schedulability analysis of synchronous digraph real-time tasks · introduction real-timetaskmodels:...

Schedulability Analysis ofSynchronous Digraph Real-Time Tasks

Morteza Mohaqeqi, Jakaria Abdullah, Nan Guan, Wang Yi

Uppsala University

ECRTS 2016

Introduction

Real-Time Task Models:

Liu & Layland

sporadicmultiframe (MF)

generalized MF (GMF)

non-cyclicGMF

recurring branching (RB)

Digraph (DRT)

Proposed by M. Stiggeet al. (2011)Real-time tasks withdifferent job types

v5 v6

v7

30

9

10

8

Synchronous Digraph Real-Time Tasks - 1/18 - Mohaqeqi, Abdullah, Guan and Yi

The Digraph Real-Time (DRT) Task Model

Job Types• WCET• Relative deadline

Conditional flow(Branch)

v1

v2

v3

v4〈4, 15〉

〈1, 10〉

〈2, 5〉

〈1, 20〉

minimuminter-release〈WCET , deadline〉

15

40

20

2510

t0 5 10 15 20 25 30 35 40 45 50 55 60

v1 v2 v4 v3

t0 5 10 15 20 25 30 35 40 45 50 55 60

v1 v2 v1


Outline

1 A Review on DRT

2 Synchronous DRT

3 Schedulability Analysis

4 Conclusion


Synchronous DRT

Synchronized Release


Semantics

v1

v2

v3

v4

4

1

1

2

s1

15

40

25

2510

Task T1:

v5 v6

v7

2 1

1

s1

20

9

10

8

Task T2:

t0 5 10 15 20 25 30 35 40

t0 5 10 15 20 25 30 35 40T1

T2

v1 v2 v3

v5 v6blocked


Overview

AssumptionsUniprocessorPreemptive schedulingFixed priority

Contributions� Schedulability analysis� Heuristics for better efficiency


Outline

1 A Review on DRT

2 Synchronous DRT


4 Conclusion


DRT Schedulability

v1 v2

v3

〈1〉〈2〉

〈1〉

15

9

10

8

t0 5 10 15 20 25 30 35

v1 v2 v3 v3

Request Function

t

rf (t)


DRT Schedulability Condition

Notation:A set of tasks τ = {T1,T2, . . . ,Tn}πi : A path in Ti ’s graph

Theorem (Stigge 2013)A job with WCET “e” and relative deadline “d” is schedulable under aset of higher priority tasks τ if and only if for all (π1, . . . , πn) ∈ Π(τ):

∃t ≤ d : e +∑Ti∈τ

rf πi (t) ≤ t (1)

rf πi (t) could be derived independently.


SDRT Schedulability

v1 v2

v3

〈1〉〈2〉

〈1〉

15

9

10

8

s1

t0 5 10 15 20 25 30 35

v1 v2 v3 v3

t

rf (t)

s1


Alignment

v1

v2

v3

v4

4

1

1

2

s1

15

40

25

2510

Task T1:

v5 v6

v7

2 1

1

s1

20

9

10

8

Task T2:

0 5 10 15 20 25 30 35 40T2

0 5 10 15 20 25 30 35 40T1

v1 v2 v3

v5 v6

Alignment

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

v1 v2 v3

v5 v6

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

v1 v2 v3

v5 v6blocked

Unsynchronized

Synchronized (Aligned)

rf 1

s1

rf 2

s1

rf 1

s1

rf 2

s1

Alignment

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

v1 v2 v3

v5 v6

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

v1 v2 v3

v5 v6blocked

Unsynchronized Synchronized (Aligned)rf 1

s1

rf 2

s1

rf 1

s1

rf 2

s1

SDRT Schedulability Condition

τ = {T1,T2, . . . ,Tn}πi : A path in Ti ’s graph

TheoremA job with WCET “e” and relative deadline “d” is schedulable under aset of tasks τ if and only if for all π = (π1, . . . , πn) ∈ Π(τ), ∀R ∈ RF π:

∃t ≤ d : e +∑

rf i∈Synch(R)Ti∈τhp

rf i (t) ≤ t

Efficient Exploration� Removing dominated request function� Search using an “abstraction and refinement” approach


Experiments: Analysis Efficiency

0 10 20 30 400

5

10

15

Number of Total Actions (Utilization = 0.5)

Run

-Tim

e(secon

ds)

0 10 20 30 400

5

10

15


Run

-Tim

e(secon

ds)

v1 v2

v3

s1

s2

v4 v5

v6

s1

s2

refinementlevel

0 1 2 3 ... n

under-approx.

over-approx.

τ

Step 1

Step 2

Over-approx. v1 v2

v3

v4 v5

v6

v1 v2

v3

s1v4 v5

v6

s1

Und

er-app

rox. v1 v2

v3

v4 v5

v6

v1 v2

v3

s1v4 v5

v6

s1


v1 v2

v3

s1

s2

v4 v5

v6

s1

s2

refinementlevel

0 1 2 3 ... n

under-approx.

over-approx.

τ

Step 1 Step 2

Over-approx. v1 v2

v3

v4 v5

v6

v1 v2

v3

s1v4 v5

v6

s1

Und

er-app

rox. v1 v2

v3

v4 v5

v6

v1 v2

v3

s1v4 v5

v6

s1


Experiments

0 10 20 30 400

5

10

15


Run

-Tim

e(secon

ds) Without abstraction and refinement

With abstraction and refinement

0 10 20 30 400

5

10

15


Run

-Tim

e(secon

ds) Without abstraction and refinement

With abstraction and refinement

Outline

1 A Review on DRT

2 Synchronous DRT


4 Conclusion


Conclusion and Future Work

SDRT as an extension of DRT

Expressivenessperodicsporadi

. . . DRT SDRT Timed Automata

Multicore Scheduling• Task-level paritioning• Job-level paritioning


Schedulability Analysis ofSynchronous Digraph Real-Time Tasks

Morteza Mohaqeqi, Jakaria Abdullah, Nan Guan, Wang Yi

Uppsala University

ECRTS 2016

Thanks!

