Compositional Analysis Framework using EDP Resource Models

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Compositional Analysis Framework using EDP
Resource Models

• Arvind Easwaran, Madhukar Anand, Insup Lee
• Real-Time Group
• University of Pennsylvania

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Outline

• Compositional schedulability analysis
• Hierarchical scheduling frameworks
• Periodic resource models
• EDP model based interfaces
• Model definition and properties
• Notions of optimality
• Techniques for interface generation

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Compositional Schedulability Analysis
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Hierarchical Scheduling Framework

• Sporadic task
• T (p,e,d), d ? p
• Notations
• Leaf ? C1, C2, C3
• Non-leaf ? C4, C5
• Uniprocessor platform

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Compositional Analysis
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Resource model interface

• Characterization of resource supply
• Partial resource supply bounded-delay, periodic
• Periodic resource model ? (?,?)
• ? units of resource in every ? time units
• Bandwidth ?/?
• ? as component interface
• Schedulability condition for component under ?

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Schedulability conditions

• A leaf component C with EDF is schedulable using
  a periodic resource model ? (?,?)

• sbf?(t) Supply bound function Minimum resource
  supply of model ? in any time interval of length
  t
• lsbf?(t) Linear lower bound of sbf?(t)

if and only if, for all t LCM, dbfC(t)
sbf?(t)
if, for all t LCM, dbfC(t) lsbf?(t)

• A leaf component C with DM is schedulable using a
  periodic resource model ? (?,?)

if and only if, for all i, exists t di,
rbfC,i(t) sbf?(t)
if, for all i, exists t di, rbfC,i(t)
lsbf?(t)
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Compositional Analysis
Task (p,e,d)
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Sub-optimality of periodic interfaces

• R1 Use of lsbf? instead of sbf?

To reduce starvation length, one must increase
bandwidth

• R2 Mapping ? (?,?) ? T? (?,?,?)

? (?,?)
?(?,??),?gt0

• R3 Dependency between starvation
• length and bandwidth in sbf?

dbf
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EDP resource model based interfaces
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What are they?

• Explicit Deadline Periodic resource
• Specification ? (?,?,?)
• Explicit deadline ?
• ? resource units in ? time units
• Repeat supply every ? time units

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Supply bound function (sbf?)

• sbf?(t)
• lsbf?(t)

Q
lsbf
(
t
)
(
t
(
2
)
)

-
P

D
-
Q
W
1
2
4
4
3
4
4
P

Bandwidth
Starvation length
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Schedulability conditions

• A leaf component C with EDF is schedulable using
  a EDP resource model ? (?,?,?)

if and only if, for all t LCM, dbfC(t)
sbf?(t)

• A leaf component C with DM is schedulable using a
  EDP resource model ? (?,?,?)

if and only if, for all i, exists t di,
rbfC,i(t) sbf?(t)
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Bandwidth optimal interface

• Given leaf component C, period ?
• Compute ? and ?
• Relation between ? and ? desired
• We use bandwidth optimality
• Minimizes resource bandwidth ?/?
• Occurs when ?? (Theorem 3.2)

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Bandwidth-deadline optimal

• Choose interface with Largest ? among all
  bandwidth optimal interfaces
• Reduced demand for composition
• Interface generation procedure
• Set ??, compute ? (?,?,?)
• Set ??, compute ? (?,?,?)

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EDP Interface Composition
Task (p,e,d)
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Addressing Sub-optimalities

• R1 Use of lsbf instead of sbf

• R2 Mapping from resource model to
  periodic task

• R3 Dependency between starvation
• length and bandwidth in sbf

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Addressing R1

• Function y takes only two values
• yk or k-1, where t k??
• Solving eqn. dbfC(t) sbf?(t) (Algorithm 1)
• Fix y, solve for ?
• No need to use lsbf?

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Addressing R2

• Mapping under EDF (Definition 5.2)
• ?(?,?,?) ? T?(?,?,??-?)
• Demand-supply optimal (Theorem 5.3)
• R ? Any uniprocessor resource supply
• sbfR ? sbf? if and only if R satisfies task T?

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Addressing R3

• Dependency between starvation length and
  bandwidth in sbf

Reduce starvation length without increasing
bandwidth
? (?,?,?)
?(?,?,?-?),?gt0
dbf
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Conclusion and future work

• Conclusion
• Introduced EDP resource model interfaces
• Developed compositional analysis techniques
• Similar techniques under DM scheduler
• Future work
• Support for incremental analysis
• Independence from order of composition
• Development of bandwidth bounds
• Comparison to different resource models

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Questions?