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Schedulability Analysis of Synchronous Digraph Real-Time Tasks - - PowerPoint PPT Presentation

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Schedulability Analysis of Synchronous Digraph Real-Time Tasks Morteza Mohaqeqi, Jakaria Abdullah, Nan Guan, Wang Yi Uppsala University ECRTS 2016 Introduction Real-Time Task Models: Digraph (DRT) recurring branching (RB) non-cyclic GMF

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SLIDE 1

Schedulability Analysis of Synchronous Digraph Real-Time Tasks

Morteza Mohaqeqi, Jakaria Abdullah, Nan Guan, Wang Yi Uppsala University ECRTS 2016

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SLIDE 2

Introduction

Real-Time Task Models:

Liu & Layland sporadic multiframe (MF) generalized MF (GMF) non-cyclic GMF recurring branching (RB) Digraph (DRT)

Synchronous Digraph Real-Time Tasks

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Introduction

Real-Time Task Models:

Liu & Layland sporadic multiframe (MF) generalized MF (GMF) non-cyclic GMF recurring branching (RB) Digraph (DRT)

Proposed by M. Stigge et al. (2011) Real-time tasks with different job types

v5 v6 v7 30 9 10 8

Synchronous Digraph Real-Time Tasks

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SLIDE 4

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 minimum inter-release WCET, deadline 15 40 20 25 10

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SLIDE 5

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 minimum inter-release WCET, deadline 15 40 20 25 10

t

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

v1 v2 v4 v3

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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 minimum inter-release WCET, deadline 15 40 20 25 10

t

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

v1 v2 v4 v3 t

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

v1 v2 v1

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Outline

1

A Review on DRT

2

Synchronous DRT

3

Schedulability Analysis

4

Conclusion

Synchronous Digraph Real-Time Tasks

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SLIDE 8

Synchronous DRT

Synchronized Release

Synchronous Digraph Real-Time Tasks

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SLIDE 9

Semantics

v1 v2 v3 v4 4 1 1 2 s1 15 40 25 25 10 Task T1: v5 v6 v7 2 1 1 s1 20 9 10 8 Task T2:

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SLIDE 10

Semantics

v1 v2 v3 v4 4 1 1 2 s1 15 40 25 25 10 Task T1: v5 v6 v7 2 1 1 s1 20 9 10 8 Task T2:

t

5 10 15 20 25 30 35 40

t

5 10 15 20 25 30 35 40

T1 T2 v1 v2 v3 v5 v6

blocked

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SLIDE 11

Overview

Assumptions

Uniprocessor Preemptive scheduling Fixed priority

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SLIDE 12

Overview

Assumptions

Uniprocessor Preemptive scheduling Fixed priority

Contributions

Schedulability analysis Heuristics for better efficiency

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Outline

1

A Review on DRT

2

Synchronous DRT

3

Schedulability Analysis

4

Conclusion

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DRT Schedulability

v1 v2 v3 1 2 1 15 9 10 8

t

5 10 15 20 25 30 35

v1 v2 v3 v3

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SLIDE 15

DRT Schedulability

v1 v2 v3 1 2 1 15 9 10 8

t

5 10 15 20 25 30 35

v1 v2 v3 v3 Request Function t rf (t)

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DRT Schedulability Condition

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

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SLIDE 17

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 a set 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)

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SLIDE 18

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 a set 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.

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SLIDE 19

SDRT Schedulability

v1 v2 v3 1 2 1 15 9 10 8 s1

t

5 10 15 20 25 30 35

v1 v2 v3 v3 t rf (t) s1

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SLIDE 20

Alignment

v1 v2 v3 v4 4 1 1 2 s1 15 40 25 25 10 Task T1: v5 v6 v7 2 1 1 s1 20 9 10 8 Task T2:

5 10 15 20 25 30 35 40

T2

5 10 15 20 25 30 35 40

T1 v1 v2 v3 v5 v6

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SLIDE 21

Alignment

5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40

v1 v2 v3 v5 v6

Unsynchronized

rf 1 s1 rf 2 s1

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SLIDE 22

Alignment

5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40

v1 v2 v3 v5 v6

5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40

v1 v2 v3 v5 v6

blocked Unsynchronized Synchronized (Aligned)

rf 1 s1 rf 2 s1 rf 1 s1 rf 2 s1

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SLIDE 23

SDRT Schedulability Condition

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

Theorem

A job with WCET “e” and relative deadline “d” is schedulable under a set 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

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SLIDE 24

SDRT Schedulability Condition

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

Theorem

A job with WCET “e” and relative deadline “d” is schedulable under a set 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

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SLIDE 25

Experiments: Analysis Efficiency

10 20 30 40 5 10 15 Number of Total Actions (Utilization = 0.5) Run-Time (seconds)

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SLIDE 26

Experiments: Analysis Efficiency

10 20 30 40 5 10 15 Number of Total Actions (Utilization = 0.5) Run-Time (seconds) 10 20 30 40 5 10 15 Number of Total Actions (Utilization = 0.7) Run-Time (seconds)

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SLIDE 27

v1 v2 v3 s1 s2 v4 v5 v6 s1 s2 Step 1 Over-approx.

v1 v2 v3 v4 v5 v6

Under-approx.

v1 v2 v3 v4 v5 v6

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v1 v2 v3 s1 s2 v4 v5 v6 s1 s2 refinement level

1 2 3 ... n

under-approx.

