Fixed-Priority Schedulability of Sporadic Tasks on Uniprocessors is - - PowerPoint PPT Presentation

Fixed-Priority Schedulability of Sporadic Tasks

• n Uniprocessors is NP-hard

Pontus Ekberg & Wang Yi

Uppsala University

RTSS 2017

Overview

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Pseudo-poly. time algorithm Pseudo-poly. time algorithm Weakly NP-hard Weakly NP-complete Weakly NP-hard Weakly NP-complete Weakly NP-hard Exponential time algorithm Utilization bounded by a constant c Pseudo-poly. time algorithm Polynomial time for c ln and RM priorities Pseudo-poly. time algorithm Weakly NP-hard for c Weakly NP-complete for c Exponential time algorithm Weakly NP-hard for c

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for c Weakly coNP-complete for c

( ) Joseph and Pandya, 1986 ( ) Liu and Layland, 1973 ( ) Lehoczky, 1990 Pontus Ekberg 2

Overview

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Pseudo-poly. time algorithm Pseudo-poly. time algorithm

∗ ∗

Weakly NP-hard Weakly NP-complete Weakly NP-hard Weakly NP-complete Weakly NP-hard Exponential time algorithm Utilization bounded by a constant c Pseudo-poly. time algorithm

∗

Polynomial time for c ln and RM priorities Pseudo-poly. time algorithm

∗

Weakly NP-hard for c Weakly NP-complete for c Exponential time algorithm Weakly NP-hard for c

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for c Weakly coNP-complete for c

(∗) Joseph and Pandya, 1986 ( ) Liu and Layland, 1973 ( ) Lehoczky, 1990 Pontus Ekberg 2

Overview

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Pseudo-poly. time algorithm Pseudo-poly. time algorithm

∗ ∗

Weakly NP-hard Weakly NP-complete Weakly NP-hard Weakly NP-complete Weakly NP-hard Exponential time algorithm Utilization bounded by a constant c Pseudo-poly. time algorithm Polynomial time for c ⩽ ln 2 and RM priorities

†

Pseudo-poly. time algorithm

∗

Weakly NP-hard for c Weakly NP-complete for c Exponential time algorithm Weakly NP-hard for c

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for c Weakly coNP-complete for c

(∗) Joseph and Pandya, 1986 (†) Liu and Layland, 1973 ( ) Lehoczky, 1990 Pontus Ekberg 2

Overview

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Pseudo-poly. time algorithm Pseudo-poly. time algorithm

∗ ∗

Weakly NP-hard Weakly NP-complete Weakly NP-hard Weakly NP-complete Weakly NP-hard Exponential time algorithm

‡

Utilization bounded by a constant c Pseudo-poly. time algorithm Polynomial time for c ⩽ ln 2 and RM priorities

†

Pseudo-poly. time algorithm

∗

Weakly NP-hard for c Weakly NP-complete for c Exponential time algorithm

‡

Weakly NP-hard for c

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for c Weakly coNP-complete for c

(∗) Joseph and Pandya, 1986 (†) Liu and Layland, 1973 (‡) Lehoczky, 1990 Pontus Ekberg 2

Overview

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Pseudo-poly. time algorithm Pseudo-poly. time algorithm Weakly NP-hard Weakly NP-complete Weakly NP-hard Weakly NP-complete Weakly NP-hard Exponential time algorithm Utilization bounded by a constant c Pseudo-poly. time algorithm Polynomial time for c ⩽ ln 2 and RM priorities Pseudo-poly. time algorithm Weakly NP-hard for c Weakly NP-complete for c Exponential time algorithm Weakly NP-hard for c

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for c Weakly coNP-complete for c

( ) Joseph and Pandya, 1986 ( ) Liu and Layland, 1973 ( ) Lehoczky, 1990 Pontus Ekberg 2

Overview

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Pseudo-poly. time algorithm Pseudo-poly. time algorithm Weakly NP-hard Weakly NP-complete Weakly NP-hard Weakly NP-complete Weakly NP-hard Exponential time algorithm Utilization bounded by a constant c Pseudo-poly. time algorithm Polynomial time for c ⩽ ln 2 and RM priorities Pseudo-poly. time algorithm Weakly NP-hard for c Weakly NP-complete for c Exponential time algorithm Weakly NP-hard for c

