Real-Time Workload Models with E cient Analysis · The Digraph Real-Time (DRT) Task Model Adaptive...
Transcript of Real-Time Workload Models with E cient Analysis · The Digraph Real-Time (DRT) Task Model Adaptive...
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Real-Time Workload Models with Efficient AnalysisAdvanced Course, 3 Lectures, September 2014
Martin Stigge
Uppsala University, Sweden
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Fahrplan
1 DRT Tasks in the Model HierarchyLiu and Layland and Sporadic TasksFrames and BranchingThe Digraph Real-Time (DRT) Task ModelAdaptive Variable-Rate (AVR) Tasks
2 Feasibility Analysis of DRTFeasibility TheoremDemand PairsTest TerminationEvaluation
3 Static Priority Schedulability Analysis of DRTResponse-Time AnalysisRequest FunctionsRefinement AlgorithmEvaluation
Martin Stigge Workload Models + Analysis 2
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Fahrplan
1 DRT Tasks in the Model HierarchyLiu and Layland and Sporadic TasksFrames and BranchingThe Digraph Real-Time (DRT) Task ModelAdaptive Variable-Rate (AVR) Tasks
2 Feasibility Analysis of DRTFeasibility TheoremDemand PairsTest TerminationEvaluation
3 Static Priority Schedulability Analysis of DRTResponse-Time AnalysisRequest FunctionsRefinement AlgorithmEvaluation
Martin Stigge Workload Models + Analysis 2
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Problem Overview
Workload Model
Task A
Task B
Task C
DRT
High priority
A1
A2
A3
A4
Medium priority
B1
B2
B3
Low priorityC
Scheduler Model
EDF/Static Prio/...
Task At
Task Bt
Feasible? Schedulable? Response times?
Our Setting:• DRT tasks
• Static Priorities
• Precise Test
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The Liu and Layland (L&L) Task Model(Liu and Layland, 1973)
• Tasks are periodic• Job WCET e• Period p (implicit deadline)
� ;
(e, p)e e
tp p
...
• Advantages: Well-known model; efficient schedulability tests
• Disadvantage: Very limited expressiveness
Martin Stigge Workload Models + Analysis 4
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A Hierarchy of Models
Liu & Layland(e, d = p)
efficient
Sch
edu
lab
ility
An
alys
is
difficult
low
Exp
ress
iven
ess
high
Martin Stigge Workload Models + Analysis 5
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The Sporadic Task Model(Mok, 1983)
• Deadlines 6= periods; releases sporadic
• Each tasks defined by:• Job WCET e• Relative deadline d• Minimum inter-release delay p
� ;
(e, d , p)e e
t6 p 6 p
d d...
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The General Multiframe (GMF) Task Model
• Behavior is not always periodic
Frame 0 Frame 1 Frame 2 Frame 3 Frame 0
t
• Task is split into frames, each with own• Execution time e(j)
• Inter-release separation p(j)
• Deadline d (j) for the job
Martin Stigge Workload Models + Analysis 7
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The General Multiframe (GMF) Task Model
• Behavior is not always periodic
Frame 0 Frame 1 Frame 2 Frame 3 Frame 0
t
• Task is split into frames, each with own• Execution time e(j)
• Inter-release separation p(j)
• Deadline d (j) for the job
Martin Stigge Workload Models + Analysis 7
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The General Multiframe (GMF) Task Model (cont.)(Baruah et al., 1999)
• Tasks cycle through job types• Vector for WCET (e(1), . . . , e(n))• Vector for deadlines (d (1), . . . , d (n))• Vector for minimum inter-release delays (p(1), . . . , p(n))
J1
J2
J3
J4
J5
p(1)p(2)
p(3)
p(4)p(5)
;e(1) e(2)
t
6 p(1) 6 p(2)
d (1) d (2)
...
Martin Stigge Workload Models + Analysis 8
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A Hierarchy of Models
Liu & Layland(e, d = p)
sporadic(e, d , p)
multiframe(ei , d = p)
generalized multiframe (GMF)(ei , di , pi )
efficient
Sch
edu
lab
ility
An
alys
is
difficult
low
Exp
ress
iven
ess
high
Martin Stigge Workload Models + Analysis 9
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The Non-Cyclic GMF Task Model(Moyo et al., 2010)
• Frame order unknown a priori
• Syntax similar to GMF:• Vector for WCET (e(1), . . . , e(n))• Vector for deadlines (d (1), . . . , d (n))• Vector for minimum inter-release delays (p(1), . . . , p(n))
J1
J2
J3
J4
J5
;e(1) e(2)
t
6 p(1) 6 p(2)
d (1) d (2)
...
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The Recurring Branching (RB) Task Model(Baruah, 1998)
• Introduces branching structures
• Tree for tasks• Vertices J: jobs to be released (with WCET and deadline)• Edges (Ji , Jj): minimum inter-release delays p(Ji , Jj)• General period parameter P
J1
J2
J3
J4
J5
J6
Period P = 57
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The Recurring Branching (RB) Task Model(Baruah, 1998)
• Introduces branching structures
• Tree for tasks• Vertices J: jobs to be released (with WCET and deadline)• Edges (Ji , Jj): minimum inter-release delays p(Ji , Jj)• General period parameter P
J1
J2
J3
J4
J5
J6
Period P = 57
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The Recurring Real-Time (RRT) Task Model(Baruah, 1998)
• Compact Branching representation
• Directed acyclic graph (DAG) for tasks• Vertices J: jobs to be released (with WCET and deadline)• Edges (Ji , Jj): minimum inter-release delays p(Ji , Jj)• General period parameter P
J1
J2
J3
J4
J5
Period P = 57
Martin Stigge Workload Models + Analysis 12
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A Hierarchy of Models
Liu & Layland(e, d = p)
sporadic(e, d , p)
multiframe(ei , d = p)
generalized multiframe (GMF)(ei , di , pi )
recurring branching (RB)(tree, p)
recurring RT (RRT)(DAG, p)
non-cyclic GMF(order arbitrary)
non-cyclic RRT(DAG, pi )
efficient
Sch
edu
lab
ility
An
alys
is
difficult
low
Exp
ress
iven
ess
high
Martin Stigge Workload Models + Analysis 13
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Restrictions of RRT
• Tasks are still recurrent
• Always revisit source J1
• No cycles allowed!
J1
J2
J3
J4
J5
J6
• Consequences:
• No local loops• Not compositional
(for modes etc.)
Martin Stigge Workload Models + Analysis 14
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Restrictions of RRT
• Tasks are still recurrent
• Always revisit source J1
• No cycles allowed!
J1
J2
J3
J4
J5
J6
• Consequences:
• No local loops• Not compositional
(for modes etc.)
Martin Stigge Workload Models + Analysis 14
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Restrictions of RRT
• Tasks are still recurrent
• Always revisit source J1
• No cycles allowed!
J1
J2
J3
J4
J5
J6
• Consequences:
• No local loops• Not compositional
(for modes etc.)
Martin Stigge Workload Models + Analysis 14
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The Digraph Real-Time (DRT) Task Model(S. et al., 2011)
• Generalizes periodic, sporadic, GMF, RRT, . . .
