Simulation Relations, Interface Complexity, and Resource Optimality for RealTime Hierarchical System

1
Simulation Relations, Interface Complexity, and
Resource Optimality for Real-Time Hierarchical
Systems

• Insup Lee
• PRECISE Center
• Department 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

2
Hierarchical scheduling framework
Local
scheduler
Local
Local
Local
scheduler
scheduler
scheduler
Subsystem
Subsystem
Subsystem
1
2
n
HRT
HRT
HRT
i
2
1
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

4
Resource Satisfiability Analysis

• Given a component and a resource model, resource
  satisfiability analysis is to determine if, for
  every time interval,

(maximum possible) resource demand that the
components task set needs under its scheduling
algorithm
(minimum possible) resource supply that resource
model provides

5
Motivation

• Compositional schedulability analysis
• Resource model based analysis
• Periodic resource models

Periodic resource model
EDP resource model
Incremental analysis
Resource optimality
Multiprocessor clustering
Conclusions
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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)

demand
t
0 1 2 3 4
5 6 7 8 9
10
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Demand Bound - EDF

• For a periodic workload set W Ti(pi,ei),
• dbf (W,A,t) for EDF algorithm Baruah et
  al.,90

demand
t
0 1 2 3 4
5 6 7 8 9
10
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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.,
  G(3,2)
• we can identify the worst-case resource allocation

supply
t
0 1 2 3 4
5 6 7 8 9
10
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Demand-based Schedulability Analysis

• A periodic task set is schedulable under EDF
• if and only if dbf(t) t

over the periodic
resource model G(P,Q)
lsbf(t)
Baruah, et. al., 90
Shin and Lee, 2003
t
lsbf(t)
resource
dbf(t)
t
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Resource Models as Interfaces

• Characterization of resource supply
• Underlying components perspective Virtualizes
  scheduling hierarchy represented by all
  higher-level components
• Parent components perspective Characterizes
  resource supply to underlying component
• Fractional resource model
• b.t units of resource in every t time units (0 lt
  b lt 1)
• Not realizable (processor cannot be shared
  fractionally)

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Component Abstraction
EDF
RM
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Compositional Real-Time Guarantees
EDF
RM
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Resource Supply Models
ACSR
Recurring branching resource supply model
EDP model
Tree schedule
Bounded-delay Resource model
Periodic model
Cyclic Executive
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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
  models

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Motivation
Periodic resource model
EDP resource model
Incremental analysis
Characterization of optimality in compositional
schedulability analysis
Resource optimality
Multiprocessor clustering
Conclusions
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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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Assumptions and Notations

• Assumptions
• Workload comprised of constrained deadline
  periodic tasks
• W ?i (Ti,Ci,Di)i1n, with Di Ti for all
  i
• Ignore preemption related overheads
• Notations
• Schedulability load of component C W, EDF
• LOADC maxtgt0 dbfC(t)/t
• Schedulability load of component C W, DM
• LOADC maxi mintDi 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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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

t LOADC
Slope of line LOADC LOAD?,S
dbf of periodic task ?C
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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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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

• Suppose we abstract??1 using periodic task ?
  (O,T,C,D), where O is release offset
• O and OD must be an entry in table
• For all integers k, OkT and OkTD must also be
  entries in the table
• Only possible when T 63, which is LCM of
  periods of tasks ?1 and ?2

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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), e1n1 e2n2) e.g., (21,10)
  where n1p1 n2p2
  LCM(p1,p2)
• What is the problem?
• beh(P1) beh(P2) beh(P1P2)?
• 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
• T_1 is schedulable under S_1
• (2) T_2 is schedulable under S_2
• (3) T_1 is not schedulable under S_2

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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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Motivation
Periodic resource model
EDP resource model

• Virtual processor cluster-based
• scheduling on multiprocessors
• Need for virtual clustering
• MPR model based scheduling
• Optimal virtual clustering for
• implicit deadline task systems

Incremental analysis
Resource optimality
Multiprocessor clustering
CARTS
July 23, 2008
AFOSR PI Meeting
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Multicore Processor Virtualization

• Compositional analysis of hierarchical
  multiprocessor real-time systems, through
  component interfaces
• Using virtualization to develop new component
  interface for multiprocessor platforms

Virtual CPU
Scheduler
S
S
S
Task
Task
Task
Task
Task
Task
July 23, 2008
AFOSR PI Meeting
31
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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

partitionedschedulinglow utilization,easy to
compute
globalschedulinghigh utilization,hard to
compute
cluster-basedschedulingsmall clusters gt
partitioned,large clusters gt global
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Virtual Clustering Interface
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Virtual Clustering

• Task set and number of processors
• ????????(3,2,3), ??(6,4,6), and ??(6,3,6),
  m4
• 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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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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Virtual Cluster-based Scheduling

• Split task set ? into clusters ?x1, , ?xk
• Abstract ?xi into MPR interface ?i (for cluster
  VCi)
• Transform each ?i into periodic tasks
• Enables inter-cluster scheduler to schedule ?i

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Resource Satisfiability Condition
Pseudo-polynomial upper bound for
Al Thm. 2 in SEL08
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Inter-cluster Scheduling

• Suppose
• ?i (?) is GCD of periods/deadlines of all tasks
  in W
• Each model ?i(?, ?i, mi) is transformed to some
  periodic tasks all having period and deadline
  ?
• McNaughtons algorithm is used for inter-cluster
  scheduling

• Inter-cluster scheduling is optimal
• Successive jobs of the same task are scheduled
  in identical intervals

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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.

This work was supported in part by AFOSR and NSF.