Simulation Relations, Interface Complexity, and Resource Optimality for Real-Time Hierarchical Systems
Insup Lee
PRECISE CenterDepartment of Computer and Information Science
University of Pennsylvania
November 15, 2009
RePP Workshop, ESWeek 2009
Join work with M. Anand, A. Easwaran, L. Phan, A. Philippou, O. Sokolsky
Hierarchical scheduling framework
Global scheduler
Subsystem 1
Localscheduler
Subsystem 2
Localscheduler
Subsystem n
Localscheduler
R1
Localscheduler
Interface 1 Interface 2 Interface n
HRTiHRT2HRT1
R2 Rm
[Behnam]
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Abstraction and Composition
• Abstraction Problem: Abstract resource demand of real-time component into an interface
• Minimize resource usage: Identify minimum amount of resource required to guarantee schedulability of real-time component
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Sporadic (10,2,10)
EDF
Sporadic (15,2,15)
Component interface
Resource Satisfiability Analysis
• Given a component and a resource model, resource satisfiability analysis is to determine if, for every time interval,
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(maximum possible)resource demand
that the component’s task set needs under its scheduling
algorithm
(minimum possible) resource supplythat resource model
provides
≤
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Motivation
Resource optimality
Periodic resource model
Conclusions
EDP resource model
Incremental analysis
Multiprocessor clustering
Compositional schedulability analysis• Resource model based analysis• Periodic resource models
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Resource Demand Bound• Resource demand bound during an interval of length t
– dbf(W,A,t) computes the maximum possible resource demand that W requires under algorithm A during a time interval of length t
• Periodic task model T(p,e) [Liu & Layland, ’73]– characterizes the periodic behavior of resource demand with
period p and execution time e– Ex: T(3,2)
0 1 2 3 4 5 6 7 8 9 10
t dem
and
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• For a periodic workload set W = {Ti(pi,ei)}, – dbf (W,A,t) for EDF algorithm [Baruah et al.,‘90]
Demand Bound - EDF
€
dbf (W, EDF, t) =t
pi
⎢
⎣ ⎢
⎥
⎦ ⎥
Ti∈W
∑ ⋅ei
0 1 2 3 4 5 6 7 8 9 10
t dem
and
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Task (resource demand) representations
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Resource Supply Bound
• Resource supply during an interval of length t
– sbfR(t) : the minimum possible resource supply by resource R over all intervals of length t
• For a single periodic resource model, i.e., Γ(3,2)
– we can identify the worst-case resource allocation
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t sup
ply
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Demand-based Schedulability Analysis
• A periodic task set is schedulable under EDF
if and only if dbf(t) ≤ t ≤
€
∀t > 0
over the periodic resource model Γ(P,Q)
[Baruah, et. al., ‘90]
t
reso
urce
t
dbf(t)
lsbf(t)[Shin and Lee, 2003]
lsbf(t)
10
Resource Models as Interfaces
• Characterization of resource supply– Underlying component’s perspective: Virtualizes
scheduling hierarchy represented by all higher-level components
– Parent component’s perspective: Characterizes resource supply to underlying component
• Fractional resource model– b.t units of resource in every t time units (0 < b <= 1) – Not realizable (processor cannot be shared fractionally)
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0 1 2 3 4 5 6 7 8 9 10Fraction b
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Component Abstraction
T11(25,4)
T12(40,5)
T21(25,4)
T22(40,5)
R(?, ?)
EDF
EDF RM
R2(?, ?)R1(?, ?)R1(10, 3.01) R2(10, 4.34)
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Compositional Real-Time Guarantees
T11(25,4)
T12(40,5)
T21(25,4)
T22(40,5)
R(?, ?)
EDF
EDF RM
R2(?, ?)R1(?, ?)R1(10, 3.1) R2(10, 4.4)
T1(10, 3.1) T2(10, 4.4)
R(5, 4.4)
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Resource Supply Models
Tree schedule
Periodic model
EDP model
Recurring branching resource supply model
ACSR+
Bounded-delayResource model
Cyclic Executive
EDP Resource Model
• Explicit Deadline Periodic resource model: = (,,) resource units in time units– Repeat supply every time units
• Properties– Time-partitioned, periodic resource allocation behavior
• Benefits of realizability and implementability– Blackout interval in EDP depends on and , for fixed
• Interval can be controlled using without changing bandwidth– Smaller bandwidth required to schedule the same
component, when compared to periodic resource models2009.10.15 RePP Workshop 15
0 1 2 3 4 5 6 7 8 9 10
(5,3,4)
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Motivation
Resource optimality
Periodic resource model
Conclusions
Characterization of optimality in compositional schedulability analysis
EDP resource model
Incremental analysis
Multiprocessor clustering
What is the Problem?
• Existing models (periodic and EDP) have resource overheads– At least in comparison to total
demand of elementary components
• Is total elementary workload a good measure?– What about overhead of DM?
