schedulability analysis of synchronous digraph real-time tasks · introduction real-timetaskmodels:...
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Schedulability Analysis ofSynchronous Digraph Real-Time Tasks
Morteza Mohaqeqi, Jakaria Abdullah, Nan Guan, Wang Yi
Uppsala University
ECRTS 2016
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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
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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
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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〉
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
Synchronous Digraph Real-Time Tasks - 2/18 - Mohaqeqi, Abdullah, Guan and Yi
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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〉
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
Synchronous Digraph Real-Time Tasks - 2/18 - Mohaqeqi, Abdullah, Guan and Yi
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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〉
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
Synchronous Digraph Real-Time Tasks - 2/18 - Mohaqeqi, Abdullah, Guan and Yi
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Outline
1 A Review on DRT
2 Synchronous DRT
3 Schedulability Analysis
4 Conclusion
Synchronous Digraph Real-Time Tasks - 3/18 - Mohaqeqi, Abdullah, Guan and Yi
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Synchronous DRT
Synchronized Release
Synchronous Digraph Real-Time Tasks - 4/18 - Mohaqeqi, Abdullah, Guan and Yi
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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
Synchronous Digraph Real-Time Tasks - 5/18 - Mohaqeqi, Abdullah, Guan and Yi
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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
Synchronous Digraph Real-Time Tasks - 5/18 - Mohaqeqi, Abdullah, Guan and Yi
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Overview
AssumptionsUniprocessorPreemptive schedulingFixed priority
Contributions� Schedulability analysis� Heuristics for better efficiency
Synchronous Digraph Real-Time Tasks - 6/18 - Mohaqeqi, Abdullah, Guan and Yi
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Overview
AssumptionsUniprocessorPreemptive schedulingFixed priority
Contributions� Schedulability analysis� Heuristics for better efficiency
Synchronous Digraph Real-Time Tasks - 6/18 - Mohaqeqi, Abdullah, Guan and Yi
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Outline
1 A Review on DRT
2 Synchronous DRT
3 Schedulability Analysis
4 Conclusion
Synchronous Digraph Real-Time Tasks - 7/18 - Mohaqeqi, Abdullah, Guan and Yi
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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)
Synchronous Digraph Real-Time Tasks - 8/18 - Mohaqeqi, Abdullah, Guan and Yi
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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)
Synchronous Digraph Real-Time Tasks - 8/18 - Mohaqeqi, Abdullah, Guan and Yi
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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.
Synchronous Digraph Real-Time Tasks - 9/18 - Mohaqeqi, Abdullah, Guan and Yi
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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.
Synchronous Digraph Real-Time Tasks - 9/18 - Mohaqeqi, Abdullah, Guan and Yi
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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.
Synchronous Digraph Real-Time Tasks - 9/18 - Mohaqeqi, Abdullah, Guan and Yi
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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
Synchronous Digraph Real-Time Tasks - 10/18 - Mohaqeqi, Abdullah, Guan and Yi
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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
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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
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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
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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