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for 0 < c < 1 Weakly coNP-complete for 0 < c < 1

( ) Joseph and Pandya, 1986 ( ) Liu and Layland, 1973 ( ) Lehoczky, 1990 Pontus Ekberg 2

Overview

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Pseudo-poly. time algorithm Pseudo-poly. time algorithm Weakly NP-hard Weakly NP-complete Weakly NP-hard Weakly NP-complete Weakly NP-hard Exponential time algorithm Utilization bounded by a constant c Pseudo-poly. time algorithm Polynomial time for c ⩽ ln 2 and RM priorities Pseudo-poly. time algorithm Weakly NP-hard for c Weakly NP-complete for c Exponential time algorithm Weakly NP-hard for c

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for 0 < c < 1 Weakly coNP-complete for 0 < c < 1

( ) Joseph and Pandya, 1986 ( ) Liu and Layland, 1973 ( ) Lehoczky, 1990 Pontus Ekberg 2

Overview

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Pseudo-poly. time algorithm Pseudo-poly. time algorithm Weakly NP-hard Weakly NP-complete Weakly NP-hard Weakly NP-complete Weakly NP-hard Exponential time algorithm Utilization bounded by a constant c Pseudo-poly. time algorithm Polynomial time for c ⩽ ln 2 and RM priorities Pseudo-poly. time algorithm Weakly NP-hard for c Weakly NP-complete for c Exponential time algorithm Weakly NP-hard for c

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for 0 < c < 1 Weakly coNP-complete for 0 < c < 1

( ) Joseph and Pandya, 1986 ( ) Liu and Layland, 1973 ( ) Lehoczky, 1990 Pontus Ekberg 2

Overview

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Pseudo-poly. time algorithm Pseudo-poly. time algorithm Weakly NP-hard Weakly NP-complete Weakly NP-hard Weakly NP-complete Weakly NP-hard Exponential time algorithm Utilization bounded by a constant c Pseudo-poly. time algorithm Polynomial time for c ⩽ ln 2 and RM priorities Pseudo-poly. time algorithm Weakly NP-hard for c Weakly NP-complete for c Exponential time algorithm Weakly NP-hard for c

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for 0 < c < 1 Weakly coNP-complete for 0 < c < 1

( ) Joseph and Pandya, 1986 ( ) Liu and Layland, 1973 ( ) Lehoczky, 1990 Pontus Ekberg 2

Overview

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Pseudo-poly. time algorithm Pseudo-poly. time algorithm Weakly NP-hard Weakly NP-complete Weakly NP-hard Weakly NP-complete Weakly NP-hard Exponential time algorithm Utilization bounded by a constant c Pseudo-poly. time algorithm Polynomial time for c ⩽ ln 2 and RM priorities Pseudo-poly. time algorithm Weakly NP-hard for c Weakly NP-complete for c Exponential time algorithm Weakly NP-hard for c

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for 0 < c < 1 Weakly coNP-complete for 0 < c < 1

( ) Joseph and Pandya, 1986 ( ) Liu and Layland, 1973 ( ) Lehoczky, 1990 Pontus Ekberg 2

Overview

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Pseudo-poly. time algorithm Pseudo-poly. time algorithm Weakly NP-hard Weakly NP-complete Weakly NP-hard Weakly NP-complete Weakly NP-hard Exponential time algorithm Utilization bounded by a constant c Pseudo-poly. time algorithm Polynomial time for c ⩽ ln 2 and RM priorities Pseudo-poly. time algorithm Weakly NP-hard for 0 < c < 1 Weakly NP-complete for c Exponential time algorithm Weakly NP-hard for c

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for 0 < c < 1 Weakly coNP-complete for 0 < c < 1

( ) Joseph and Pandya, 1986 ( ) Liu and Layland, 1973 ( ) Lehoczky, 1990 Pontus Ekberg 2