  • ver-approx.

τ

Step 1 Step 2 Over-approx.

v1 v2 v3 v4 v5 v6 v1 v2 v3 s1 v4 v5 v6 s1

Under-approx.

v1 v2 v3 v4 v5 v6 v1 v2 v3 s1 v4 v5 v6 s1

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Experiments

10 20 30 40 5 10 15 Number of Total Actions (Utilization = 0.5) Run-Time (seconds) Without abstraction and refinement With abstraction and refinement 10 20 30 40 5 10 15 Number of Total Actions (Utilization = 0.7) Run-Time (seconds) Without abstraction and refinement With abstraction and refinement

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Outline

1

A Review on DRT

2

Synchronous DRT

3

Schedulability Analysis

4

Conclusion

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SLIDE 31

Conclusion and Future Work

SDRT as an extension of DRT

Expressiveness

perodic sporadi . . . DRT SDRT Timed Automata

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SLIDE 32

Conclusion and Future Work

SDRT as an extension of DRT

Expressiveness

perodic sporadi . . . DRT SDRT Timed Automata Multicore Scheduling

  • Task-level paritioning
  • Job-level paritioning

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SLIDE 33

Schedulability Analysis of Synchronous Digraph Real-Time Tasks

Morteza Mohaqeqi, Jakaria Abdullah, Nan Guan, Wang Yi Uppsala University ECRTS 2016

Thanks!

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SLIDE 34

Appendix

Request Function Dominance Abstraction and Refinement Experiment Setting Experiments: Path Combinations (RF Dominance) Experiments: Acceptance Ratio Why Synchronized Release? Multirate Tasks Critical Instant SDRT vs. DAG

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SLIDE 35

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]

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SLIDE 36

Number of Path Combinations

Number of path combinations that should be considered in schedulability 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 0.2 0.4 0.6 0.8 1 Total combinations Utilization

Total combinations SDRT Dominance (3n actions) SDRT Dominance (n actions) SDRT Dominance (No action)

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SLIDE 37

Schedulability Analysis Results

Schedulability analysis results for different number of synchronizations

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 0.05 178 372 19167

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SLIDE 38

Why Execution-Independent Synchronization?

Separation of Computation and Communication

  • More predictability
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SLIDE 39

Why Execution-Independent Synchronization?

Separation of Computation and Communication

  • More predictability

Ada’s Rendezvous mechanism Fixed input/output instants

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SLIDE 40

SDRT Modeling Usage

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

Rate-dependent behaviour

f1 f2 f3 s1 p1 p2 p3 v1 v2 s1 p4 p5

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SLIDE 41

SDRT Modeling Usage

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

Rate-dependent behaviour

f1 f1 f2 f3 s1 p1 p2 p3 v1 v1 v2 s1 p4 p5

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SLIDE 42

SDRT Modeling Usage

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

Rate-dependent behaviour

f2 f1 f2 f3 s1 p1 p2 p3 v2 v1 v2 s1 p4 p5

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SLIDE 43

Request Function Dominance

t rf (t)

1 2 3 4 5 6 7 8 1 2 3 4 5

rf 2 s rf 1 s ts t′

s

t rf (t)

1 2 3 4 5 6 7 8 1 2 3 4 5

rf 2 s rf 1

Lemma

A 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).

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SLIDE 44

Abstraction and Refinement

Abstraction:

t rf (t)

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6

rf 1 rf 2

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SLIDE 45

Abstraction and Refinement

Abstraction:

t rf (t)

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6

rf 1 ⊔ rf 2 rf 1 rf 2

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SLIDE 46

Abstraction and Refinement

Abstraction: Refinement:

t rf (t)

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6

rf 1 ⊔ rf 2 rf 1 rf 2

The most abstract request function Concrete request functions

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SLIDE 47

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 v3 v3 v3 v3 t

1 2 3 4

J deadline miss

  • f J

The critical instant for J is not necessarily when all the tasks are released simultaneously with J.

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SLIDE 48

Future Work

Broadcast synchronization Critical instant for the general case

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SLIDE 49

References

[Stigge-2013] M. Stigge and W. Yi, “Combinatorial abstraction refinement for feasibility analysis,” 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 protocols for 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 tasks with variable rate-dependent behaviour under fixed priority scheduling,” Real-Time and 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.