Synchronous Digraph Real-Time Tasks - 13/18 - Mohaqeqi, Abdullah, Guan and Yi
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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
Synchronous Digraph Real-Time Tasks - 13/18 - Mohaqeqi, Abdullah, Guan and Yi
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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
Number of Total Actions (Utilization = 0.7)
Run
-Tim
e(secon
ds)
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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
Number of Total Actions (Utilization = 0.7)
Run
-Tim
e(secon
ds)
![Page 27: Schedulability Analysis of Synchronous Digraph Real-Time Tasks · Introduction Real-TimeTaskModels: Liu&Layland multiframe(MF) sporadic generalizedMF(GMF) non-cyclic GMF recurringbranching(RB)](https://reader033.vdocument.in/reader033/viewer/2022042112/5e8d49c562fadd2e941b521a/html5/thumbnails/27.jpg)
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
Synchronous Digraph Real-Time Tasks - 15/18 - Mohaqeqi, Abdullah, Guan and Yi
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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
Synchronous Digraph Real-Time Tasks - 15/18 - Mohaqeqi, Abdullah, Guan and Yi
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Experiments
0 10 20 30 400
5
10
15
Number of Total Actions (Utilization = 0.5)
Run
-Tim
e(secon
ds) Without abstraction and refinement
With abstraction and refinement
0 10 20 30 400
5
10
15
Number of Total Actions (Utilization = 0.7)
Run
-Tim
e(secon
ds) 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
Synchronous Digraph Real-Time Tasks - 17/18 - Mohaqeqi, Abdullah, Guan and Yi
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Conclusion and Future Work
SDRT as an extension of DRT
Expressivenessperodicsporadi
. . . DRT SDRT Timed Automata
Multicore Scheduling• Task-level paritioning• Job-level paritioning
Synchronous Digraph Real-Time Tasks - 18/18 - Mohaqeqi, Abdullah, Guan and Yi
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Conclusion and Future Work
SDRT as an extension of DRT
Expressivenessperodicsporadi
. . . DRT SDRT Timed Automata
Multicore Scheduling• Task-level paritioning• Job-level paritioning
Synchronous Digraph Real-Time Tasks - 18/18 - Mohaqeqi, Abdullah, Guan and Yi
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Schedulability Analysis ofSynchronous Digraph Real-Time Tasks
Morteza Mohaqeqi, Jakaria Abdullah, Nan Guan, Wang Yi
Uppsala University
ECRTS 2016
Thanks!
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Appendix
Request Function DominanceAbstraction and RefinementExperiment SettingExperiments: Path Combinations (RF Dominance)Experiments: Acceptance RatioWhy Synchronized Release?Multirate TasksCritical InstantSDRT vs. DAG
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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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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)
![Page 37: Schedulability Analysis of Synchronous Digraph Real-Time Tasks · Introduction Real-TimeTaskModels: Liu&Layland multiframe(MF) sporadic generalizedMF(GMF) non-cyclic GMF recurringbranching(RB)](https://reader033.vdocument.in/reader033/viewer/2022042112/5e8d49c562fadd2e941b521a/html5/thumbnails/37.jpg)
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
![Page 38: Schedulability Analysis of Synchronous Digraph Real-Time Tasks · Introduction Real-TimeTaskModels: Liu&Layland multiframe(MF) sporadic generalizedMF(GMF) non-cyclic GMF recurringbranching(RB)](https://reader033.vdocument.in/reader033/viewer/2022042112/5e8d49c562fadd2e941b521a/html5/thumbnails/38.jpg)
Why Execution-Independent Synchronization?
Separation of Computationand Communication• More predictability
Ada’s Rendezvous mechanism
Fixed input/output instants
![Page 39: Schedulability Analysis of Synchronous Digraph Real-Time Tasks · Introduction Real-TimeTaskModels: Liu&Layland multiframe(MF) sporadic generalizedMF(GMF) non-cyclic GMF recurringbranching(RB)](https://reader033.vdocument.in/reader033/viewer/2022042112/5e8d49c562fadd2e941b521a/html5/thumbnails/39.jpg)
Why Execution-Independent Synchronization?
