Bo Peng , Nathan Fisher, and Marko Bertogna

Explicit Preemption Placement for Real-Time Conditional Code via Graph Grammars and Dynamic Programming

Bo Peng, Nathan Fisher, and Marko Bertogna

Department of Computer Science, Wayne State University, USA

Algorithmic Research Group, University of Modena, Italy

211

1

2

This research has been supported by US National Science Foundation and the European Commission.

Graph grammars; Dynamic programming.

Optimize the WCET+CRPD of the flowgraphs.

Series-parallel flowgraphs.

Limited-preemption scheduling model in real -time code.

22

Outline

Background

Model

Problem Statement

Solution

33

Outline

Background

Model

Problem Statement

Conclude the presentation

Solution

44

IntroductionBackgroundBackgroundModelModelProblem StatementProblem StatementSolutionsSolutionsConclusionsConclusions

WCET+CRPD analysis

Real-time schedulability analysis

Limited preemption scheduling

Reduce WCET+CRPD

Write A Read A

Write B

55

Limited Preemption SchedulingBackgroundBackgroundModelModelProblem StatementProblem StatementSolutionsSolutionsConclusionsConclusions

CRPD

Cache evictions by preempting higher-priority tasks.

Precise upper bounds on the cache-related preemption delays (CRPD).Delay the preemption to maintain a job's locality.

Reduce WCET when arbitrarily preempt

Cache

TAG AB

Cache eviction

Cache reload

66

Limited Preemption SchedulingBackgroundBackgroundModelModelProblem StatementProblem StatementSolutionsSolutionsConclusionsConclusions

Precise upper bounds on the cache-related preemption delays (CRPD).Delay the preemption to maintain a job's locality.

Reduce WCET when arbitrarily preempt

Reduce CRPD via limited preemption while preserving system schedulability.

Our goal

77

Related WorkBackgroundBackgroundModelModelProblem StatementProblem StatementSolutionsSolutionsConclusionsConclusions

Fixed preemption

point models

Floating preemption point

models

EDF scheduled systems

Burns ['94]Baruah ['05]Bertogna and Baruah ['10]

Fixed Priority scheduled systems

Burns ['94], Bril et al. ['09],Bertogna et al. ['11], Davis et al. ['12]

Yao et al. ['09]

Preemption model

Scheduling algorithm

Code structure

Linear code structure

Bertogna et al.['10, '11]

Conditional code structure

Our work ？

Control flowgraph: : set of basic blocks (BBs) : set of edges : WCET function of BBs : CRPD function of edges : Potential Preemption Point : Effective Preemption Point

88

Model),,,( zsP EVG

},...,,{ 21 nV

VVE ),( 21

0:

0:

E

VC

BackgroundBackgroundModelModelProblem StatementProblem StatementSolutionsSolutionsConclusionsConclusions

PPP

EPP

1

2

3

4s z

99

Can handle "if else" and "switch" statements.

Observation

Model Series-parallel graphs:

Graph Creation

Series Composition

Parallel Composition


1010

Problem StatementBackgroundBackgroundModelModelProblem StatementProblem StatementSolutionsSolutionsConclusionsConclusions

Series-parallel graphs

Sum of BBs' WCETSum of selected

edges' CRPDChoose the path with max WCET+CRPD

NPR from e1 to xUpper bound of NPR from schedulability

analysis

1111

Problem Statement

Find a selection of EPPs that minimizes the WCET+CRPD of a flowgraph.

The cost of any non-preemptive region should less than Q.

A B C D E F G H I3 4 1 2 1 2 1 3 2

1 3 5 5 1 6 3 1


1NPR 2NPR 3NPR 4NPR

Cost=7 Cost=7 Cost=7 Cost=3

Cost = 24

Q=8

Bertogna et al.['11]

Optimal selection of EPPsin sequential flowgraphs

1212

Previous algorithm is not optimal for conditional structure.Use graph grammar and dynamic programming technique.

