c4(t)

tdtci i

i

kk

1

)(

40

Complexity

EDD

Scheduler (queue ordering):Feasibility Test (guarantee test):

EDF

Scheduler (insertion in the queue):Feasibility Test (guarantee single task):

O(n log n)

O(n)

O(n)

O(n)

41

EDF optimality

In the sense of feasibility [Dertouzos 1974]

An algorithm A is optimal in the sense offeasibility if it generates a feasible schedule, ifthere exists one.

Demonstration method

It is sufficient to prove that, given an arbitraryfeasible schedule, the schedule generated byEDF is also feasible.

42

0 2 4 6 8 10 12 14 16

t = 4

EDF

1

2

3

4

Dertouzos Transformationfor (t = 0 to Dmax–1)

if ((t) E(t)) {(tE) = (t);(t) = E(t);

}

(t) = task executing at time t

E(t) = task with min d at time t

tE = time at which E is executed

(t) = 4E(t) = 2tE = 6

Pierre

Texte surligné

14/10/2015

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37

Problem

Algorithm [Horn 74]


Assumptions





1 preem. Lmax


38

EDF Example

39


t

c1(t)

c2(t)

c3(t)

c4(t)

tdtci i

i

kk

1

)(

40

Complexity

EDD


EDF


O(n log n)

O(n)

O(n)

O(n)

41

EDF optimality





42

0 2 4 6 8 10 12 14 16

t = 4

EDF

1

2

3

4


if ((t) E(t)) {(tE) = (t);(t) = E(t);

}




(t) = 4E(t) = 2tE = 6

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Problem

Algorithm [Horn 74]


Assumptions





1 preem. Lmax


38

EDF Example

39


t

c1(t)

c2(t)

c3(t)

c4(t)

tdtci i

i

kk

1

)(

40

Complexity

EDD


EDF


O(n log n)

O(n)

O(n)

O(n)

41

EDF optimality





42

0 2 4 6 8 10 12 14 16

t = 4

EDF

1

2

3

4


if ((t) E(t)) {(tE) = (t);(t) = E(t);

}




(t) = 4E(t) = 2tE = 6

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37

Problem

Algorithm [Horn 74]


Assumptions





1 preem. Lmax


38

EDF Example

39


t

c1(t)

c2(t)

c3(t)

c4(t)

tdtci i

i

kk

1

)(

40

Complexity

EDD


EDF


O(n log n)

O(n)

O(n)

O(n)

41

EDF optimality





42

0 2 4 6 8 10 12 14 16

t = 4

EDF

1

2

3

4


if ((t) E(t)) {(tE) = (t);(t) = E(t);

}




(t) = 4E(t) = 2tE = 6

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Problem

Algorithm [Horn 74]


Assumptions





1 preem. Lmax


38

EDF Example

39


t

c1(t)

c2(t)

c3(t)

c4(t)

tdtci i

i

kk

1

)(

40

Complexity

EDD


EDF


O(n log n)

O(n)

O(n)

O(n)

41

EDF optimality





42

0 2 4 6 8 10 12 14 16

t = 4

EDF

1

2

3

4


if ((t) E(t)) {(tE) = (t);(t) = E(t);

}




(t) = 4E(t) = 2tE = 6

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43

0 2 4 6 8 10 12 14 16

t = 4

1

2

3

4

(t) = 4E(t) = 2tE = 6





if ((t) E(t)) {(tE) = (t);(t) = E(t);

}

44

0 2 4 6 8 10 12 14 16

t = 5

1

2

3

4

(t) = 3E(t) = 2tE = 7





if ((t) E(t)) {(tE) = (t);(t) = E(t);

}

45

0 2 4 6 8 10 12 14 16

t = 5

1

2

3

4

(t) = 3E(t) = 2tE = 7





if ((t) E(t)) {(tE) = (t);(t) = E(t);

}

46

0 2 4 6 8 10 12 14 16

t = 5

1

2

3

4

Dertouzos TransformationDertouzos transformation preserves schedulability, in fact:

for the advanced slice, this is obvious!

for the postponed slice, the slack cannot decrease:

47

A property of optimal algorithms

If a task set is not schedulable by anoptimal algorithm, then cannot bescheduled by any other algorithm.

If an algorithm A minimizes Lmax then A isalso optimal in the sense of feasibility.The opposite is not true.