• Directed graph for each task• Vertices v : jobs to be released (with WCET and deadline)• Edges (u, v): minimum inter-release delays p(u, v)
v1〈2, 5〉
v2
〈1, 8〉v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
Martin Stigge Workload Models + Analysis 15
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DRT: Semantics
v1〈2, 5〉
v2
〈1, 8〉v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
Path π = (v4)Path π = (v4, v2)Path π = (v4, v2, v3)
0 10 20 28
20
35 43
15
5
10
1
8
3
8...
t
Martin Stigge Workload Models + Analysis 16
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DRT: Semantics
v1〈2, 5〉
v2
〈1, 8〉v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
Path π = (v4)
Path π = (v4, v2)Path π = (v4, v2, v3)
0 10
20 28
20
35 43
15
5
10
1
8
3
8...
t
Martin Stigge Workload Models + Analysis 16
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DRT: Semantics
v1〈2, 5〉
v2
〈1, 8〉v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
Path π = (v4)
Path π = (v4, v2)
Path π = (v4, v2, v3)
0 10 20 28
20
35 43
15
5
10
1
8
3
8...
t
Martin Stigge Workload Models + Analysis 16
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DRT: Semantics
v1〈2, 5〉
v2
〈1, 8〉v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
Path π = (v4)Path π = (v4, v2)
Path π = (v4, v2, v3)
0 10 20 28
20
35 43
15
5
10
1
8
3
8...
t
Martin Stigge Workload Models + Analysis 16
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A Hierarchy of Models
Liu & Layland(e, d = p)
sporadic(e, d , p)
multiframe(ei , d = p)
generalized multiframe (GMF)(ei , di , pi )
recurring branching (RB)(tree, p)
recurring RT (RRT)(DAG, p)
non-cyclic GMF(order arbitrary)
non-cyclic RRT(DAG, pi )
Digraph (DRT)(arbitrary graph)
efficient
Sch
edu
lab
ility
An
alys
is
difficult
low
Exp
ress
iven
ess
high
Martin Stigge Workload Models + Analysis 17
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A Hierarchy of Models
Liu & Layland(e, d = p)
sporadic(e, d , p)
multiframe(ei , d = p)
generalized multiframe (GMF)(ei , di , pi )
recurring branching (RB)(tree, p)
recurring RT (RRT)(DAG, p)
non-cyclic GMF(order arbitrary)
non-cyclic RRT(DAG, pi )
Digraph (DRT)(arbitrary graph)
efficient
Sch
edu
lab
ility
An
alys
is
difficult
low
Exp
ress
iven
ess
high
Strongly (co)NP-hard
Pseudo-Polynomial
Martin Stigge Workload Models + Analysis 17
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Extended DRT (EDRT)
• Extends DRT with global delay constraints
• Directed graph for each task• Vertices v : jobs to be released (with WCET and deadline)• Edges (u, v): minimum inter-release delays p(u, v)• k global constraints (u, v , γ)
v1
v2
v3
v4
v5
52
23
72
2
9
6
Theorem (S. et al., 2011)
For k-EDRT task systems with bounded utilization, feasibility is
1 decidable in pseudo-polynomial time if k is constant, and
2 strongly coNP-hard in general.
Martin Stigge Workload Models + Analysis 18
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Extended DRT (EDRT)
• Extends DRT with global delay constraints
• Directed graph for each task• Vertices v : jobs to be released (with WCET and deadline)• Edges (u, v): minimum inter-release delays p(u, v)• k global constraints (u, v , γ)
v1
v2
v3
v4
v5
52
23
72
2
9
6
Theorem (S. et al., 2011)
For k-EDRT task systems with bounded utilization, feasibility is
1 decidable in pseudo-polynomial time if k is constant, and
2 strongly coNP-hard in general.
Martin Stigge Workload Models + Analysis 18
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A Hierarchy of Models
Liu & Layland(e, d = p)
sporadic(e, d , p)
multiframe(ei , d = p)
generalized multiframe (GMF)(ei , di , pi )
recurring branching (RB)(tree, p)
recurring RT (RRT)(DAG, p)
non-cyclic GMF(order arbitrary)
non-cyclic RRT(DAG, pi )
Digraph (DRT)(arbitrary graph)
k-EDRT(k constraints)
Extended DRT (EDRT)(constraints)
efficient
Sch
edu
lab
ility
An
alys
is
difficult
low
Exp
ress
iven
ess
high
Strongly (co)NP-hard
Pseudo-Polynomial
Martin Stigge Workload Models + Analysis 19
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DRT Examples
v
〈e, d〉
p
Sporadic Task
v
〈e, p〉
p
Sporadic Task(implicit deadline)
u
〈e1, d1〉
v
〈e2, p2〉
p1 p2
p1
p2
Sporadic Taskwith 2 modes
v1
v2
v3
v4
v5
GMF Task
v1
v2
v3 v4
v5
Branching Task
Martin Stigge Workload Models + Analysis 20
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DRT Examples
v
〈e, d〉
p
Sporadic Task
v
〈e, p〉
p
Sporadic Task(implicit deadline)
u
〈e1, d1〉
v
〈e2, p2〉
p1 p2
p1
p2
Sporadic Taskwith 2 modes
v1
v2
v3
v4
v5
GMF Task
v1
v2
v3 v4
v5
Branching Task
Martin Stigge Workload Models + Analysis 20
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DRT Examples
v
〈e, d〉
p
Sporadic Task
v
〈e, p〉
p
Sporadic Task(implicit deadline)
u
〈e1, d1〉
v
〈e2, p2〉
p1 p2
p1
p2
Sporadic Taskwith 2 modes
v1
v2
v3
v4
v5
GMF Task
v1
v2
v3 v4
v5
Branching Task
Martin Stigge Workload Models + Analysis 20
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DRT Examples
v
〈e, d〉
p
Sporadic Task
v
〈e, p〉
p
Sporadic Task(implicit deadline)
u
〈e1, d1〉
v
〈e2, p2〉
p1 p2
p1
p2
Sporadic Taskwith 2 modes
v1
v2
v3
v4
v5
GMF Task
v1
v2
v3 v4
v5
Branching Task
Martin Stigge Workload Models + Analysis 20
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DRT Examples
v
〈e, d〉
p
Sporadic Task
v
〈e, p〉
p
Sporadic Task(implicit deadline)
u
〈e1, d1〉
v
〈e2, p2〉
p1 p2
p1
p2
Sporadic Taskwith 2 modes
v1
v2
v3
v4
v5
GMF Task
v1
v2
v3 v4
v5
Branching Task
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Adaptive Variable-Rate (AVR) Tasks
• Rate-Adaptive Tasks (Buttazzo et al., DATE 2014)
• Variable Rate-dependent Behaviour (VRB) (Davis et al., RTAS 2014)
• Adaptive Variable-Rate (AVR) Tasks (Biondi et al., ECRTS 2014)
• . . .
