Improving the Performance of Periodic Real-time Processes: a Graph - - PowerPoint PPT Presentation

▶
improving the performance of periodic real time processes
SMART_READER_LITE
LIVE PREVIEW

Improving the Performance of Periodic Real-time Processes: a Graph - - PowerPoint PPT Presentation

Dec 30, 2023 391 likes •806 views

Improving the Performance of Periodic Real-time Processes: a Graph Theoretical Approach Ton Boode Hajo Broersma Jan Broenink Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, The Netherlands August

slide-1
SLIDE 1

Improving the Performance of Periodic Real-time Processes: a Graph Theoretical Approach

Ton Boode Hajo Broersma Jan Broenink Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, The Netherlands August 26, 2013

slide-2
SLIDE 2

Overview

➓ Periodic real-time processes represented by graphs ➓

❧

➓

❛

➓

❞

➓

♥

➓ ➓ Performance of Periodic Real-time Processes 2/26

slide-3
SLIDE 3

Overview

➓ Periodic real-time processes represented by graphs ➓ Cartesian product Hi❧Hj ➓

❛

➓

❞

➓

♥

➓ ➓ Performance of Periodic Real-time Processes 2/26

slide-4
SLIDE 4

Overview

➓ Periodic real-time processes represented by graphs ➓ Cartesian product Hi❧Hj ➓ Weak synchronised product Hi ❛ Hj ➓

❞

➓

♥

➓ ➓ Performance of Periodic Real-time Processes 2/26

slide-5
SLIDE 5

Overview

➓ Periodic real-time processes represented by graphs ➓ Cartesian product Hi❧Hj ➓ Weak synchronised product Hi ❛ Hj ➓ Reduced weak synchronised product Hi ❞ Hj ➓

♥

➓ ➓ Performance of Periodic Real-time Processes 2/26

slide-6
SLIDE 6

Overview

➓ Periodic real-time processes represented by graphs ➓ Cartesian product Hi❧Hj ➓ Weak synchronised product Hi ❛ Hj ➓ Reduced weak synchronised product Hi ❞ Hj ➓ Synchronised product Hi ♥ Hj ➓ ➓ Performance of Periodic Real-time Processes 2/26

slide-7
SLIDE 7

Overview

➓ Periodic real-time processes represented by graphs ➓ Cartesian product Hi❧Hj ➓ Weak synchronised product Hi ❛ Hj ➓ Reduced weak synchronised product Hi ❞ Hj ➓ Synchronised product Hi ♥ Hj ➓ Performance gain, necessary and sufficient conditions ➓ Performance of Periodic Real-time Processes 2/26

slide-8
SLIDE 8

Overview

➓ Periodic real-time processes represented by graphs ➓ Cartesian product Hi❧Hj ➓ Weak synchronised product Hi ❛ Hj ➓ Reduced weak synchronised product Hi ❞ Hj ➓ Synchronised product Hi ♥ Hj ➓ Performance gain, necessary and sufficient conditions ➓ Future work Performance of Periodic Real-time Processes 2/26

slide-9
SLIDE 9

Parallel processes represented by graphs

OBJECT DISTANCE = read distance sensors → compute object distance → distance meas → SKIP ROBOT SPEED = distance meas → compute robot speed → robot speed → SKIP MOTOR SPEED = robot speed → compute motor speed → write motor speed setpoint → SKIP SEQUENCE CONTROL = (OBJECT DISTANCE

  • ROBOT SPEED
  • MOTOR SPEED);

SEQUENCE CONTROL;

Performance of Periodic Real-time Processes 3/26

slide-10
SLIDE 10

Parallel processes represented by graphs

rs c m s wmss rds cod dm dm crs r s v1 v2 v3 v4 v9 v10 v11 v12 v8 v7 v6 v5 MS RS OD

+ +

SQ = asynchronous synchronous

Performance of Periodic Real-time Processes 4/26

slide-11
SLIDE 11

Parallel processes represented by graphs

MS = (V (H1), A(H1), {λ(a)|a P A(H1)}) = ({v1, v2, v3, v4}, {v1v2, v2v3, v3v4}, {(v1v2, rs), (v2v3, cms), (v3v4, wmss)}) RS = (V (H2), A(H2), {λ(a)|a P A(H2)}) = ({v5, v6, v7, v8}, {v5v6, v6v7, v7v8}, {(v5v6, dm), (v6v7, crs), (v7v8, rs)}) OD = (V (H3), A(H3), {λ(a)|a P A(H3)}) = ({v9, v10, v11, v12}, {v9v10, v10v11, v11v12, }, {(v9v10, rds), (v10v11, cod), (v11v12, dm)})