Overview

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Pseudo-poly. time algorithm Pseudo-poly. time algorithm Weakly NP-hard Weakly NP-complete Weakly NP-hard Weakly NP-complete Weakly NP-hard Exponential time algorithm Utilization bounded by a constant c Pseudo-poly. time algorithm Polynomial time for c ⩽ ln 2 and RM priorities Pseudo-poly. time algorithm Weakly NP-hard for 0 < c < 1 Weakly NP-complete for c Exponential time algorithm Weakly NP-hard for c

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for 0 < c < 1 Weakly coNP-complete for 0 < c < 1

( ) Joseph and Pandya, 1986 ( ) Liu and Layland, 1973 ( ) Lehoczky, 1990 Pontus Ekberg 2

Overview

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Pseudo-poly. time algorithm Pseudo-poly. time algorithm Weakly NP-hard Weakly NP-complete Weakly NP-hard Weakly NP-complete Weakly NP-hard Exponential time algorithm Utilization bounded by a constant c Pseudo-poly. time algorithm Polynomial time for c ⩽ ln 2 and RM priorities Pseudo-poly. time algorithm Weakly NP-hard for 0 < c < 1 Weakly NP-complete for c Exponential time algorithm Weakly NP-hard for 0 < c < 1

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for 0 < c < 1 Weakly coNP-complete for 0 < c < 1

( ) Joseph and Pandya, 1986 ( ) Liu and Layland, 1973 ( ) Lehoczky, 1990 Pontus Ekberg 2

Overview

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Pseudo-poly. time algorithm Pseudo-poly. time algorithm Weakly NP-hard Weakly NP-complete Weakly NP-hard Weakly NP-complete Weakly NP-hard Exponential time algorithm Utilization bounded by a constant c Pseudo-poly. time algorithm Polynomial time for c ⩽ ln 2 and RM priorities Pseudo-poly. time algorithm Weakly NP-hard for c Weakly NP-complete for 0 < c < 1 Exponential time algorithm Weakly NP-hard for 0 < c < 1

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for 0 < c < 1 Weakly coNP-complete for 0 < c < 1

( ) Joseph and Pandya, 1986 ( ) Liu and Layland, 1973 ( ) Lehoczky, 1990 Pontus Ekberg 2