(Biondi, ECRTS 2014)
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AVR Tasks: Execution Modes
(Biondi, ECRTS 2014)
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AVR Business
(This morning in Pisa)
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Fahrplan
1 DRT Tasks in the Model HierarchyLiu and Layland and Sporadic TasksFrames and BranchingThe Digraph Real-Time (DRT) Task ModelAdaptive Variable-Rate (AVR) Tasks
2 Feasibility Analysis of DRTFeasibility TheoremDemand PairsTest TerminationEvaluation
3 Static Priority Schedulability Analysis of DRTResponse-Time AnalysisRequest FunctionsRefinement AlgorithmEvaluation
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Fahrplan
1 DRT Tasks in the Model HierarchyLiu and Layland and Sporadic TasksFrames and BranchingThe Digraph Real-Time (DRT) Task ModelAdaptive Variable-Rate (AVR) Tasks
2 Feasibility Analysis of DRTFeasibility TheoremDemand PairsTest TerminationEvaluation
3 Static Priority Schedulability Analysis of DRTResponse-Time AnalysisRequest FunctionsRefinement AlgorithmEvaluation
Martin Stigge Workload Models + Analysis 24
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Feasibility Theorem
Theorem
For a task set τ , the following three properties are equivalent.
1 Task set τ is feasible.
2 Task set τ is EDF schedulable.
3 The following condition holds: ∀t > 0 :∑
T∈τ dbfT (t) 6 t
What is dbfT (t)? How to compute it?
Martin Stigge Workload Models + Analysis 25
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Feasibility Theorem
Theorem
For a task set τ , the following three properties are equivalent.
1 Task set τ is feasible.
2 Task set τ is EDF schedulable.
3 The following condition holds: ∀t > 0 :∑
T∈τ dbfT (t) 6 t
What is dbfT (t)? How to compute it?
Martin Stigge Workload Models + Analysis 25
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Demand-Bound Functions
dbfT (t): Maximal demand in any window of size t
0 10 20 28 35 43
5 1 3...
t
43
Demand: 5 + 1 + 3 = 9
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Feasibility Test
Theorem (Feasibility Theorem)
For a task set τ , the following three properties are equivalent.
1 Task set τ is feasible.
2 Task set τ is EDF schedulable.
3 The following condition holds: ∀t > 0 :∑
T∈τ dbfT (t) 6 t∑dbfT (t)
t
1 How to calculate dbfT (t)?
2 How to check existence of violating t?
Martin Stigge Workload Models + Analysis 27
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Feasibility Test
Theorem (Feasibility Theorem)
For a task set τ , the following three properties are equivalent.
1 Task set τ is feasible.
2 Task set τ is EDF schedulable.
3 The following condition holds: ∀t > 0 :∑
T∈τ dbfT (t) 6 t∑dbfT (t)
t
t
1 How to calculate dbfT (t)?
2 How to check existence of violating t?
Martin Stigge Workload Models + Analysis 27
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Feasibility Test
Theorem (Feasibility Theorem)
For a task set τ , the following three properties are equivalent.
1 Task set τ is feasible.
2 Task set τ is EDF schedulable.
3 The following condition holds: ∀t > 0 :∑
T∈τ dbfT (t) 6 t∑dbfT (t)
t
t
1 How to calculate dbfT (t)?
2 How to check existence of violating t?
Martin Stigge Workload Models + Analysis 27
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Feasibility Test
Theorem (Feasibility Theorem)
For a task set τ , the following three properties are equivalent.
1 Task set τ is feasible.
2 Task set τ is EDF schedulable.
3 The following condition holds: ∀t > 0 :∑
T∈τ dbfT (t) 6 t∑dbfT (t)
t
t
1 How to calculate dbfT (t)?
2 How to check existence of violating t?
Martin Stigge Workload Models + Analysis 27
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dbf for Sporadic Tasks
Sporadic task T = (e, d , p)∑dbfT (t)
te
d p
dbfT (t) = max
{0,
⌊t − d
p+ 1
⌋· e}
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dbf for Sporadic Tasks
Sporadic task T = (e, d , p)∑dbfT (t)
te
d p
dbfT (t) = max
{0,
⌊t − d
p+ 1
⌋· e}
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dbf for DRT: From G (T ) to dbfT
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
Path Abstraction: Demand Pair
0 10 20 30 40 50 600369
12
dbfT (t)
t
〈9, 43〉
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dbf for DRT: From G (T ) to dbfT
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
Path Abstraction: Demand Pair
0 10 20 30 40 50 600369
12
dbfT (t)
t
〈9, 43〉
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dbf for DRT: From G (T ) to dbfT
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
Path Abstraction: Demand Pair
0 10 20 30 40 50 600369
12
dbfT (t)
t
〈9, 43〉
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dbf for DRT: From G (T ) to dbfT
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
Path Abstraction: Demand Pair
0 10 20 30 40 50 600369
12
dbfT (t)
t
〈9, 43〉
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dbf for DRT: From G (T ) to dbfT
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
Path Abstraction: Demand Pair
0 10 20 30 40 50 600369
12
dbfT (t)
t
〈9, 43〉
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Demand Pairs
Formally:
• Given path π = (π0, . . . , πl)
• Execution demand: e(π) :=∑l
i=0 e(πi )
• Deadline: d(π) :=∑l−1
i=0 p(πi , πi+1) + d(πl)
• 〈e(π), d(π)〉 is a demand pair for π
dbfT (t) = max {e | 〈e, d〉 demand pair with d 6 t}
How to compute all demand pairs?
• Enumerate paths: Too expensive! (Exponential..)
• Better: Iteration using abstraction
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Demand Pairs
Formally:
• Given path π = (π0, . . . , πl)
• Execution demand: e(π) :=∑l
i=0 e(πi )
• Deadline: d(π) :=∑l−1
i=0 p(πi , πi+1) + d(πl)
• 〈e(π), d(π)〉 is a demand pair for π
dbfT (t) = max {e | 〈e, d〉 demand pair with d 6 t}
How to compute all demand pairs?
• Enumerate paths: Too expensive! (Exponential..)
• Better: Iteration using abstraction
Martin Stigge Workload Models + Analysis 30
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Demand Triples• Idea: Start with 0-paths (one vertex), extend stepwise• We need: Abstraction which
1 allows to extend paths,2 contains demand pair information,3 without visiting/storing all paths
• Idea: Demand triples• Execution demand e(π)• Deadline d(π)• Last vertex πl
• Demand triple 〈e(π), d(π), πl〉 is another abstraction!
v1〈2, 5〉
v2
〈1, 8〉v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
Path (v4)
; 〈5, 10, v4〉
Path (v4, v2)
; 〈6, 28, v2〉
Path (v4, v2, v3)
; 〈9, 43, v3〉
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Demand Triples• Idea: Start with 0-paths (one vertex), extend stepwise• We need: Abstraction which
1 allows to extend paths,2 contains demand pair information,3 without visiting/storing all paths
• Idea: Demand triples• Execution demand e(π)• Deadline d(π)• Last vertex πl
• Demand triple 〈e(π), d(π), πl〉 is another abstraction!
v1〈2, 5〉
v2
〈1, 8〉v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
Path (v4)
; 〈5, 10, v4〉
Path (v4, v2)
; 〈6, 28, v2〉
Path (v4, v2, v3)
; 〈9, 43, v3〉
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Demand Triples• Idea: Start with 0-paths (one vertex), extend stepwise• We need: Abstraction which
1 allows to extend paths,2 contains demand pair information,3 without visiting/storing all paths
• Idea: Demand triples• Execution demand e(π)• Deadline d(π)• Last vertex πl
• Demand triple 〈e(π), d(π), πl〉 is another abstraction!