Performance of Periodic Real-time Processes 5/26

slide-12
SLIDE 12

Cartesian product

r s cms w m s s rds cod dm dm crs rs v1v5v9 v4v8v12 v4v8v9 v4v5v12 v1v5v12 v1v8v9 MS RS OD

Performance of Periodic Real-time Processes 6/26

slide-13
SLIDE 13

Weak synchronised product

rs c m s w m s s rds cod dm dm crs rs v1v5v9 v4v8v12 v4v8v9 v4v5v12 v1v5v12 v1v8v9 MS RS OD

Performance of Periodic Real-time Processes 7/26

slide-14
SLIDE 14

Reduced weak synchronised product

rs c m s wmss rds cod dm dm crs rs v1v5v9 v4v8v12 v4v8v11 v4v7v12 v1v5v12 v1v8v11 MS RS OD

Performance of Periodic Real-time Processes 8/26

slide-15
SLIDE 15

Synchronised product intermediate stage

rs dm c m s w m s s rds cod d m crs v1v5v9 v4v8v12 v4v8v11 v4v7v12 v1v5v12 v1v8v11 rs Intermediate stage

Performance of Periodic Real-time Processes 9/26

slide-16
SLIDE 16

Synchronised product

rs c m s wmss rds cod dm crs v1v5v9 v4v8v12 v1v5v11 v2v8v12 MS RS OD v1v6v12 v1v7v12

Performance of Periodic Real-time Processes 10/26

slide-17
SLIDE 17

Multi dimensional pathological example

⇒

a b b c b a b

H2 H1 H2 H1

c a c a

H1 + H2 + H3 H3 + H3

c b b b b

H ) 2 (H1 H3

a c c a b b b b

H 2 H1 H3

c c a a a a c c b b a b b

H2 H1 + H3

c a c

⇐ ⇒ ⇒

Performance of Periodic Real-time Processes 11/26

slide-18
SLIDE 18

Lemma 5

Lemma

Let Hi be an acyclic graph for i = 1, 2, . . . , k, where k ➙ 2. Then ℓ(♥Hi) = ℓ(H1) + ℓ(H2) + . . . + ℓ(Hk) if and only if every Hi has at least one longest path without synchronising arcs.

Performance of Periodic Real-time Processes 12/26

slide-19
SLIDE 19

Lemma 6

Lemma

Let Hi be an acyclic graph for i = 1, 2, . . . , k, where k ➙ 2. Then ℓ(♥Hi) ➔ ℓ(❞Hi) if there exists Hn, Hm, n ✘ m, 1 ↕ n, m ↕ k, such that each longest path in Hn, Hm, contains at least one same labelled synchronising arc.

Performance of Periodic Real-time Processes 13/26

slide-20
SLIDE 20

Theorem 1

Theorem

Let Hi be an acyclic graph for i = 1, 2, . . . , k, where k ➙ 2. Then ℓ(♥Hi) ➔ ℓ(❞Hi) if there exists Hn, Hm, n ✘ m, 1 ↕ n, m ↕ k, such that each longest path in Hn, contains at least one synchronising arc and there is at least one longest path with a same labelled synchronisation arc in Hm.

Performance of Periodic Real-time Processes 14/26

slide-21
SLIDE 21

Future work

➓ Algorithms for optimising the performance gain ➓ ➓ ➓ ➓ ➓ ➓ ➓ ➓ Performance of Periodic Real-time Processes 15/26

slide-22
SLIDE 22

Future work

➓ Algorithms for optimising the performance gain ➓ The number of longest paths in a graph is exponential ➓ ➓ ➓ ➓ ➓ ➓ ➓ Performance of Periodic Real-time Processes 15/26

slide-23
SLIDE 23

Future work

➓ Algorithms for optimising the performance gain ➓ The number of longest paths in a graph is exponential ➓ Scheduling of the synchronised product with internal