Upper path: 28Lower path: 25Combined result: 32(U)Optimal result: 27(L)

Observation

Problem Statement

How about the previous algorithm for conditional structure?

A B C

D E F

J K L

3 4 1

2 1 2

1 3 2

1 3

55 1 6

3 1

G H I1 2 3

1 6

7 5


WCET+CRPD

Q=8

1313

Graph Grammar Decompose control flowgraphs (Linear time

parsing) Extended Backus-Naur Form (EBNF)


04/21/23

1414

<SB> <SB>

Dynamic Programming:Sequential Block


0 1 2 3 4 5 6 70 4 4 4 4 4 INF INF INF1 4 4 4 4 INF INF INF INF2 4 4 4 INF INF INF INF INF3 4 4 INF INF INF INF INF INF4 4 INF INF INF INF INF INF INF5 INF INF INF INF INF INF INF INF6 INF INF INF INF INF INF INF INF7 INF INF INF INF INF INF INF INF

0 1 2 3 4 5 6 70 7 7 8 8 INF INF INF INF1 7 8 8 8 INF INF INF INF2 8 8 8 8 INF INF INF INF3 8 8 8 8 INF INF INF INF4 8 8 8 8 INF INF INF INF5 8 8 8 8 INF INF INF INF6 INF INF INF INF INF INF INF INF7 INF INF INF INF INF INF INF INF

Q=84

s z3 1

s A zB

]][[ zsCost Blocks contains and :A B ]][[ zsCost Blocks only contains :B

Cost matrices reflect optimal substructures

4]0][4[]][[:2

4]0][1[]][[:1

CostCost

Cost），，（Cost

ZAS

ZBBA

743)(]][[:2

8413)(]][[:1

prevAZS

prevBAAZS

CostCCost

Cost），（CCost

1515

Corollary

(1)For each pathFour possible situations: whether take the preemption before and after the middle block;Choose the min WCET+CRPD from the four situations.(2)Choose the max WCET+CRPD among all paths.

Theorem

A optimal EPP selection of a CB is combined by optimal selections of all sequential paths.

Theorem

A optimal EPP selection of a larger block is combined by the optimal selections of left block and right block.

Dynamic Programming:Conditional Block and Block Union

Corollary

(1)Preempt at the edge between the two blocks.Left block: cost[i][0];Right block: cost[edge_value][j];WCET+CRPD: cost[i][0]+ edge_value + cost[edge_value][j].(2)Do not preempt at the edge between the two blocks.Leftblock: cost[i][x];Rightblock: cost[Q-x][j];x from 1to Q;WCET+CRPD:min{cost[i][x]+cost[Q-x][j]}. min{cost[i][x]+cost[Q-x][j]}.(3)Choose the smaller WCET+CRPD between (1) and (2) as cost[i][j] of the larger block.

A B C

D E F

J K L

3 4 1

2 1 2

1 3 2

1 3

55 1 6

3 1

G H I1 2 3

1 6

7 5

]0][0[CostQQ

)|(| 3QVO

<SB> <SB>

B4

z3

s

1

A

Q=8


(1) Get the optimal value of when preempt.

(2) Get the optimal value of when do not preempt.

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Dynamic Programming:Conditional Block and Block Union

A B C

D E F

J K L

3 4 1

2 1 2

1 3 2

1 3

55 1 6

3 1

G H I1 2 3

1 6

7 5

]0][0[Cost )|(| 3QVO

Q=8


In conditional structure:Optimal EPPs selection in pseudo-polynomial time.

Contribution

Approximate flexible matrix The WCET+CRPD will be larger than the optimal one. Reduce running time. Time complexity:

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An Alternative HeuristicBackgroundBackgroundModelModelProblem StatementProblem StatementSolutionsSolutionsConclusionsConclusions

... ...

...

...