v1〈2, 5〉
v2
〈1, 8〉v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
Path (v4)
; 〈5, 10, v4〉
Path (v4, v2)
; 〈6, 28, v2〉
Path (v4, v2, v3)
; 〈9, 43, v3〉
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Demand Triples• Idea: Start with 0-paths (one vertex), extend stepwise• We need: Abstraction which
1 allows to extend paths,2 contains demand pair information,3 without visiting/storing all paths
• Idea: Demand triples• Execution demand e(π)• Deadline d(π)• Last vertex πl
• Demand triple 〈e(π), d(π), πl〉 is another abstraction!
v1〈2, 5〉
v2
〈1, 8〉v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
Path (v4)
; 〈5, 10, v4〉
Path (v4, v2)
; 〈6, 28, v2〉
Path (v4, v2, v3)
; 〈9, 43, v3〉
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Iterative Procedure
• Create all demand triples up to some D:
1 Start with all 0-paths, i.e., 〈e(v), d(v), v〉 for all vertices v
2 Pick some stored demand triple 〈e, d , u〉3 Create new demand triple:
• Choose successor vertex v of u• e′ = e + e(v)• d ′ = d − d(u) + p(u, v) + d(v)• 〈e′, d ′, v〉 is new demand triple!
4 Store 〈e′, d ′, v〉 if• not stored yet, and• d ′ 6 D
5 Repeat from 2 until no change
• Efficient procedure!• Note: Actual paths never stored• Optimizations: Discard non-critical triples along the way
• Exercise: What’s dbfτi (26) for graph on previous slide?
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Example
0 10 20 30 40 50 60 700
3
6
9
12
dbfT (t)
t
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
D
Optimization: Discard dominated triples
(e, d , v) < (e ′, d ′, v) ⇐⇒ e > e ′ ∧ d 6 d ′
Martin Stigge Workload Models + Analysis 33
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Example
0 10 20 30 40 50 60 700
3
6
9
12
dbfT (t)
t
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
D
Optimization: Discard dominated triples
(e, d , v) < (e ′, d ′, v) ⇐⇒ e > e ′ ∧ d 6 d ′
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Example
0 10 20 30 40 50 60 700
3
6
9
12
dbfT (t)
t
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
D
Optimization: Discard dominated triples
(e, d , v) < (e ′, d ′, v) ⇐⇒ e > e ′ ∧ d 6 d ′
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Example
0 10 20 30 40 50 60 700
3
6
9
12
dbfT (t)
t
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
〈6, 30, v4〉〈6, 20, v4〉
D
Optimization: Discard dominated triples
(e, d , v) < (e ′, d ′, v) ⇐⇒ e > e ′ ∧ d 6 d ′
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Example
0 10 20 30 40 50 60 700
3
6
9
12
dbfT (t)
t
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
〈6, 30, v4〉〈6, 20, v4〉
D
Optimization: Discard dominated triples
(e, d , v) < (e ′, d ′, v) ⇐⇒ e > e ′ ∧ d 6 d ′
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Example
0 10 20 30 40 50 60 700
3
6
9
12
dbfT (t)
t
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
D
Optimization: Discard dominated triples
(e, d , v) < (e ′, d ′, v) ⇐⇒ e > e ′ ∧ d 6 d ′
Martin Stigge Workload Models + Analysis 33
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Example
0 10 20 30 40 50 60 700
3
6
9
12
dbfT (t)
t
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
D
Optimization: Discard dominated triples
(e, d , v) < (e ′, d ′, v) ⇐⇒ e > e ′ ∧ d 6 d ′
Martin Stigge Workload Models + Analysis 33
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Example
0 10 20 30 40 50 60 700
3
6
9
12
dbfT (t)
t
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
D
Optimization: Discard dominated triples
(e, d , v) < (e ′, d ′, v) ⇐⇒ e > e ′ ∧ d 6 d ′
Martin Stigge Workload Models + Analysis 33
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Example
0 10 20 30 40 50 60 700
3
6
9
12
dbfT (t)
t
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
D
Optimization: Discard dominated triples
(e, d , v) < (e ′, d ′, v) ⇐⇒ e > e ′ ∧ d 6 d ′
Martin Stigge Workload Models + Analysis 33
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Example
0 10 20 30 40 50 60 700
3
6
9
12
dbfT (t)
t
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
D
Optimization: Discard dominated triples
(e, d , v) < (e ′, d ′, v) ⇐⇒ e > e ′ ∧ d 6 d ′
Martin Stigge Workload Models + Analysis 33
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Example
0 10 20 30 40 50 60 700
3
6
9
12
dbfT (t)
t
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
D
Optimization: Discard dominated triples
(e, d , v) < (e ′, d ′, v) ⇐⇒ e > e ′ ∧ d 6 d ′
Martin Stigge Workload Models + Analysis 33
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Example
0 10 20 30 40 50 60 700
3
6
9
12
dbfT (t)
t
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
D
Optimization: Discard dominated triples
(e, d , v) < (e ′, d ′, v) ⇐⇒ e > e ′ ∧ d 6 d ′
Martin Stigge Workload Models + Analysis 33
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Feasibility Test Revisited
Theorem (Feasibility Theorem)
For a task set τ , the following three properties are equivalent.
1 Task set τ is feasible.
2 Task set τ is EDF schedulable.
3 The following condition holds: ∀t > 0 :∑
T∈τ dbfT (t) 6 t∑dbfT (t)
t
t
1 How to calculate dbfT (t)?
2 How to check existence of violating t?
Martin Stigge Workload Models + Analysis 34
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Feasibility Test Revisited
Theorem (Feasibility Theorem)
For a task set τ , the following three properties are equivalent.
1 Task set τ is feasible.
2 Task set τ is EDF schedulable.
3 The following condition holds: ∀t > 0 :∑
T∈τ dbfT (t) 6 t∑dbfT (t)
t
t
D
1 How to calculate dbfT (t)?
2 How to check existence of violating t?
Martin Stigge Workload Models + Analysis 34
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Calculating the Bound
∑dbfT (t)
t
dbf(t)
t
D
Linear bou
nd for dbf(t)
• Linear bound for dbf(t)• Slope: Less than 1
• Intersection with t gives bound D
• Check only up to D
J1
J2
J3 J4
J5
J6
J7
J8
“Most dense” cycle
dbf(t) 6 t · U(τ) + esum
Martin Stigge Workload Models + Analysis 35
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Calculating the Bound
∑dbfT (t)
t
dbf(t)
t
D
Linear bou
nd for dbf(t)
• Linear bound for dbf(t)• Slope: Less than 1
• Intersection with t gives bound D
• Check only up to D
J1
J2
J3 J4
J5
J6
J7
J8
“Most dense” cycle
dbf(t) 6 t · U(τ) + esum
Martin Stigge Workload Models + Analysis 35
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Calculating the Bound
∑dbfT (t)
t
dbf(t)
t
D
Linear bou
nd for dbf(t)
• Linear bound for dbf(t)• Slope: Less than 1
• Intersection with t gives bound D
• Check only up to D
J1
J2
J3 J4
J5
J6
J7
J8
“Most dense” cycle
dbf(t) 6 t · U(τ) + esum
Martin Stigge Workload Models + Analysis 35
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Complexity Result
Theorem (S. et al., 2011)
For DRT task systems τ with a utilization bounded by any c < 1,feasibility can be decided in pseudo-polynomial time.