A Tale of Two Reductions

EDF-schedulability

• Constrained

deadlines

• Bounded utilization

EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

YES NO YES NO YES NO coNP-hard coNP-hard NP-hard

Pontus Ekberg 3

A Tale of Two Reductions

EDF-schedulability

• Constrained

deadlines

• Bounded utilization

EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

YES NO YES NO YES NO coNP-hard coNP-hard NP-hard

Pontus Ekberg 3

A Tale of Two Reductions

EDF-schedulability

• Constrained

deadlines

• Bounded utilization

EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

YES NO YES NO YES NO coNP-hard coNP-hard NP-hard

Pontus Ekberg 3

A Tale of Two Reductions

EDF-schedulability

• Constrained

deadlines

• Bounded utilization

EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

YES NO YES NO YES NO coNP-hard coNP-hard NP-hard

Pontus Ekberg 3

A Tale of Two Reductions

EDF-schedulability

• Constrained

deadlines

• Bounded utilization

EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

YES NO YES NO YES NO coNP-hard coNP-hard NP-hard

Pontus Ekberg 3

A Tale of Two Reductions

EDF-schedulability

• Constrained

deadlines

• Bounded utilization

EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

YES NO YES NO YES NO coNP-hard coNP-hard NP-hard

Pontus Ekberg 3

A Tale of Two Reductions

EDF-schedulability

• Constrained

deadlines

• Bounded utilization

EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

YES NO YES NO YES NO coNP-hard coNP-hard NP-hard

Pontus Ekberg 3

EDF-schedulability ⇝ FP-schedulability

D dbf dbf dbf dbf D D

x x

rbf rbf rbf

x x

Pontus Ekberg 4

EDF-schedulability ⇝ FP-schedulability

D dbf(τ3) dbf(τ2) dbf(τ1) dbf D D

x x

rbf rbf rbf

x x

Pontus Ekberg 4

EDF-schedulability ⇝ FP-schedulability

D dbf(τ3) dbf(τ2) dbf(τ1) dbf D D

x x

rbf rbf rbf

x x

Pontus Ekberg 4

EDF-schedulability ⇝ FP-schedulability

D dbf(τ3) dbf(τ2) + dbf(τ1) + ∑ dbf

=

D D

x x

rbf rbf rbf

x x

Pontus Ekberg 4

EDF-schedulability ⇝ FP-schedulability

D dbf(τ3) dbf(τ2) + dbf(τ1) + ∑ dbf

=

D

⟲

D

x x

rbf rbf rbf

x x

Pontus Ekberg 4

EDF-schedulability ⇝ FP-schedulability

D dbf(τ3) dbf(τ2) + dbf(τ1) + ∑ dbf

=

D

⟲

D

x x

rbf rbf rbf

x x

Pontus Ekberg 4

EDF-schedulability ⇝ FP-schedulability

D dbf(τ3) dbf(τ2) + dbf(τ1) + ∑ dbf

=

D

⟲

D

x x

rbf rbf rbf

x x

Pontus Ekberg 4

EDF-schedulability ⇝ FP-schedulability

D dbf(τ3) dbf(τ2) + dbf(τ1) + ∑ dbf

=

D

⟲

D

=

x x

rbf(τ3) rbf(τ2) + rbf(τ1) + +

x x

Pontus Ekberg 4

EDF-schedulability ⇝ FP-schedulability

D dbf(τ3) dbf(τ2) + dbf(τ1) + ∑ dbf

=

D

⟲

D

=

x x

rbf(τ3) rbf(τ2) + rbf(τ1) + +

x x

Pontus Ekberg 4

EDF-schedulability ⇝ FP-schedulability

D dbf(τ3) dbf(τ2) + dbf(τ1) + ∑ dbf

=

D

⟲

D

=

x x

rbf(τ3) rbf(τ2) + rbf(τ1) + +

x x + 1

Pontus Ekberg 4

EDF-schedulability ⇝ FP-schedulability

D dbf(τ3) dbf(τ2) + dbf(τ1) + ∑ dbf

=

D

⟲

D

=

x x +1

rbf(τ3) rbf(τ2) + rbf(τ1) + +

x x + 1

Pontus Ekberg 4

A Tale of Two Reductions

EDF-schedulability

• Constrained

deadlines

• Bounded utilization

EDF-schedulability

• Constrained

deadlines

• Bounded utilization

EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

YES NO YES NO YES NO coNP-hard coNP-hard NP-hard

Pontus Ekberg 5

A Tale of Two Reductions

EDF-schedulability

• Constrained

deadlines

• Bounded utilization

EDF-schedulability

• Constrained

deadlines

• Bounded utilization

EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

YES NO YES NO YES NO coNP-hard coNP-hard NP-hard

Pontus Ekberg 5

Reducing EDF-schedulability to a Special Case

EDF-schedulability with constrained deadlines and bounded utilization

coNP-hard

With pairwise coprime periods

coNP-hard?

With harmonic periods

Polynomial time Bonifaci et al., 2013

Pontus Ekberg 6

Reducing EDF-schedulability to a Special Case

EDF-schedulability with constrained deadlines and bounded utilization

coNP-hard

With pairwise coprime periods

coNP-hard?

With harmonic periods

Polynomial time Bonifaci et al., 2013

Pontus Ekberg 6

Reducing EDF-schedulability to a Special Case

EDF-schedulability with constrained deadlines and bounded utilization

coNP-hard

With pairwise coprime periods

coNP-hard?

With harmonic periods

Polynomial time Bonifaci et al., 2013

Pontus Ekberg 6

Reducing EDF-schedulability to a Special Case

EDF-schedulability with constrained deadlines and bounded utilization

coNP-hard

With pairwise coprime periods

coNP-hard?

With harmonic periods

Polynomial time Bonifaci et al., 2013

Pontus Ekberg 6

Reducing EDF-schedulability to a Special Case

EDF-schedulability with constrained deadlines and bounded utilization

coNP-hard

With pairwise coprime periods

coNP-hard?

With harmonic periods

Polynomial time Bonifaci et al., 2013

Pontus Ekberg 6

Reducing EDF-schedulability to a Special Case

EDF-schedulability with constrained deadlines and bounded utilization

coNP-hard

With pairwise coprime periods

coNP-hard?