• How to account for DM overhead in component C5
– Depends on interfaces of C4 and C3
• Do we really need resource models for this analysis?– Final goal is to abstract
components into tasks and jobs
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C1
EDFC2
DM
C4
EDFC3
EDF
C5
DM
Sporadic tasks Sporadic tasks
Sporadic tasks
Assumptions and Notations
• Assumptions– Workload comprised of constrained deadline periodic
tasks• W = {i = (Ti,Ci,Di)i=1…n}, with Di ≤ Ti for all i
– Ignore preemption related overheads
• Notations– Schedulability load of component C = {W, EDF}
• LOADC = maxt>0 {dbfC(t)/t}– Schedulability load of component C = {W, DM}
• LOADC = maxi mint≤Di {rbfC,i(t)/t}
– Feasibility load of workload W• LOADW = LOADC, where C = {W, EDF}
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Load Optimal Interfaces
• Match feasibility load of interface C with schedulability load of component C C = (1,LOADC,1) is a periodic task– Release of first job in C is synchronized with release of first job
in
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CS
C
Interface set
t × LOADC
dbfC
dbf of periodic task C
Slope of lineLOADC = LOAD{,S}
Significance of Load Optimality
• Proportionate fair scheduling of components
• Comparison to resource model based interfaces– Periodic model =(,) is load optimal only when =0– EDP model =(,,) is load optimal when
= and is GCD of deadlines and periods of all tasks in the system
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dbfC
sbf
+ - (+-)
-+- (-+-)
Are Load Optimal Interfaces Really Optimal?
• Consider– C1 has workload {(6,1,6),(12,1,12)} and uses EDF– C2 has workload {(5,1,3),(10,1,7)} and uses EDF– C3 has workload {C1,C2} and uses EDF
• argmaxt dbfC1(t)/t ≠ argmaxt dbfC2
(t)/t
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dbfC1+ dbfC2
t × (LOADC1 + LOADC2
)
76
4
2
1
5 6 10 12 15
Example (1)
• Zero slack assumption in open systems– Let 1=(7,1,7), 2=(9,1,9), C1={(1,2), DM}, C3={(C1,C2), EDF} for
some C2
– Since C2 is unknown to C1, assume max. interference for 1 and 2
from C2
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Release Deadline Release Deadline
0 7 0 9
7 2 9 9
14 4 18 9
21 6 27 8
28 7 36 9
35 7 45 9
42 3 54 9
49 5
56 7
Task 1 Task 2
Example (2)
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Questions
• Hardness of achieving demand optimal interfaces– Can we classify the hardness?
• Classification may lead us to some approximation schemes
• Load optimal interfaces and resource model based techniques offer one extreme solution– Abstract component into a single periodic task
• Can we trade interface size for resource utilization?– What about logarithmically or polynomially large interfaces?
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Task (resource demand) representations
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Non-composable periodic models?
• What are right abstraction levels for real-time components?
(period, execution time)
• P1 = (p1,e1); e.g., (3,1)• P2 = (p2,e2); e.g., (7,1)• What is P1 || P2?
– (LCM(p1,p2), e1*n1 + e2*n2); e.g., (21,10) where n1*p1 = n2*p2 = LCM(p1,p2)
• What is the problem?
– beh(P1) || beh(P2) = beh(P1||P2)?• Compositionality
– (P1 || P2) || P3 = P || P3, where P = P1 || P2
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ACSR
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ACSR+ for supply partition specification
Notion of “schedulable under”(1) T_1 is schedulable under S_1
(2) T_2 is schedulable under S_2
(3) T_1 is not schedulable under S_2
Current work
• Simulation and schedulability Relations (preliminary results with Anna Philippou)– Between demand and supply– Between demand processes– Between supplies
• Semantic characterization of demand-supply based schedulability analysis
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July 23, 2008 AFOSR PI Meeting
Motivation
Resource optimality
Periodic resource model
CARTS
EDP resource model
Incremental analysis
Multiprocessor clustering
Virtual processor cluster-based scheduling on multiprocessors
• Need for virtual clustering • MPR model based scheduling• Optimal virtual clustering for
implicit deadline task systems
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July 23, 2008 AFOSR PI Meeting 31
Multicore Processor Virtualization
Scheduler
CPU
1. Compositional analysis of hierarchical multiprocessor real-time systems, through component interfaces
2. Using virtualization to develop new component interface for multiprocessor platforms
TaskTask
S
interface
TaskTask
S
interface
TaskTask
S
interface
CPU CPUVirtual CPU
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Virtual Clusters
• Use platform virtualization to provide a trade-off between resource utilization and scheduling complexity
– Cluster interface: (,m)• is the resource model, m is the maximum number
of physical processors available
– Inter-cluster scheduling is optimal
partitionedscheduling:
low utilization,easy to compute
globalscheduling:
high utilization,hard to compute
cluster-basedscheduling:
small clusters => partitioned,large clusters => global
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33
Virtual Clustering Interface
For each i, mi (<= m) is maximum number of physical processors that can be assigned to i at any instant
1(m1) 2(m2) k(mk). . .