Pseudo-polynomial time = Tractable/efficient
Martin Stigge Workload Models + Analysis 36
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Evaluation: Runtime vs. Utilization
0% 20% 40% 60% 80% 100%Task Set Utilization
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Ana
lysi
sR
un-T
ime
(sec
onds
)EDF
Setting:
• Randomly generated task sets
• 1-30 tasks, 5-10 vertices per task, branching degree 1-3, . . .
Martin Stigge Workload Models + Analysis 37
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Fahrplan
1 DRT Tasks in the Model HierarchyLiu and Layland and Sporadic TasksFrames and BranchingThe Digraph Real-Time (DRT) Task ModelAdaptive Variable-Rate (AVR) Tasks
2 Feasibility Analysis of DRTFeasibility TheoremDemand PairsTest TerminationEvaluation
3 Static Priority Schedulability Analysis of DRTResponse-Time AnalysisRequest FunctionsRefinement AlgorithmEvaluation
Martin Stigge Workload Models + Analysis 38
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Fahrplan
1 DRT Tasks in the Model HierarchyLiu and Layland and Sporadic TasksFrames and BranchingThe Digraph Real-Time (DRT) Task ModelAdaptive Variable-Rate (AVR) Tasks
2 Feasibility Analysis of DRTFeasibility TheoremDemand PairsTest TerminationEvaluation
3 Static Priority Schedulability Analysis of DRTResponse-Time AnalysisRequest FunctionsRefinement AlgorithmEvaluation
Martin Stigge Workload Models + Analysis 38
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Complexity of Schedulability Tests
• Pseudo-polynomial schedulability tests possible?
EDF SP
L&L YesGMF YesDRT Yes
EDRT No
• ∗ = Takada & Sakamura, 1997
• Flawed!
• Replaced by:
Theorem (S. et al., 2012)
For GMF task systems, the schedulability problem for static priorityschedulers is strongly coNP-hard.
Martin Stigge Workload Models + Analysis 39
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Complexity of Schedulability Tests
• Pseudo-polynomial schedulability tests possible?
EDF SP
L&L Yes YesGMF Yes Yes∗
DRT Yes ?EDRT No ?
• ∗ = Takada & Sakamura, 1997
• Flawed!
• Replaced by:
Theorem (S. et al., 2012)
For GMF task systems, the schedulability problem for static priorityschedulers is strongly coNP-hard.
Martin Stigge Workload Models + Analysis 39
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Complexity of Schedulability Tests
• Pseudo-polynomial schedulability tests possible?
EDF SP
L&L Yes YesGMF Yes Yes∗No!DRT Yes No
EDRT No No
• ∗ = Takada & Sakamura, 1997
• Flawed!
• Replaced by:
Theorem (S. et al., 2012)
For GMF task systems, the schedulability problem for static priorityschedulers is strongly coNP-hard.
Martin Stigge Workload Models + Analysis 39
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Hardness Result: Proof Sketch
3-PARTITION instance I :
m bins 3m items
Possible to (exactly) fit all items? (strongly NP-hard)
Reduction to GMF schedulability:
One GMF task
for each item
Packing possible ⇐⇒ Busy interval
t
Thus: τ(I ) unsched. ⇐⇒ I ∈ 3-PARTITION
Martin Stigge Workload Models + Analysis 40
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Hardness Result: Proof Sketch
3-PARTITION instance I :
m bins 3m items
Possible to (exactly) fit all items? (strongly NP-hard)
Reduction to GMF schedulability:
One GMF task
for each item
Packing possible ⇐⇒ Busy interval
t
Thus: τ(I ) unsched. ⇐⇒ I ∈ 3-PARTITION
Martin Stigge Workload Models + Analysis 40
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Model Hierarchy Revisited
coNP-hard
p.p.
L&L
sporadicMF
GMF
RB
RRT
ncGMF
ncRRT
DRT
k-EDRT
EDRT
EDF SP
coNP-hard
p.p.
L&L
sporadicMF
GMF
RB
RRT
ncGMF
ncRRT
DRT
k-EDRT
EDRT
Martin Stigge Workload Models + Analysis 41
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Response-Time Analysis (RTA)
Use RTA for SP Schedulability Analysis.
Response time
Standard RTA for static priorities + periodic/sporadic tasks:
Rj = Cj +∑
i∈hp(j)
⌈Rj
Ti
⌉Ci
Martin Stigge Workload Models + Analysis 42
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Problem: Path Combinations
v1
v2
v3
v4
v5
u1
u2
u3
↓
Response time
v1
v2
v3
v4
v5
u1
u2
u3
↓
Response time
Combinatorial Explosion!
Martin Stigge Workload Models + Analysis 43
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Problem: Path Combinations
v1
v2
v3
v4
v5
u1
u2
u3
↓
Response time
v1
v2
v3
v4
v5
u1
u2
u3
↓
Response time
Combinatorial Explosion!
Martin Stigge Workload Models + Analysis 43
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Intuition: No Task-Local Worst Cases
Task Thigh:v1〈2, 5〉 v2 〈6, 10〉
5
10
Which path is worse: (v1, v2) or (v2, v1)?
It depends! T1: u〈2, 7〉 versus T2: w〈4, 10〉
v1 v2
Thigh:
v2 v1
Thigh:
T1:
0 2 4 6 8 10 12
T1:
0 2 4 6 8 10 12
T2:
0 2 4 6 8 10 12
T2:
0 2 4 6 8 10 12
Martin Stigge Workload Models + Analysis 44
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Intuition: No Task-Local Worst Cases
Task Thigh:v1〈2, 5〉 v2 〈6, 10〉
5
10
Which path is worse: (v1, v2) or (v2, v1)?
It depends! T1: u〈2, 7〉 versus T2: w〈4, 10〉
v1 v2
Thigh:
v2 v1
Thigh:
T1:
0 2 4 6 8 10 12
T1:
0 2 4 6 8 10 12
T2:
0 2 4 6 8 10 12
T2:
0 2 4 6 8 10 12
Martin Stigge Workload Models + Analysis 44
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Intuition: No Task-Local Worst Cases
Task Thigh:v1〈2, 5〉 v2 〈6, 10〉
5
10
Which path is worse: (v1, v2) or (v2, v1)?
It depends! T1: u〈2, 7〉 versus T2: w〈4, 10〉
v1 v2
Thigh:
v2 v1
Thigh:
T1:
0 2 4 6 8 10 12
T1:
0 2 4 6 8 10 12
T2:
0 2 4 6 8 10 12
T2:
0 2 4 6 8 10 12
Martin Stigge Workload Models + Analysis 44
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Intuition: No Task-Local Worst Cases
Task Thigh:v1〈2, 5〉 v2 〈6, 10〉
5
10
Which path is worse: (v1, v2) or (v2, v1)?