With harmonic periods

Polynomial time†

† Bonifaci et al., 2013

Pontus Ekberg 6

Reducing EDF-schedulability to a Special Case

EDF-schedulability with constrained deadlines and bounded utilization

coNP-hard

With pairwise coprime periods

coNP-hard?

With harmonic periods

Polynomial time†

† Bonifaci et al., 2013

T

Pontus Ekberg 6

Reducing EDF-schedulability to a Special Case

EDF-schedulability with constrained deadlines and bounded utilization

coNP-hard

With pairwise coprime periods

coNP-hard?

With harmonic periods

Polynomial time†

† Bonifaci et al., 2013

T T′

Pontus Ekberg 6

Outline of the Reduction

1 Scale all task parameters uniformly by a huge number

.

2 Add small numbers i to each period so that the periods become

pairwise coprime.

e1 d1 p1 e d p e2 d2 p2 e d p e3 d3 p3 e d p

· · ·

en dn pn en dn pn

n

• Tie

i can be found in polynomial time.

• Tie

i are so small relative to

that schedulability is unafgected.

Pontus Ekberg 7

Outline of the Reduction

1 Scale all task parameters uniformly by a huge number κ. 2 Add small numbers i to each period so that the periods become

pairwise coprime.

e1 d1 p1 e d p e2 d2 p2 e d p e3 d3 p3 e d p

· · ·

en dn pn en dn pn

n

• Tie

i can be found in polynomial time.

• Tie

i are so small relative to

that schedulability is unafgected.

Pontus Ekberg 7

Outline of the Reduction

1 Scale all task parameters uniformly by a huge number κ. 2 Add small numbers i to each period so that the periods become

pairwise coprime.

e d p κe1 κd1 κp1 e d p κe2 κd2 κp2 e d p κe3 κd3 κp3

· · ·

en dn pn κen κdn κpn

n

• Tie

i can be found in polynomial time.

• Tie

i are so small relative to

that schedulability is unafgected.

Pontus Ekberg 7

Outline of the Reduction

1 Scale all task parameters uniformly by a huge number κ. 2 Add small numbers δi to each period so that the periods become

pairwise coprime.

e d p κe1 κd1 κp1 e d p κe2 κd2 κp2 e d p κe3 κd3 κp3

· · ·

en dn pn κen κdn κpn

n

• Tie

i can be found in polynomial time.

• Tie

i are so small relative to

that schedulability is unafgected.

Pontus Ekberg 7

Outline of the Reduction

1 Scale all task parameters uniformly by a huge number κ. 2 Add small numbers δi to each period so that the periods become

pairwise coprime.

e d p κe1 κd1 κp1 e d p κe2 κd2 κp2 δ2 e d p κe3 κd3 κp3

· · ·

en dn pn κen κdn κpn

n

• Tie

i can be found in polynomial time.

• Tie

i are so small relative to

that schedulability is unafgected.

Pontus Ekberg 7

Outline of the Reduction

1 Scale all task parameters uniformly by a huge number κ. 2 Add small numbers δi to each period so that the periods become

pairwise coprime.

e d p κe1 κd1 κp1 e d p κe2 κd2 κp2 δ2 e d p κe3 κd3 κp3 δ3

· · ·

en dn pn κen κdn κpn

n

• Tie

i can be found in polynomial time.

• Tie

i are so small relative to

that schedulability is unafgected.

Pontus Ekberg 7

Outline of the Reduction

1 Scale all task parameters uniformly by a huge number κ. 2 Add small numbers δi to each period so that the periods become

pairwise coprime.

e d p κe1 κd1 κp1 e d p κe2 κd2 κp2 δ2 e d p κe3 κd3 κp3 δ3

· · ·

en dn pn κen κdn κpn δn

• Tie

i can be found in polynomial time.

• Tie

i are so small relative to

that schedulability is unafgected.

Pontus Ekberg 7

Outline of the Reduction

1 Scale all task parameters uniformly by a huge number κ. 2 Add small numbers δi to each period so that the periods become

pairwise coprime.

e d p κe1 κd1 κp1 e d p κe2 κd2 κp2 δ2 e d p κe3 κd3 κp3 δ3

· · ·

en dn pn κen κdn κpn δn

• Tie δi can be found in polynomial time.
• Tie

i are so small relative to

that schedulability is unafgected.