48

Non Preemptive Scheduling

Under non preemptive execution, EDF is not optimal:

1

2

1

2

Feasible schedule

EDF

0 2 431 5 6 7 8 9

0 2 431 5 6 7 8 9

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8

43

0 2 4 6 8 10 12 14 16

t = 4

1

2

3

4

(t) = 4E(t) = 2tE = 6





if ((t) E(t)) {(tE) = (t);(t) = E(t);

}

44

0 2 4 6 8 10 12 14 16

t = 5

1

2

3

4

(t) = 3E(t) = 2tE = 7





if ((t) E(t)) {(tE) = (t);(t) = E(t);

}

45

0 2 4 6 8 10 12 14 16

t = 5

1

2

3

4

(t) = 3E(t) = 2tE = 7





if ((t) E(t)) {(tE) = (t);(t) = E(t);

}

46

0 2 4 6 8 10 12 14 16

t = 5

1

2

3

4

Dertouzos TransformationDertouzos transformation preserves schedulability, in fact:

for the advanced slice, this is obvious!

for the postponed slice, the slack cannot decrease:

47

A property of optimal algorithms

If a task set is not schedulable by anoptimal algorithm, then cannot bescheduled by any other algorithm.

If an algorithm A minimizes Lmax then A isalso optimal in the sense of feasibility.The opposite is not true.

48


Under non preemptive execution, EDF is not optimal:

1

2

1

2

Feasible schedule

EDF

0 2 431 5 6 7 8 9

0 2 431 5 6 7 8 9

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9

49

1

2

0 2 431 5 6 7 8 9


To achieve optimality, an algorithm should be clairvoyant,and decide to leave the CPU idle in the presence of readytasks:

If we forbid leaving the CPU idle in the presence of readytasks, then EDF is optimal. We say that:

NP-EDF is optimal among non-idle scheduling algorithms

50

empty schedule

partial schedule

complete schedule

F N FNN N N N

depth = n

# leaves = n!

complexity: O(n n!)


The problem of finding a feasible schedule is NP hard andis treated off-line with tree search algorithms:

51

Bratley’s Algorithm [Bratley 71]

( 1 | no-preem | Lmax )

Reduces the average complexity by a pruning rule:

Do not expand unless the partial schedule isfound to be strongly feasible .

A partial schedule is said to be stronglyfeasible if adding any of the remaining nodesit remains feasible.

52

6 51 2

6 531

6 61 3

6 3 431 21 3 4

6 4 4

Example

4 2 71 1 51 2 60 2 4

1

2

3

4

ai Ci di

16

Scheduled task

finishing time

1 2 3 46 2 3 2

2

3 4

3 1

4

2

3 7

12

7

1

1 = {4, 2, 3, 1}2 = {4, 3, 2, 1}



53

Spring algorithm [Stankovic & Ramamritham 87]

min H

min H

min H

min H

Backtrackingis possible

Heuristic search

1. The schedule for a set of N tasks is constructed in N steps

2. The search is driven by a heuristic function H

3. At each step the algorithm selects the task that minimizes the heuristic function

54

H = ri FCFSH = Ci SJFH = Di DMH = di EDF

H = w1 ri + w2 Di

H = w1 Ci + w2 di

H = w1 Vi + w2 di


Heuristic functions

Example of heuristic functions:

Composite heuristic functions:

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9

49

1

2

0 2 431 5 6 7 8 9


To achieve optimality, an algorithm should be clairvoyant,and decide to leave the CPU idle in the presence of readytasks:

If we forbid leaving the CPU idle in the presence of readytasks, then EDF is optimal. We say that:

NP-EDF is optimal among non-idle scheduling algorithms

50

empty schedule

partial schedule

complete schedule

F N FNN N N N

depth = n

# leaves = n!

complexity: O(n n!)


The problem of finding a feasible schedule is NP hard andis treated off-line with tree search algorithms:

51



Reduces the average complexity by a pruning rule:

Do not expand unless the partial schedule isfound to be strongly feasible .

A partial schedule is said to be stronglyfeasible if adding any of the remaining nodesit remains feasible.

52

6 51 2

6 531

6 61 3

6 3 431 21 3 4

6 4 4

Example

4 2 71 1 51 2 60 2 4

1

2

3

4

ai Ci di

16

Scheduled task

finishing time

1 2 3 46 2 3 2

2

3 4

3 1

4

2

3 7

12

7

1

1 = {4, 2, 3, 1}2 = {4, 3, 2, 1}



53


min H

min H

min H

min H

Backtrackingis possible

Heuristic search

1. The schedule for a set of N tasks is constructed in N steps

2. The search is driven by a heuristic function H

3. At each step the algorithm selects the task that minimizes the heuristic function

54

H = ri FCFSH = Ci SJFH = Di DMH = di EDF

H = w1 ri + w2 Di

H = w1 Ci + w2 di

H = w1 Vi + w2 di


Heuristic functions

Example of heuristic functions:

Composite heuristic functions:

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10

55

H = Ei ( w1 ri + w2 Di )H = Ei ( w1 Ci + w2 di )

Eligibility

Ei =

i

Ei = 1

i


Possibility to handle precedence constraints:

Heuristic functions:

Heuristic functions

56

Complexity:

min H

min H

min H

min H

Exhaustive search:Heuristic search:Heuristic w. k btracks:

Heuristic algorithm


O(N·N!)