It depends! T1: u〈2, 7〉 versus T2: w〈4, 10〉
v1 v2
Thigh:
v2 v1
Thigh:
T1:
0 2 4 6 8 10 12
T1:
0 2 4 6 8 10 12
T2:
0 2 4 6 8 10 12
T2:
0 2 4 6 8 10 12
Martin Stigge Workload Models + Analysis 44
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Request Functions
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
rf (t)
t0 5 10 15 20 25 30 35 40
02468
10 rf (v4,v2,v3)
rf π(t) := max{e(π′) | π′ is prefix of π and p(π′) < t
}Martin Stigge Workload Models + Analysis 45
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Request Functions (cont.)
Useful for deriving response time:
RSP(v , r̄f ) = min
{t > 0 | e(v) +
∑T ′>T
rf (T ′)(t) 6 t
}RSP(v) = max
r̄f ∈RF (τ)RSP(v , r̄f )
Combinatorial Explosion?!
Martin Stigge Workload Models + Analysis 46
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Request Functions (cont.)
Useful for deriving response time:
RSP(v , r̄f ) = min
{t > 0 | e(v) +
∑T ′>T
rf (T ′)(t) 6 t
}RSP(v) = max
r̄f ∈RF (τ)RSP(v , r̄f )
Combinatorial Explosion?!
Martin Stigge Workload Models + Analysis 46
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Abstract Request Functions
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
rf (t)
t0 5 10 15 20 25 30 35 40
02468
10 rf (v4,v2,v3)
rf (v5,v4,v2)
arf
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Abstract Request Functions
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
rf (t)
t0 5 10 15 20 25 30 35 40
02468
10 rf (v4,v2,v3)
rf (v5,v4,v2)
arf
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Abstract Request Functions
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
rf (t)
t0 5 10 15 20 25 30 35 40
02468
10 rf (v4,v2,v3)
rf (v5,v4,v2)
arf
Martin Stigge Workload Models + Analysis 47
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Abstraction Tree
rf 1 rf 2
rf 3 rf 4
rf 5
Define an abstraction tree per task:
• Leaves are concrete rf
• Each node: maximum function of child nodes
• Root is maximum of all rf
Allows stepwise refinement!
Martin Stigge Workload Models + Analysis 48
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Abstraction Tree
rf 1 rf 2
rf 3 rf 4
rf 5
Define an abstraction tree per task:
• Leaves are concrete rf
• Each node: maximum function of child nodes
• Root is maximum of all rf
Allows stepwise refinement!
Martin Stigge Workload Models + Analysis 48
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Abstraction Tree
rf 1 rf 2
rf 3 rf 4
rf 5
Define an abstraction tree per task:
• Leaves are concrete rf
• Each node: maximum function of child nodes
• Root is maximum of all rf
Allows stepwise refinement!
Martin Stigge Workload Models + Analysis 48
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Abstraction Tree
rf 1 rf 2
rf 3 rf 4
rf 5
Define an abstraction tree per task:
• Leaves are concrete rf
• Each node: maximum function of child nodes
• Root is maximum of all rf
Allows stepwise refinement!
Martin Stigge Workload Models + Analysis 48
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Abstraction Tree
rf 1 rf 2
rf 3 rf 4
rf 5
Define an abstraction tree per task:
• Leaves are concrete rf
• Each node: maximum function of child nodes
• Root is maximum of all rf
Allows stepwise refinement!
Martin Stigge Workload Models + Analysis 48
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Abstraction Tree
rf 1 rf 2
rf 3 rf 4
rf 5
Define an abstraction tree per task:
• Leaves are concrete rf
• Each node: maximum function of child nodes
• Root is maximum of all rf
Allows stepwise refinement!
Martin Stigge Workload Models + Analysis 48
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Refinement Algorithm
r̄f = (rf (T1), rf (T2), rf (T3))Tuple:
↓
RSP(v , r̄f ) = 23Response time:
r̄f 1 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 2 = (rf ′(T1), rf (T2), rf (T3))
r̄f 3 = (rf ′′(T1), rf (T2), rf (T3))
Step:
→ 18
→ 21
r̄f 2 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 4 = (rf (T1), rf ′(T2), rf (T3))
r̄f 5 = (rf (T1), rf ′′(T2), rf (T3))
→ 20
→ 17
Initialization:• Most abstract functions
Each iteration:
• Replace functions along abstraction trees
Termination:
• All functions are concreteOR:
• Estimate is already safeIn T1:
rf ′ rf ′′
rfIn T2:
rf ′ rf ′′
rf
Store
(23, r̄f 1)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
...
Using: RSP(v , r̄f ) = min{t > 0 | e(v) +
∑T ′>T rf (T ′)(t) 6 t
}
Martin Stigge Workload Models + Analysis 49
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Refinement Algorithm
r̄f = (rf (T1), rf (T2), rf (T3))Tuple:
↓
RSP(v , r̄f ) = 23Response time:
r̄f 1 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 2 = (rf ′(T1), rf (T2), rf (T3))
r̄f 3 = (rf ′′(T1), rf (T2), rf (T3))
Step:
→ 18
→ 21
r̄f 2 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 4 = (rf (T1), rf ′(T2), rf (T3))
r̄f 5 = (rf (T1), rf ′′(T2), rf (T3))
→ 20
→ 17
Initialization:• Most abstract functions
Each iteration:
• Replace functions along abstraction trees
Termination:
• All functions are concreteOR:
• Estimate is already safeIn T1:
rf ′ rf ′′
rfIn T2:
rf ′ rf ′′
rf
Store
(23, r̄f 1)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
...
Using: RSP(v , r̄f ) = min{t > 0 | e(v) +
∑T ′>T rf (T ′)(t) 6 t
}
Martin Stigge Workload Models + Analysis 49
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Refinement Algorithm
r̄f = (rf (T1), rf (T2), rf (T3))Tuple:
↓
RSP(v , r̄f ) = 23Response time:
r̄f 1 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 2 = (rf ′(T1), rf (T2), rf (T3))
r̄f 3 = (rf ′′(T1), rf (T2), rf (T3))
Step:
→ 18
→ 21
r̄f 2 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 4 = (rf (T1), rf ′(T2), rf (T3))
r̄f 5 = (rf (T1), rf ′′(T2), rf (T3))
→ 20
→ 17
Initialization:• Most abstract functions
Each iteration:
• Replace functions along abstraction trees
Termination:
• All functions are concreteOR:
• Estimate is already safeIn T1:
rf ′ rf ′′
rfIn T2:
rf ′ rf ′′
rf
Store
(23, r̄f 1)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
...
Using: RSP(v , r̄f ) = min{t > 0 | e(v) +
∑T ′>T rf (T ′)(t) 6 t
}
Martin Stigge Workload Models + Analysis 49
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Refinement Algorithm
r̄f = (rf (T1), rf (T2), rf (T3))Tuple:
↓
RSP(v , r̄f ) = 23Response time:
r̄f 1 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 2 = (rf ′(T1), rf (T2), rf (T3))
r̄f 3 = (rf ′′(T1), rf (T2), rf (T3))
Step:
→ 18
→ 21
r̄f 2 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 4 = (rf (T1), rf ′(T2), rf (T3))
r̄f 5 = (rf (T1), rf ′′(T2), rf (T3))
→ 20
→ 17
Initialization:• Most abstract functions
Each iteration:
• Replace functions along abstraction trees
Termination:
• All functions are concreteOR:
• Estimate is already safe
In T1:
rf ′ rf ′′
rf
In T2:
rf ′ rf ′′
rf
Store
(23, r̄f 1)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
...