Pontus Ekberg 7

Outline of the Reduction

1 Scale all task parameters uniformly by a huge number κ. 2 Add small numbers δi to each period so that the periods become

pairwise coprime.

e d p κe1 κd1 κp1 e d p κe2 κd2 κp2 δ2 e d p κe3 κd3 κp3 δ3

· · ·

en dn pn κen κdn κpn δn

• Tie δi can be found in polynomial time.
• Tie δi are so small relative to κ that schedulability is unafgected.

Pontus Ekberg 7

Some Number Theory

Tie Jacobsthal function g(n) gives the largest gap between numbers that are coprime to n. Tie Jacobsthal function Coprime to ? g g n log n

Iwaniec, 1978

Pontus Ekberg 8

Some Number Theory

Tie Jacobsthal function g(n) gives the largest gap between numbers that are coprime to n. Tie Jacobsthal function Coprime to 100?

5 10 15 20 25 30 35

g g n log n

Iwaniec, 1978

Pontus Ekberg 8

Some Number Theory

Tie Jacobsthal function g(n) gives the largest gap between numbers that are coprime to n. Tie Jacobsthal function Coprime to 100?

5 10 15 20 25 30 35

g g n log n

Iwaniec, 1978

Pontus Ekberg 8

Some Number Theory

Tie Jacobsthal function g(n) gives the largest gap between numbers that are coprime to n. Tie Jacobsthal function Coprime to 100?

5 10 15 20 25 30 35

g(100) = 4 g n log n

Iwaniec, 1978

Pontus Ekberg 8

Some Number Theory

Tie Jacobsthal function g(n) gives the largest gap between numbers that are coprime to n. Tie Jacobsthal function Coprime to 100?

5 10 15 20 25 30 35

g(100) = 4 g(n) ∈ O(log2 n)

Iwaniec, 1978

Pontus Ekberg 8

Outline of the Reduction

1 Scale all task parameters uniformly by a huge number κ. 2 Add small numbers δi to each period so that the periods become

pairwise coprime.

κe1 κd1 κp1 κe2 κd2 κp2 δ2 κe3 κd3 κp3 δ3

· · ·

κen κdn κpn δn

• Poly. length
• Tie δi can be found in polynomial time.
• Tie δi are so small relative to κ that schedulability is unafgected.

Pontus Ekberg 9

Outline of the Reduction

1 Scale all task parameters uniformly by a huge number κ. 2 Add small numbers δi to each period so that the periods become

pairwise coprime.

κe1 κd1 κp1 κe2 κd2 κp2 δ2 κe3 κd3 κp3 δ3

· · ·

κen κdn κpn δn

• Poly. length
• Tie δi can be found in polynomial time.
• Tie δi are so small relative to κ that schedulability is unafgected.

Pontus Ekberg 9

Outline of the Reduction

1 Scale all task parameters uniformly by a huge number κ. 2 Add small numbers δi to each period so that the periods become

pairwise coprime.

κe1 κd1 κp1 κe2 κd2 κp2 δ2 κe3 κd3 κp3 δ3

· · ·

κen κdn κpn δn

• Poly. length
• Tie δi can be found in polynomial time.

✓

• Tie δi are so small relative to κ that schedulability is unafgected.