1
x1
2 m. . . Physical processors
x2xk
. . . Task clusters
Virtual processors
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Virtual Clustering
• Task set and number of processors ====(3,2,3), =(6,4,6), and =(6,3,6), m=4
• Schedule under clustered scheduling 1, 2, 3 scheduled on processors 1 and 2
4, 5, 6 scheduled on processors 3 and 4
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0 1 2 3 4 5 6
Processor 1
Processor 2
Processor 3
Processor 4
11
5 5
2
2
3 3
4 4
1 1
2
2
33
4 4
55
6
6 6
34
Multiprocessor Periodic Resource (MPR) model
= (, ,,m’)
units of resource supply guaranteed in every time units, with concurrency at most m’ in any time instant
• Consider = (5, 12, 3)
• Why MPR model?– Periodicity enables transformation of MPR model to periodic tasks
which can be scheduled using standard algorithms
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Processor 2
Processor 3
Processor 4
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Virtual Cluster-based Scheduling
1. Split task set into clusters x1, …, xk
2. Abstract xi into MPR interface i (for cluster VCi)
3. Transform each i into periodic tasks1. Enables inter-cluster scheduler to schedule i
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Resource Satisfiability Condition
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)()ˆ(ˆarg1
llliestlm
jjj DAsbfCmIIIi
ij
Workload upper bound for Type (1) intervalsliCm
Workload upper bound for Type(2) intervalsassuming no carry-in demand at tidle
[BCL05]
j
jI
ˆ
mi-1 largest carry-in demand values at tidle
(at most mi-1 tasks are active at tidle) [Baruah07]
estlm
jj
i
IIarg1
)ˆ(
Pseudo-polynomial upper bound for Al [Thm. 2 in SEL08]
ll DA
37
Inter-cluster Scheduling
• Suppose. i (=) is GCD of periods/deadlines of all tasks in W
2. Each model i=(, i, mi) is transformed to some periodic tasks all having period and deadline
3. McNaughton’s algorithm is used for inter-cluster scheduling
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• Inter-cluster scheduling is optimal
• Successive jobs of the same task are scheduled in identical intervals
0 2
Processor 1
Processor 2
Processor 3
1 2
3
4
42
1
3
4
42
2
38
Questions
• Open issues– Efficient clustering techniques for constrained and arbitrary
deadline task systems– Interface optimality for multiprocessor systems
• Hierarchical/compositional multi-mode systems with resource constraints
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References• A Compositional Scheduling Framework for Digital Avionics Systems, Arvind Easwaran, Insup Lee, Oleg Sokolsky, Steve
Vestal, IEEE RTCSA 2009.• Optimal Virtual Cluster-based Multiprocessor Scheduling, Arvind Easwaran, Insik Shin, and Insup Lee, to be published in Real-
Time Systems Journal (RTSJ).• On the complexity of generating optimal interfaces for hierarchical systems, Arvind Easwaran, Madhukar Anand, Insup Lee,
Oleg Sokolsky, Workshop on Compositional Theory and Technology for Real-Time Embedded Systems (CRTS 2008).• Hierarchical Scheduling Framework for Virtual Clustering of Multiprocessors, Insik Shin, Arvind Easwaran, Insup Lee, ECRTS,
Prague, Czech Republic, July 2-4, 2008 (Runner-up in the best paper award)• Robust and Sustainable Schedulability Analysis of Embedded Software, Madhukar Anand and Insup Lee, LCTES, Tucson, AZ,
Jun 12-13, 2008 • Compositional Feasibility Analysis for Conditional Task Models, Madhukar Anand, Arvind Easwaran, Sebastian Fischmeister,
and Insup Lee, ISORC, Orlando, Florida, May 5-7, 2008• Compositional Real-Time Scheduling Framework with Periodic Model, Insik Shin and Insup Lee, ACM Transactions on
Embedded Computing Systems (TECS), vol 7, no 3, April 2008• Interface Algebra for Analysis of Hierarchical Real-Time Systems, Arvind Easwaran, Insup Lee, Oleg Sokolsky, Foundations of
Interface Technologies (FIT) workshop, Budapest, Hungary, April 5, 2008• Compositional Analysis Framework using EDP Resource Models, Arvind Easwaran, Madhukar Anand, and Insup Lee, Tucson,
Arizona, Dec 3-6, 2007 • Resources in process algebra, Insup Lee, Anna Philippou, Oleg Sokolsky, Journal of Logic and Algebraic Programming, Vol.
72, pp. 98-122,2007 • Incremental Schedulability Analysis of Hierarchical Real-Time Components, Arvind Easwaran, Insik Shin, Oleg Sokolsky, Insup
Lee, ACM EMSOFT 2006.
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This work was supported in part by AFOSR and NSF.