Using: RSP(v , r̄f ) = min{t > 0 | e(v) +
∑T ′>T rf (T ′)(t) 6 t
}
Martin Stigge Workload Models + Analysis 49
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Refinement Algorithm
r̄f = (rf (T1), rf (T2), rf (T3))Tuple:
↓
RSP(v , r̄f ) = 23Response time:
r̄f 1 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 2 = (rf ′(T1), rf (T2), rf (T3))
r̄f 3 = (rf ′′(T1), rf (T2), rf (T3))
Step:
→ 18
→ 21
r̄f 2 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 4 = (rf (T1), rf ′(T2), rf (T3))
r̄f 5 = (rf (T1), rf ′′(T2), rf (T3))
→ 20
→ 17
Initialization:• Most abstract functions
Each iteration:
• Replace functions along abstraction trees
Termination:
• All functions are concreteOR:
• Estimate is already safe
In T1:
rf ′ rf ′′
rf
In T2:
rf ′ rf ′′
rf
Store
(23, r̄f 1)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
...
Using: RSP(v , r̄f ) = min{t > 0 | e(v) +
∑T ′>T rf (T ′)(t) 6 t
}
Martin Stigge Workload Models + Analysis 49
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Refinement Algorithm
r̄f = (rf (T1), rf (T2), rf (T3))Tuple:
↓
RSP(v , r̄f ) = 23Response time:
r̄f 1 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 2 = (rf ′(T1), rf (T2), rf (T3))
r̄f 3 = (rf ′′(T1), rf (T2), rf (T3))
Step:
→ 18
→ 21
r̄f 2 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 4 = (rf (T1), rf ′(T2), rf (T3))
r̄f 5 = (rf (T1), rf ′′(T2), rf (T3))
→ 20
→ 17
Initialization:• Most abstract functions
Each iteration:
• Replace functions along abstraction trees
Termination:
• All functions are concreteOR:
• Estimate is already safe
In T1:
rf ′ rf ′′
rf
In T2:
rf ′ rf ′′
rf
Store
(23, r̄f 1)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
...
Using: RSP(v , r̄f ) = min{t > 0 | e(v) +
∑T ′>T rf (T ′)(t) 6 t
}
Martin Stigge Workload Models + Analysis 49
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Refinement Algorithm
r̄f = (rf (T1), rf (T2), rf (T3))Tuple:
↓
RSP(v , r̄f ) = 23Response time:
r̄f 1 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 2 = (rf ′(T1), rf (T2), rf (T3))
r̄f 3 = (rf ′′(T1), rf (T2), rf (T3))
Step:
→ 18
→ 21
r̄f 2 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 4 = (rf (T1), rf ′(T2), rf (T3))
r̄f 5 = (rf (T1), rf ′′(T2), rf (T3))
→ 20
→ 17
Initialization:• Most abstract functions
Each iteration:
• Replace functions along abstraction trees
Termination:
• All functions are concreteOR:
• Estimate is already safeIn T1:
rf ′ rf ′′
rfIn T2:
rf ′ rf ′′
rf
Store
(23, r̄f 1)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
...
Using: RSP(v , r̄f ) = min{t > 0 | e(v) +
∑T ′>T rf (T ′)(t) 6 t
}
Martin Stigge Workload Models + Analysis 49
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Refinement Algorithm
r̄f = (rf (T1), rf (T2), rf (T3))Tuple:
↓
RSP(v , r̄f ) = 23Response time:
r̄f 1 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 2 = (rf ′(T1), rf (T2), rf (T3))
r̄f 3 = (rf ′′(T1), rf (T2), rf (T3))
Step:
→ 18
→ 21
r̄f 2 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 4 = (rf (T1), rf ′(T2), rf (T3))
r̄f 5 = (rf (T1), rf ′′(T2), rf (T3))
→ 20
→ 17
Initialization:• Most abstract functions
Each iteration:
• Replace functions along abstraction trees
Termination:
• All functions are concreteOR:
• Estimate is already safeIn T1:
rf ′ rf ′′
rfIn T2:
rf ′ rf ′′
rf
Store
(23, r̄f 1)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
...
Using: RSP(v , r̄f ) = min{t > 0 | e(v) +
∑T ′>T rf (T ′)(t) 6 t
}
Martin Stigge Workload Models + Analysis 49
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Refinement Algorithm
r̄f = (rf (T1), rf (T2), rf (T3))Tuple:
↓
RSP(v , r̄f ) = 23Response time:
r̄f 1 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 2 = (rf ′(T1), rf (T2), rf (T3))
r̄f 3 = (rf ′′(T1), rf (T2), rf (T3))
Step:
→ 18
→ 21
r̄f 2 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 4 = (rf (T1), rf ′(T2), rf (T3))
r̄f 5 = (rf (T1), rf ′′(T2), rf (T3))
→ 20
→ 17
Initialization:• Most abstract functions
Each iteration:
• Replace functions along abstraction trees
Termination:
• All functions are concreteOR:
• Estimate is already safeIn T1:
rf ′ rf ′′
rf
In T2:
rf ′ rf ′′
rf
Store
(23, r̄f 1)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
...
Using: RSP(v , r̄f ) = min{t > 0 | e(v) +
∑T ′>T rf (T ′)(t) 6 t
}
Martin Stigge Workload Models + Analysis 49
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Refinement Algorithm
r̄f = (rf (T1), rf (T2), rf (T3))Tuple:
↓
RSP(v , r̄f ) = 23Response time:
r̄f 1 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 2 = (rf ′(T1), rf (T2), rf (T3))
r̄f 3 = (rf ′′(T1), rf (T2), rf (T3))
Step:
→ 18
→ 21
r̄f 2 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 4 = (rf (T1), rf ′(T2), rf (T3))
r̄f 5 = (rf (T1), rf ′′(T2), rf (T3))
→ 20
→ 17
Initialization:• Most abstract functions
Each iteration:
• Replace functions along abstraction trees
Termination:
• All functions are concreteOR:
• Estimate is already safeIn T1:
rf ′ rf ′′
rf
In T2:
rf ′ rf ′′
rf
Store
(23, r̄f 1)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
...
Using: RSP(v , r̄f ) = min{t > 0 | e(v) +
∑T ′>T rf (T ′)(t) 6 t
}
Martin Stigge Workload Models + Analysis 49
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Refinement Algorithm
r̄f = (rf (T1), rf (T2), rf (T3))Tuple:
↓
RSP(v , r̄f ) = 23Response time:
r̄f 1 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 2 = (rf ′(T1), rf (T2), rf (T3))
r̄f 3 = (rf ′′(T1), rf (T2), rf (T3))
Step:
→ 18
→ 21
r̄f 2 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 4 = (rf (T1), rf ′(T2), rf (T3))
r̄f 5 = (rf (T1), rf ′′(T2), rf (T3))
→ 20
→ 17
Initialization:• Most abstract functions
Each iteration:
• Replace functions along abstraction trees
Termination:
• All functions are concreteOR:
• Estimate is already safeIn T1:
rf ′ rf ′′
rf
In T2:
rf ′ rf ′′
rf
Store
(23, r̄f 1)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
...
