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Real-Time Workload Models with Efficient AnalysisAdvanced Course, 3 Lectures, September 2014
Martin Stigge
Uppsala University, Sweden
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
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
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
Martin Stigge Workload Models + Analysis 3
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
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
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...
Martin Stigge Workload Models + Analysis 6
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
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
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
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
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)
...
Martin Stigge Workload Models + Analysis 10
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
Martin Stigge Workload Models + Analysis 11
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
Martin Stigge Workload Models + Analysis 11
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
Martin Stigge Workload Models + Analysis 21
AVR Tasks: Execution Modes
(Biondi, ECRTS 2014)
Martin Stigge Workload Models + Analysis 22
AVR Business
(This morning in Pisa)
Martin Stigge Workload Models + Analysis 23
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
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
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
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
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
Martin Stigge Workload Models + Analysis 26
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
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
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
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
dbf for Sporadic Tasks
Sporadic task T = (e, d , p)∑dbfT (t)
te
d p
dbfT (t) = max
{0,
⌊t − d
p+ 1
⌋· e}
Martin Stigge Workload Models + Analysis 28
dbf for Sporadic Tasks
Sporadic task T = (e, d , p)∑dbfT (t)
te
d p
dbfT (t) = max
{0,
⌊t − d
p+ 1
⌋· e}
Martin Stigge Workload Models + Analysis 28
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〉
Martin Stigge Workload Models + Analysis 29
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〉
Martin Stigge Workload Models + Analysis 29
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〉
Martin Stigge Workload Models + Analysis 29
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〉
Martin Stigge Workload Models + Analysis 29
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〉
Martin Stigge Workload Models + Analysis 29
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
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
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〉
Martin Stigge Workload Models + Analysis 31
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〉
Martin Stigge Workload Models + Analysis 31
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〉
Martin Stigge Workload Models + Analysis 31
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〉
Martin Stigge Workload Models + Analysis 31
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?
Martin Stigge Workload Models + Analysis 32
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
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
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
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 ′
Martin Stigge Workload Models + Analysis 33
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 ′
Martin Stigge Workload Models + Analysis 33
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
}
Martin Stigge Workload Models + Analysis 49
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
}
Martin Stigge Workload Models + Analysis 49
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
Martin Stigge Workload Models + Analysis 50
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
Martin Stigge Workload Models + Analysis 51
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
Martin Stigge Workload Models + Analysis 52
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 53
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
Martin Stigge Workload Models + Analysis 54
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
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
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
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
Q & A
Thanks!
Martin Stigge Workload Models + Analysis 55
Backup Slides Coming Up . . .
Martin Stigge Workload Models + Analysis 56
Path Abstractions: SP + EDF
Martin Stigge Workload Models + Analysis 57
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
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
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
Recommended