deadlines

➓ ➓ ➓ ➓ ➓ ➓ Performance of Periodic Real-time Processes 15/26

slide-24
SLIDE 24

Future work

➓ Algorithms for optimising the performance gain ➓ The number of longest paths in a graph is exponential ➓ Scheduling of the synchronised product with internal deadlines ➓ Memory usage ➓ ➓ ➓ ➓ ➓ Performance of Periodic Real-time Processes 15/26

slide-25
SLIDE 25

Future work

➓ Algorithms for optimising the performance gain ➓ The number of longest paths in a graph is exponential ➓ Scheduling of the synchronised product with internal deadlines ➓ Memory usage ➓ Synchronised product, associativity and commutativity ➓ ➓ ➓ ➓ Performance of Periodic Real-time Processes 15/26

slide-26
SLIDE 26

Future work

➓ Algorithms for optimising the performance gain ➓ The number of longest paths in a graph is exponential ➓ Scheduling of the synchronised product with internal deadlines ➓ Memory usage ➓ Synchronised product, associativity and commutativity ➓ Decomposition of a component into its prime factors ➓ ➓ ➓ Performance of Periodic Real-time Processes 15/26

slide-27
SLIDE 27

Future work

➓ Algorithms for optimising the performance gain ➓ The number of longest paths in a graph is exponential ➓ Scheduling of the synchronised product with internal deadlines ➓ Memory usage ➓ Synchronised product, associativity and commutativity ➓ Decomposition of a component into its prime factors ➓ Constraints for the prime factors of the synchronised

product

➓ ➓ Performance of Periodic Real-time Processes 15/26

slide-28
SLIDE 28

Future work

➓ Algorithms for optimising the performance gain ➓ The number of longest paths in a graph is exponential ➓ Scheduling of the synchronised product with internal deadlines ➓ Memory usage ➓ Synchronised product, associativity and commutativity ➓ Decomposition of a component into its prime factors ➓ Constraints for the prime factors of the synchronised product ➓ Algorithm that calculates prime factors. ➓ Performance of Periodic Real-time Processes 15/26

slide-29
SLIDE 29

Future work

➓ Algorithms for optimising the performance gain ➓ The number of longest paths in a graph is exponential ➓ Scheduling of the synchronised product with internal deadlines ➓ Memory usage ➓ Synchronised product, associativity and commutativity ➓ Decomposition of a component into its prime factors ➓ Constraints for the prime factors of the synchronised product ➓ Algorithm that calculates prime factors. ➓ An example of the decomposition of a graph Performance of Periodic Real-time Processes 15/26

slide-30
SLIDE 30

Decomposition of the original graph into its prime factors H c a b

Performance of Periodic Real-time Processes 16/26

slide-31
SLIDE 31

Decomposition of the original graph into its prime factors

H=H1+H2+H3 H1 H3 H2 a b c c a b ?

Performance of Periodic Real-time Processes 17/26

slide-32
SLIDE 32

Decomposition of the original graph into its prime factors

H1 H2 H2 a b c c a b c b c a c c a b c b c a c c a b c b c a a b a b a b

Performance of Periodic Real-time Processes 18/26

slide-33
SLIDE 33

Decomposition of the original graph into its prime factors

H1 H2 H2 a b c c a b c b c a c c a c c a c c b c b c a b a b a b

Performance of Periodic Real-time Processes 19/26

slide-34
SLIDE 34

Decomposition of the original graph into its prime factors

H1 H2 H2 a b c c a b c b c a c c a c c a c c b c b c a b a b a b

Performance of Periodic Real-time Processes 20/26

slide-35
SLIDE 35

Decomposition of the original graph into its prime factors

intermediate stage (H1 H2 H2) b c a c a c b a

Performance of Periodic Real-time Processes 21/26

slide-36
SLIDE 36

Decomposition of the original graph into its prime factors

H ≠ H2 H3 b c a H1

Performance of Periodic Real-time Processes 22/26

slide-37
SLIDE 37

Decomposition of a component into its prime factors

Performance of Periodic Real-time Processes 23/26

slide-38
SLIDE 38

Memory usage versus performance using decomposition

time memory usage d m

Performance of Periodic Real-time Processes 24/26

slide-39
SLIDE 39

Thank you for listening!

Performance of Periodic Real-time Processes 25/26

slide-40
SLIDE 40

Improving the Performance of Periodic Real-time Processes: a Graph Theoretical Approach

Ton Boode Hajo Broersma Jan Broenink Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, The Netherlands August 26, 2013