Using: RSP(v , r̄f ) = min{t > 0 | e(v) +
∑T ′>T rf (T ′)(t) 6 t
}
Martin Stigge Workload Models + Analysis 49
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Refinement Algorithm
r̄f = (rf (T1), rf (T2), rf (T3))Tuple:
↓
RSP(v , r̄f ) = 23Response time:
r̄f 1 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 2 = (rf ′(T1), rf (T2), rf (T3))
r̄f 3 = (rf ′′(T1), rf (T2), rf (T3))
Step:
→ 18
→ 21
r̄f 2 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 4 = (rf (T1), rf ′(T2), rf (T3))
r̄f 5 = (rf (T1), rf ′′(T2), rf (T3))
→ 20
→ 17
Initialization:• Most abstract functions
Each iteration:
• Replace functions along abstraction trees
Termination:
• All functions are concreteOR:
• Estimate is already safeIn T1:
rf ′ rf ′′
rfIn T2:
rf ′ rf ′′
rf
Store
(23, r̄f 1)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
...
Using: RSP(v , r̄f ) = min{t > 0 | e(v) +
∑T ′>T rf (T ′)(t) 6 t
}
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Refinement Algorithm
r̄f = (rf (T1), rf (T2), rf (T3))Tuple:
↓
RSP(v , r̄f ) = 23Response time:
r̄f 1 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 2 = (rf ′(T1), rf (T2), rf (T3))
r̄f 3 = (rf ′′(T1), rf (T2), rf (T3))
Step:
→ 18
→ 21
r̄f 2 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 4 = (rf (T1), rf ′(T2), rf (T3))
r̄f 5 = (rf (T1), rf ′′(T2), rf (T3))
→ 20
→ 17
Initialization:• Most abstract functions
Each iteration:
• Replace functions along abstraction trees
Termination:
• All functions are concreteOR:
• Estimate is already safe
In T1:
rf ′ rf ′′
rfIn T2:
rf ′ rf ′′
rf
Store
(23, r̄f 1)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
...
Using: RSP(v , r̄f ) = min{t > 0 | e(v) +
∑T ′>T rf (T ′)(t) 6 t
}
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Refinement Algorithm
r̄f = (rf (T1), rf (T2), rf (T3))Tuple:
↓
RSP(v , r̄f ) = 23Response time:
r̄f 1 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 2 = (rf ′(T1), rf (T2), rf (T3))
r̄f 3 = (rf ′′(T1), rf (T2), rf (T3))
Step:
→ 18
→ 21
r̄f 2 = (rf (T1), rf (T2), rf (T3))
↓
r̄f 4 = (rf (T1), rf ′(T2), rf (T3))
r̄f 5 = (rf (T1), rf ′′(T2), rf (T3))
→ 20
→ 17
Initialization:• Most abstract functions
Each iteration:
• Replace functions along abstraction trees
Termination:
• All functions are concreteOR:
• Estimate is already safe
In T1:
rf ′ rf ′′
rfIn T2:
rf ′ rf ′′
rf
Store
(23, r̄f 1)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(18, r̄f 3)
(21, r̄f 2)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
(20, r̄f 4)
(18, r̄f 3)
(17, r̄f 5)
...
Using: RSP(v , r̄f ) = min{t > 0 | e(v) +
∑T ′>T rf (T ′)(t) 6 t
}
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Evaluation: Run-time Scaling
0% 20% 40% 60% 80% 100%Task Set Utilization
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4A
naly
sis
Run
-Tim
e(s
econ
ds)
EDFSP
1-30 tasks with 5-10 vertices each, branching degree 1-3
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Evaluation: Precision Improvement
0% 20% 40% 60% 80% 100%Fraction of improved RT estimates
5%
10%
15%
20%A
vera
geim
prov
emen
t Type AType B
Type A: lower parameter varianceType B: higher parameter variance
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Generality
• Exact solution for NP-hard problem• Efficient method• Iterative refinement
• General!• Apply to any combinatorial problem• ... with monotonic abstraction lattice
rf 1 rf 2 rf 5
rf 3 rf 4
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Fahrplan
1 DRT Tasks in the Model HierarchyLiu and Layland and Sporadic TasksFrames and BranchingThe Digraph Real-Time (DRT) Task ModelAdaptive Variable-Rate (AVR) Tasks
2 Feasibility Analysis of DRTFeasibility TheoremDemand PairsTest TerminationEvaluation
3 Static Priority Schedulability Analysis of DRTResponse-Time AnalysisRequest FunctionsRefinement AlgorithmEvaluation
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Summary
coNP-hard
p.p.
L&L
sporadicMF
GMF
RB
RRT
ncGMF
ncRRT
EDF SP
coNP-hard
p.p.
L&L
sporadicMF
GMF
RB
RRT
ncGMF
ncRRT
DRT
k-EDRT
EDRT
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Summary
coNP-hard
p.p.
L&L
sporadicMF
GMF
RB
RRT
ncGMF
ncRRT
DRT
k-EDRT
EDRT
EDF SP
coNP-hard
p.p.
L&L
sporadicMF
GMF
RB
RRT
ncGMF
ncRRT
DRT
k-EDRT
EDRT
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Summary
coNP-hard
p.p.
L&L
sporadicMF
GMF
RB
RRT
ncGMF
ncRRT
DRT
k-EDRT
EDRT
EDF
SP
coNP-hard
p.p.
L&L
sporadicMF
GMF
RB
RRT
ncGMF
ncRRT
DRT
k-EDRT
EDRT
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Summary
coNP-hard
p.p.
L&L
sporadicMF
GMF
RB
RRT
ncGMF
ncRRT
DRT
k-EDRT
EDRT
EDF SP
coNP-hard
p.p.
L&L
sporadicMF
GMF
RB
RRT
ncGMF
ncRRT
DRT
k-EDRT
EDRT
Martin Stigge Workload Models + Analysis 54
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Summary
coNP-hard
p.p.
L&L
sporadicMF
GMF
RB
RRT
ncGMF
ncRRT
DRT
k-EDRT
EDRT
EDF SP
coNP-hard
p.p.
L&L
sporadicMF
GMF
RB
RRT
ncGMF
ncRRT
DRT
k-EDRT
EDRT
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Q & A
Thanks!
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Backup Slides Coming Up . . .
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Path Abstractions: SP + EDF
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Path Abstractions: Static Priorities
v1〈2, 5〉
v2
〈1, 8〉 v3 〈3, 8〉
v4 〈5, 10〉
v5
〈1, 5〉
10
15
20
20
20
11
10
rf (t)
t0 5 10 15 20 25 30 35 40
02468
10 rf (v4,v2,v3)
rf π(t) := max{e(π′) | π′ is prefix of π and p(π′) < t
}Martin Stigge Workload Models + Analysis 58
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Path Abstractions: EDF
T2
v
t ′
T1
v
t
wf π(t, t ′) := max{e(π′) | π′ is prefix of π,
p(π′) < t and d(π′) 6 t ′}.
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Path Abstractions: EDF
T2
v
t ′
T1
v
t
wf π(t, t ′) := max{e(π′) | π′ is prefix of π,
p(π′) < t and d(π′) 6 t ′}.
Martin Stigge Workload Models + Analysis 59