Appendix

Request Function DominanceAbstraction and RefinementExperiment SettingExperiments: Path Combinations (RF Dominance)Experiments: Acceptance RatioWhy Synchronized Release?Multirate TasksCritical InstantSDRT vs. DAG

Experiment Settings

Table: Task set parameters

Task Type Small Medium Large

Vertices [3, 5] [5, 9] [7, 13]

Branching degree [1, 3] [1, 4] [1, 5]

p [50, 100] [100, 200] [200, 400]

e [1, 2] [1, 4] [1, 8]

d [25, 100] [50, 200] [100, 400]

Number of Path Combinations

Number of path combinations that should be considered inschedulability analysis

1.00E+00

1.00E+06

1.00E+12

1.00E+18

1.00E+24

1.00E+30

1.00E+36

1.00E+42

1.00E+48

1.00E+54

1.00E+60

1.00E+66

0 0.2 0.4 0.6 0.8 1

To

tal co

mbin

atio

ns

Utilization

Total combinations

SDRT Dominance (3n actions)

SDRT Dominance (n actions)

SDRT Dominance (No action)

Schedulability Analysis Results

Schedulability analysis results for different number ofsynchronizations

Acceptance Ratio Tested Combinations

Util. No act. n act. 3n act. No act. n act. 3n act.

0.35 1 1 1 37 37 37

0.4 1 1 1 52 52 52

0.45 1 1 1 70 70 70

0.5 0.94 0.96 0.96 116 165 14768

0.55 0.6 0.77 0.85 154 218 46694

0.6 0.1 0.19 0.26 225 392 59114

0.65 0 0 0.05 178 372 19167

Why Execution-Independent Synchronization?

Separation of Computationand Communication• More predictability

Ada’s Rendezvous mechanism

Fixed input/output instants

SDRT Modeling Usage

Engine control tasks(Davis-2014, Biondi-2014)

Multirate controllers Rate-dependent behaviour

f1 f2

f1 f2

f3

s1

p1 p2

p3

v1 v2

v1 v2

s1p4 p5

SDRT Modeling Usage



f1

f2

f1 f2

f3

s1

p1 p2

p3

v1

v2

v1 v2

s1p4 p5

SDRT Modeling Usage



f1

f2f1 f2

f3

s1

p1 p2

p3

v1

v2v1 v2

s1p4 p5

Request Function Dominance

t

rf (t)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

rf 2s

rf 1

s

ts t′s

t

rf (t)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

rf 2s

rf 1

LemmaA request function rf 1 dominates a request function rf 2 if:

1 ∀t : rf 1(t) ≥ rf 2(t),

2 rf 1 and rf 2 contain the same sequence of actions, and

3 (ASrf1 is empty) or (ts ≤ t′s and rf 1(ts) ≥ rf 2(t′s) and rf ′1 dominates rf ′

2), where(s, ts) = ASrf 1 [0], (s, t

′s) = ASrf 2 [0], and rf ′

1 and rf ′2 are obtained by

Align_and_Pop(rf 1 , rf 2 , s).

Abstraction and Refinement

Abstraction:

Refinement:

t

rf (t)

0 1 2 3 4 5 6 7 8 9 10 11 120

1

2

3

4

5

6 rf 1rf 2

The most abstractrequest function

Concreterequest functions

Abstraction and Refinement

Abstraction:

Refinement:

t

rf (t)

0 1 2 3 4 5 6 7 8 9 10 11 120

1

2

3

4

5

6rf 1 t rf 2

rf 1rf 2

The most abstractrequest function

Concreterequest functions

Critical (Scheduling) Instant

v1 v2〈1, 1〉〈1, 5〉

1

5

s1

T1

v3

〈1, 3〉

3

T2

J

〈1, 3〉

3, s1

T3

v1 v2

v1 v2

v3v3

v3 v3

t0 1 2 3 4

Jdeadline missof J

The critical instant for J is not

necessarily when all the tasks are

released simultaneously with J.

Future Work

Broadcast synchronizationCritical instant for the general case

References

[Stigge-2013] M. Stigge and W. Yi, “Combinatorial abstraction refinement for feasibilityanalysis,” Real-Time Systems Symposium (RTSS), 2013.

[Sun-2016] J. Sun, N. Guan, Y. Wang, Q. Deng, P. Zeng, and W. Yi, “Feasibility of fork-join real-time task graph models: hardness and algorithms,” ACM Trans. Embed.Comput. Syst. (TECS) 2016.

[Guan-2011] N. Guan, P. Ekberg, M. Stigge and W. Yi, “Resource sharing protocolsfor real-time task graph systems,” Euromicro Conference on Real-Time Systems(ECRTS), 2011.

[Biondi-2104] R. I. Davis, T. Feld, V. Pollex and F. Slomka,“Schedulability tests for taskswith variable rate-dependent behaviour under fixed priority scheduling,” Real-Timeand Embedded Technology and Applications Symposium (RTAS), 2014.

[Davis-2104] A. Biondi, A. Melani, M. Marinoni, M. D. Natale and G. Buttazzo, “Ex-act interference of adaptive variable-rate tasks under fixed-priority scheduling,”Euromicro Conference on Real-Time Systems, Madrid, 2014.

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