Separation of Computationand Communication• More predictability
Ada’s Rendezvous mechanism
Fixed input/output instants
![Page 40: Schedulability Analysis of Synchronous Digraph Real-Time Tasks · Introduction Real-TimeTaskModels: Liu&Layland multiframe(MF) sporadic generalizedMF(GMF) non-cyclic GMF recurringbranching(RB)](https://reader033.vdocument.in/reader033/viewer/2022042112/5e8d49c562fadd2e941b521a/html5/thumbnails/40.jpg)
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
![Page 41: Schedulability Analysis of Synchronous Digraph Real-Time Tasks · Introduction Real-TimeTaskModels: Liu&Layland multiframe(MF) sporadic generalizedMF(GMF) non-cyclic GMF recurringbranching(RB)](https://reader033.vdocument.in/reader033/viewer/2022042112/5e8d49c562fadd2e941b521a/html5/thumbnails/41.jpg)
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
![Page 42: Schedulability Analysis of Synchronous Digraph Real-Time Tasks · Introduction Real-TimeTaskModels: Liu&Layland multiframe(MF) sporadic generalizedMF(GMF) non-cyclic GMF recurringbranching(RB)](https://reader033.vdocument.in/reader033/viewer/2022042112/5e8d49c562fadd2e941b521a/html5/thumbnails/42.jpg)
SDRT Modeling Usage
Engine control tasks(Davis-2014, Biondi-2014)
Multirate controllers Rate-dependent behaviour
f1
f2f1 f2
f3
s1
p1 p2
p3
v1
v2v1 v2
s1p4 p5
![Page 43: Schedulability Analysis of Synchronous Digraph Real-Time Tasks · Introduction Real-TimeTaskModels: Liu&Layland multiframe(MF) sporadic generalizedMF(GMF) non-cyclic GMF recurringbranching(RB)](https://reader033.vdocument.in/reader033/viewer/2022042112/5e8d49c562fadd2e941b521a/html5/thumbnails/43.jpg)
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).
![Page 44: Schedulability Analysis of Synchronous Digraph Real-Time Tasks · Introduction Real-TimeTaskModels: Liu&Layland multiframe(MF) sporadic generalizedMF(GMF) non-cyclic GMF recurringbranching(RB)](https://reader033.vdocument.in/reader033/viewer/2022042112/5e8d49c562fadd2e941b521a/html5/thumbnails/44.jpg)
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
![Page 45: Schedulability Analysis of Synchronous Digraph Real-Time Tasks · Introduction Real-TimeTaskModels: Liu&Layland multiframe(MF) sporadic generalizedMF(GMF) non-cyclic GMF recurringbranching(RB)](https://reader033.vdocument.in/reader033/viewer/2022042112/5e8d49c562fadd2e941b521a/html5/thumbnails/45.jpg)
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
![Page 46: Schedulability Analysis of Synchronous Digraph Real-Time Tasks · Introduction Real-TimeTaskModels: Liu&Layland multiframe(MF) sporadic generalizedMF(GMF) non-cyclic GMF recurringbranching(RB)](https://reader033.vdocument.in/reader033/viewer/2022042112/5e8d49c562fadd2e941b521a/html5/thumbnails/46.jpg)
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
![Page 47: Schedulability Analysis of Synchronous Digraph Real-Time Tasks · Introduction Real-TimeTaskModels: Liu&Layland multiframe(MF) sporadic generalizedMF(GMF) non-cyclic GMF recurringbranching(RB)](https://reader033.vdocument.in/reader033/viewer/2022042112/5e8d49c562fadd2e941b521a/html5/thumbnails/47.jpg)
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.
![Page 48: Schedulability Analysis of Synchronous Digraph Real-Time Tasks · Introduction Real-TimeTaskModels: Liu&Layland multiframe(MF) sporadic generalizedMF(GMF) non-cyclic GMF recurringbranching(RB)](https://reader033.vdocument.in/reader033/viewer/2022042112/5e8d49c562fadd2e941b521a/html5/thumbnails/48.jpg)
Future Work
Broadcast synchronizationCritical instant for the general case
![Page 49: Schedulability Analysis of Synchronous Digraph Real-Time Tasks · Introduction Real-TimeTaskModels: Liu&Layland multiframe(MF) sporadic generalizedMF(GMF) non-cyclic GMF recurringbranching(RB)](https://reader033.vdocument.in/reader033/viewer/2022042112/5e8d49c562fadd2e941b521a/html5/thumbnails/49.jpg)
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.