)|(| 3VO:0i:0j

A B C

D E F

J K L

3 4 1

2 1 2

1 3 2

1 3

55 1 6

3 1

G H I1 2 3

1 6

7 5

]0][0[Cost QQ

Q

i

Q

Q

i0

Q

2

Q

Q

2

0

Q step each time

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Computational Complexity Consider parse tree and cost matrix:

Optimal solution: Heuristic solution:

)|(| 3QVO

)|(| 3VO


Randomly generate control flowgraphs: number of BBs. number of CBs.

WCET of BBs: Gaussian distribution.

CRPD: Correlates adjacent EPPs. Randomly generated

with a gaussian factor.

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Simulations

Bertogna et al.['11]

Similar to a realistic distribution


2020

Simulations: WCETBackgroundBackgroundModelModelProblem StatementProblem StatementSolutionsSolutionsConclusionsConclusions

OPT (green mesh)<Alternative heuristic (black mesh)<SEQ (red mesh)

WCET trend

CRPD +

2121

Simulations: Running timeBackgroundBackgroundModelModelProblem StatementProblem StatementSolutionsSolutionsConclusionsConclusions

Alternative heuristic (black mesh) dominates over OPT and SEQ

Time trend

2222

Simulations: WCETBackgroundBackgroundModelModelProblem StatementProblem StatementSolutionsSolutionsConclusionsConclusions

Smaller size of matrices does not significantly increase WCET+CRPD

Comparison of different setting for heuristics

CRPD +

2323

Simulations: Running timeBackgroundBackgroundModelModelProblem StatementProblem StatementSolutionsSolutionsConclusionsConclusions

Smaller size of matrices significantly decreases running time

Comparison of different setting for heuristics

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Additional Structure

Unrolled LoopNon-Unrolled Loop

(a) Loops:

Calculate the cost matrix of FiEmbed Fi to the main control flowgraph

(b) Function call:

Conlusions: Extend the structure to conditional blocks Optimal algorithm for selection of EPPs An alternative heuristic to reduce running time Exhaustive simulation

Future work: Planar separator theory Parameterized theory NP-Completeness Or, is the problem NP-

Complete?

Do optimal algorithms using polynomial time exist?

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Conlusions and Future WorkBackgroundBackgroundModelModelProblem StatementProblem StatementSolutionsSolutionsFuture WorkFuture Work

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Thanks!

2727

Related WorkEDF scheduled

systemsFixed

preemption point models

Burns ['94] : [11] A. Burns. Preemptive priority based scheduling: An appropriate engineering approach

EDF scheduled systemsFloating


Baruah ['05] : [4] S. Baruah. The limited-preemption uniprocessor scheduling of sporadic task systems.Bertogna and S. Baruah ['10] : [5] M. Bertogna and S. Baruah. Limited preemption EDF scheduling of sporadic task systems.

Fixed Priority scheduled

systemFixed


Burns ['94] : [11] A. Burns. Preemptive priority based scheduling: An appropriate engineering approachBril et al.['12]: [10]Worst-case response time analysis of real-time tasks under fixed-priority scheduling with deferred preemption.Bertogna et al. ['11]: [7] M. Bertogna et al. Improving feasibility of fixed priority tasks using non-preemptive regions.Davis et al. ['12] : [13] R. Davis and M. Bertogna. Optimal fixed priority scheduling with deferred preemption.

Fixed Priority scheduled

systemFloating preemption

point models

Yao et al. ['09]: [19] G. Yao, G. Buttazzo, and M. Bertogna. Bounding the maximum length of non-preemptive regions under fixed priority scheduling.

linear code structure

Fixed preemption

point models

Bertogna et al. ['10]: [6] M. Bertogna et al. Preemption points placement for sporadic task sets.Bertogna et al. ['11]: [8] M. Bertogna et al. Optimal selection of preemption points to minimize preemption overhead.

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Additional Structure: Loops

Whether preempt at ei, ej, eloop : eight possible situationsChoose the smallest one as the value of the cost matrix

(a) Non-Unrolled Loop:

Preemption places inside the loop is not fixed Integrate this structure to conditional structure

(b) Unrolled Loop:

Documents

Bo Peng , Nathan Fisher, and Marko Bertogna