O(N2)

O(kN2)

57

Task ID

start time

length

next

Implementation


If a feasible schedule is not found, does not meanthat there not exist one.

If a feasible solution is found, the schedule isstored into a disptach list:

58

Handling precedence constraints

Given a precedence graph, it constructs the schedule fromthe tail: among the nodes with no successors, LDF selectsthe task with the latest deadline:

Latest Deadline First (LDF) [Lawler 73]

5

A

B C

D E F3 6

5 4

2 deadline

FECDBALDF

dA dD dC dB,dE dF

EDF FEC DBA

D misses its deadline

1 | prec, sync | Lmax

59


Assumes that arrival times are known a priori;

Transforms precedence constraints into timingconstraints by modifying arrival times and deadlinesbased on the precedence graph;

Applies EDF to the modified task set.

EDF* [Chetto & Chetto 89]

1 | prec, preem | Lmax

60


EDF* [Chetto & Chetto 89] 1 | prec, preem | Lmax

A

B

A

B

CA

CB

The idea is to:

postpone the arrival time of a successor

advance the deadine of a predecessor

rB dB

rA dA

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10

55


Eligibility

Ei =

i

Ei = 1

i




Heuristic functions

56

Complexity:

min H

min H

min H

min H


Heuristic algorithm


O(N·N!)

O(N2)

O(kN2)

57

Task ID

start time

length

next

Implementation




58




5

A

B C

D E F3 6

5 4

2 deadline

FECDBALDF

dA dD dC dB,dE dF

EDF FEC DBA



59







60



A

B

A

B

CA

CB

The idea is to:



rB dB

rA dA

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10

55


Eligibility

Ei =

i

Ei = 1

i




Heuristic functions

56

Complexity:

min H

min H

min H

min H


Heuristic algorithm


O(N·N!)

O(N2)

O(kN2)

57

Task ID

start time

length

next

Implementation




58




5

A

B C

D E F3 6

5 4

2 deadline

FECDBALDF

dA dD dC dB,dE dF

EDF FEC DBA



59







60



A

B

A

B

CA

CB

The idea is to:



rB dB

rA dA

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10

55


Eligibility

Ei =

i

Ei = 1

i




Heuristic functions

56

Complexity:

min H

min H

min H

min H


Heuristic algorithm


O(N·N!)

O(N2)

O(kN2)

57

Task ID

start time

length

next

Implementation




58




5

A

B C

D E F3 6

5 4

2 deadline

FECDBALDF

dA dD dC dB,dE dF

EDF FEC DBA



59







60



A

B

A

B

CA

CB

The idea is to:



rB dB

rA dA

Pierre

Note

Interesting!

14/10/2015

10

55


Eligibility

Ei =

i

Ei = 1

i




Heuristic functions

56

Complexity:

min H

min H

min H

min H


Heuristic algorithm


O(N·N!)

O(N2)

O(kN2)

57

Task ID

start time

length

next

Implementation




58




5

A

B C

D E F3 6

5 4

2 deadline

FECDBALDF

dA dD dC dB,dE dF

EDF FEC DBA



59







60



A

B

A

B

CA

CB

The idea is to:



rB dB

rA dA

14/10/2015

10

55


Eligibility

Ei =

i

Ei = 1

i




Heuristic functions

56

Complexity:

min H

min H

min H

min H


Heuristic algorithm


O(N·N!)

O(N2)

O(kN2)

57

Task ID

start time

length

next

Implementation




58




5

A

B C

D E F3 6

5 4

2 deadline

FECDBALDF

dA dD dC dB,dE dF

EDF FEC DBA



59







60



A

B

A

B

CA

CB

The idea is to:



rB dB

rA dA

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11

61



A

B

A

B

CA

CB

The idea is to:

postpone the arrival time of a successor: r*B = rA + CA

advance the deadine of a predecessor: d*A = dB – CB

rB dB

rA dA

r*B d*A

62



1. For all root nodes, set r*i = ri.

2. Select a task i such that all its immediate predecessors have been modified, else exit.

3. Set r*i = max { ri , max (r*k + Ck) }.

4. Repeat from line 2.

Arrival time modification

ki

63



1. For all leaves, set d*i = di.

2. Select a task i such that all its immediate successors have been modified, else exit.

3. Set d*i = min { di , min (d*k – Ck) }.


Deadline modification

ik

Summary

activ prec preem algorithm authors complexity

Sync

Asyn

Asyn

Asyn

Asyn

Sync

Asyn

N

N

N

Y

Y

Y

Y

*

*

N

N

N

Y

Y

EDD

EDF

NP-EDF

Tree search

Spring

LDF

EDF*

Jackson ‘55

Horn ‘74

Jeffay ‘91

Bratley ‘71

Stankovic ‘87

Lawler ‘73

Chetto2 ‘89

O(n logn)

O(n) task

O(n·n!)

O(n2)

O(n logn)

O(n logn)

O(n logn)

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11

61



A

B

A

B

CA

CB

The idea is to:



rB dB

rA dA

r*B d*A

62








ki

63








ik

Summary


Sync

Asyn

Asyn

Asyn

Asyn

Sync

Asyn

N

N

N

Y

Y

Y

Y

*

*

N

N

N

Y

Y

EDD

EDF

NP-EDF

Tree search

Spring

LDF

EDF*

Jackson ‘55

Horn ‘74

Jeffay ‘91

Bratley ‘71

Stankovic ‘87

Lawler ‘73

Chetto2 ‘89

O(n logn)

O(n) task

O(n·n!)

O(n2)

O(n logn)

O(n logn)

O(n logn)

14/10/2015

11

61



A

B

A

B

CA

CB

The idea is to:



rB dB

rA dA

r*B d*A

62








ki

63








ik

Summary


Sync

Asyn

Asyn

Asyn

Asyn

Sync

Asyn

N

N

N

Y

Y

Y

Y

*

*

N

N

N

Y

Y

EDD

EDF

NP-EDF

Tree search

Spring

LDF

EDF*

Jackson ‘55

Horn ‘74

Jeffay ‘91

Bratley ‘71

Stankovic ‘87

Lawler ‘73

Chetto2 ‘89

O(n logn)

O(n) task

O(n·n!)

O(n2)

O(n logn)

O(n logn)

O(n logn)

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11

61



A

B

A

B

CA

CB

The idea is to:



rB dB

rA dA

r*B d*A

62








ki

63








ik

Summary


Sync

Asyn

Asyn

Asyn

Asyn

Sync

Asyn

N

N

N

Y

Y

Y

Y

*

*

N

N

N

Y

Y

EDD

EDF

NP-EDF

Tree search

Spring

LDF

EDF*

Jackson ‘55

Horn ‘74

Jeffay ‘91

Bratley ‘71

Stankovic ‘87

Lawler ‘73

Chetto2 ‘89

O(n logn)

O(n) task

O(n·n!)

O(n2)

O(n logn)

O(n logn)

O(n logn)

Pierre

Note

Pas vu Bratley et Stankovic

530054

Cross-Out

530054

Cross-Out

530054

Cross-Out

530054

Cross-Out

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11

61



A

B

A

B

CA

CB

The idea is to:



rB dB

rA dA

r*B d*A

62








ki

63








ik

Summary


Sync

Asyn

Asyn

Asyn

Asyn

Sync

Asyn

N

N

N

Y

Y

Y

Y

*

*

N

N

N

Y

Y

EDD

EDF

NP-EDF

Tree search

Spring

LDF

EDF*

Jackson ‘55

Horn ‘74

Jeffay ‘91

Bratley ‘71

Stankovic ‘87

Lawler ‘73

Chetto2 ‘89

O(n logn)

O(n) task

O(n·n!)

O(n2)

O(n logn)

O(n logn)

O(n logn)

14/10/2015

11

61



A

B

A

B

CA

CB

The idea is to:



rB dB

rA dA

r*B d*A

62








ki

63








ik

Summary


Sync

Asyn

Asyn

Asyn

Asyn

Sync

Asyn

N

N

N

Y

Y

Y

Y

*

*

N

N

N

Y

Y

EDD

EDF

NP-EDF

Tree search

Spring

LDF

EDF*

Jackson ‘55

Horn ‘74

Jeffay ‘91

Bratley ‘71

Stankovic ‘87

Lawler ‘73

Chetto2 ‘89

O(n logn)

O(n) task

O(n·n!)

O(n2)

O(n logn)

O(n logn)

O(n logn)

Documents

feasible 14/10/2015 · 14/10/2015 2 Simplifying assumptions Single processor Homogeneous task sets Fully preemptive tasks Simultaneous activations No precedence constraints No resource