Pontus Ekberg 9

Schedulability unaffected

+ + + =

Shifued how much? Shifued at most maxi

i HP

Schedulability unafgected if maxi

i HP Pontus Ekberg 10

Schedulability unaffected

+ + + =

Shifued how much? Shifued at most maxi

i HP

Schedulability unafgected if maxi

i HP Pontus Ekberg 10

Schedulability unaffected

+ + + =

⩾ 1

Shifued how much? Shifued at most maxi

i HP

Schedulability unafgected if maxi

i HP Pontus Ekberg 10

Schedulability unaffected

+ + + =

⩾κ

Shifued how much? Shifued at most maxi

i HP

Schedulability unafgected if maxi

i HP Pontus Ekberg 10

Schedulability unaffected

+ + + =

⩾κ

Shifued how much? Shifued at most maxi

i HP

Schedulability unafgected if maxi

i HP Pontus Ekberg 10

Schedulability unaffected

+ + + =

⩾κ

Shifued how much? Shifued at most maxi δi · HP

Schedulability unafgected if maxi

i HP Pontus Ekberg 10

Schedulability unaffected

+ + + =

⩾κ

Shifued how much? Shifued at most maxi δi · HP

Schedulability unafgected if κ > maxi δi · HP

Pontus Ekberg 10

Outline of the Reduction

1 Scale all task parameters uniformly by a huge number κ. 2 Add small numbers δi to each period so that the periods become

pairwise coprime.

κe1 κd1 κp1 κe2 κd2 κp2 δ2 κe3 κd3 κp3 δ3

· · ·

κen κdn κpn δn

• Poly. length
• Tie δi can be found in polynomial time.

✓

• Tie δi are so small relative to κ that schedulability is unafgected.

Pontus Ekberg 11

Outline of the Reduction

1 Scale all task parameters uniformly by a huge number κ. 2 Add small numbers δi to each period so that the periods become

pairwise coprime.

κe1 κd1 κp1 κe2 κd2 κp2 δ2 κe3 κd3 κp3 δ3

· · ·

κen κdn κpn δn

• Poly. length
• Tie δi can be found in polynomial time.

✓

• Tie δi are so small relative to κ that schedulability is unafgected. ✓

Pontus Ekberg 11

A Tale of Two Reductions

EDF-schedulability

• Constrained

deadlines

• Bounded utilization

EDF-schedulability

• Constrained

deadlines

• Bounded utilization
• Pairwise coprime

periods FP-schedulability

• Implicit deadlines

—or—

• Constrained

deadlines

• Bounded utilization

YES NO YES NO YES NO coNP-hard coNP-hard NP-hard

Pontus Ekberg 12

Conclusions

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Weakly NP-complete Weakly NP-complete Weakly NP-hard Utilization bounded by a constant c Polynomial time for c ⩽ ln 2 and RM priorities Weakly NP-complete for 0 < c < 1 Weakly NP-hard for 0 < c < 1

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for 0 < c < 1 Weakly coNP-complete for 0 < c < 1 Pontus Ekberg 13

Conclusions

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Weakly NP-complete Weakly NP-complete Weakly NP-hard Utilization bounded by a constant c Polynomial time for c ⩽ ln 2 and RM priorities Weakly NP-complete for 0 < c < 1 Weakly NP-hard for 0 < c < 1

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for 0 < c < 1 Weakly coNP-complete for 0 < c < 1 Pontus Ekberg 13

Conclusions

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Weakly NP-complete Weakly NP-complete Weakly NP-hard Utilization bounded by a constant c Polynomial time for c ⩽ ln 2 and RM priorities Weakly NP-complete for 0 < c < 1 Weakly NP-hard for 0 < c < 1

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for 0 < c < 1 Weakly coNP-complete for 0 < c < 1 Pontus Ekberg 13

Conclusions

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Weakly NP-complete Weakly NP-complete Weakly NP-hard Utilization bounded by a constant c Polynomial time for c ⩽ ln 2 and RM priorities Weakly NP-complete for 0 < c < 1 Weakly NP-hard for 0 < c < 1

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for 0 < c < 1 Weakly coNP-complete for 0 < c < 1 Pontus Ekberg 13

Conclusions

Implicit deadlines (d = p) Constrained deadlines (d ⩽ p) Arbitrary deadlines (d, p unrelated)

FP

Arbitrary utilization Weakly NP-complete Weakly NP-complete Weakly NP-hard Utilization bounded by a constant c Polynomial time for c ⩽ ln 2 and RM priorities Weakly NP-complete for 0 < c < 1 Weakly NP-hard for 0 < c < 1

EDF

Arbitrary utilization Polynomial time Strongly coNP-complete Strongly coNP-complete Utilization bounded by a constant c Polynomial time Weakly coNP-complete for 0 < c < 1 Weakly coNP-complete for 0 < c < 1 Pontus Ekberg 13

∀Tiank you! ⋄ ∃Qvestions?

Pontus Ekberg 14