quasi-static scheduling of embedded software using free...

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Quasi-Static Scheduling ofEmbedded Software Using

Free-Choice Petri Nets

Marco Sgroi, Alberto Sangiovanni-Vincentelli

Luciano Lavagno

University of California at Berkeley

Cadence Berkeley Labs

Yosinori WatanabeCadence European Labs EE249 - Fall1999

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Outline

n Quasi-Static Schedulingn Motivation

n Scheduling Free-Choice Petri Netsn Algorithmn Application example

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Classes of Scheduling

n Static: schedule completely determined at compiletime

n Dynamic: schedule determined at run-time

n Quasi-Static: most of the schedule computed atcompile time, some scheduling decisions made atrun-time (but only when necessary)

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Embedded Software Synthesis

n System specification: set of concurrent functional blocks(DF actors, CFSMs, CSP, …)

n Software implementation: set of concurrent software tasksn Two sub-problems:

u Generate code for each task (software synthesis)u Schedule tasks dynamically (scheduling)

n Our goal: minimize real-time scheduling overhead

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Petri Nets Model

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Petri Nets Model

Schedule: t12, t13, t16...

a = 5c = a + b

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Petri Nets Model

Shared Processor+ RTOS

Task 1

Task 2

Task 3

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Embedded Systems Specifications

Static

Quasi-Static

Dynamic

Specification Scheduling

Data-dependent Control(if ..then ..else, while ..do)

Real-time Control(preemption, suspension)

Data Processing (+, -, *...)

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Data processing

i k2 + o

k1

Static Data Flow network

Example: 2nd order IIR filter o(n) = k2 i(n) + k1 o(n-1)

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Data Processing

i *k2 + o

*k1

Schedule: i, *k2, *k1, +, o

IIR 2nd order filtero(n)=k1 o(n-1) + k2 i(n)

Schedule: i, *k1, *k2, +, o

i k2 + o

k1

Static Data Flow network

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Data processing (Multirate)

o

Fast Fourier Transform

i FFT o256 256

Schedule: ii…i FFT oo…. o

256 256i

Sample rate conversion

Multirate Data Flow network Petri Net

A B C D E2 7 73 82

F5

Schedule: (147A) (147B) (98C) (28D) (32E) (160F)

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Data-dependent Control

i o>0

*2

/2

Schedule: i, if (i>0) then{ /2} else{ *2}, o

• Petri Nets provide a unified model for mixed control and data processing specifications• Free-Choice (Equal Conflict) Nets: the outcome of a choice depends on the value of a token (abstracted non-deterministically) rather than on its arrival time

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Boolean Data Flow and Petri Nets

i*2F

T

F

T

*2

/2o

>0

oi o>0

*2

/2

Schedule: i, if (i>0) then{ /2} else{ *2}, o

• PNs:• Most properties are decidable• Abstract Dataflow networks by representing if-then-else structures as non-deterministic choices• Non-deterministic actors (choice and merge) make thenetwork non-determinate

BDF PN

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Existing approaches

n Lee - Messerschmitt ‘86u Static Data Flow: cannot specify data-dependent control

n Buck - Lee ‘94u Boolean Data Flow: scheduling problem is undecidable

n Thoen - Goossens - De Man ‘96u Event graph: no schedulability check, no minimization of

number of tasksn Lin ‘97

u Safe Petri Net: no schedulability check, no multi-rate

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Bounded scheduling(Marked Graphs)

n A finite complete cycle is a finite sequence of transitionfirings that returns the net to its initial state

Ð Bounded memoryÐ Infinite execution

n To find a finite complete cycle solve f(σ) D = 0

t1 t2 t3

T-invariant f(σ) = (4,2,1)

2 22

t1t2

t3

No schedule

D =1 0-2 1 0 -2 f(σ) D = 0 has no solution

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Bounded scheduling(Marked Graphs)

n Existence of a T-invariant is only a necessary conditionn Verify that the net does not deadlock by simulating the

minimal T-invariant [Lee87]

t1 t2 t3


2 2

t1 t22 3

23t3


Deadlock(0,0) (0,0) t1t1t1t1t2t2t4

σ = t1t1t1t1t2t2t4

Not enough initial tokens

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Free-Choice Petri Nets (FCPN)

Marked Graph (MG)

Free-Choice Confusion (not-Free-Choice)

n Free-Choice:u choice depends on token value rather than arrival time

u easy to analyze (using structural methods)

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t1 t2 t3 t5 t6

Bounded scheduling(Free-Choice Petri Nets)

t1 t2t3

t4

t5 t6

t7

t1 t2 t3 t5 t6

n Can the “adversary” ever force token overflow?

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t1 t2 t4 t7


t1 t2t3

t4

t5 t6

t7

t1 t2 t4 t7


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t1 t2t3

t4

t5t7

t6


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t1 t2t3

t4

t5t7

t6


22


t1 t2t3

t4

t5t7

t6


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Schedulability (FCPN)

n Quasi-Static SchedulingÐ at compile time find one schedule for every

conditional branchÐ at run-time choose one of these schedules according to

the actual value of the data.

Σ={(t1 t2 t4),(t1 t3 t5)}

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n Valid schedule ΣÐ is a set of finite firing sequences that return the net

to its initial stateÐ contains one firing sequence for every combination

of outcomes of the free choices

t3

t2t1

t5

t4

SchedulableΣ={(t1 t2 t4),(t1 t3 t5)}

t3

t2t1

t5

t4(t1 t2 t4)

t3

t2t1

t5

t4

(t1 t3 t5)

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How to check schedulability

n Basic intuition: every resolution of data-dependent choices must be schedulable

n Algorithm:u Decompose (by applying the Reduction Algorithm)

the given Free-Choice Petri Nets into as manyConflict-Free components as the number of possibleresolutions of the non-deterministic choices.

u Check if every component is statically schedulableu Derive a valid schedule, i.e. a set of finite complete

cycles one for each conflict-free component

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Allocatability(Hack, Teruel)

n An Allocation is a control function that chooses whichtransition fires among several conflicting ones ( A: P T).

n A Reduction is the Conflict Free Net generated from oneAllocation by applying the Reduction Algorithm.

n A FCPN is allocatable if every Reduction generated froman allocation is consistent.

n Theorem: A FCPN is schedulable iffu it is allocatable andu every Reduction is schedulable (following Lee)

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Reduction Algorithm

t6t1

t5

t4t2

t7

t6

t7

t1

t4t2

t4t2t6t1

t1

t5

t4

t7

t2t6

t6

t7

t1

t4t2

t1

t3 t5

t4t6

t2

t7

T-allocation A1={t1,t2,t4,t5,t6,t7}

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How to find a valid schedule

t1

t2 t4

t5

t6

t7 t9

t8 t10

t3

Conflict Relation Sets:{t2,t3},{t7,t8}

T-allocations:

A1={t1,t2,t4,t5,t6,t7,t9,t10}

A2={t1,t3,t4,t5,t6,t7,t9,t10}

A3={t1,t2,t4,t5,t6,t8,t9,t10}

A4={t1,t3,t4,t5,t6,t8,t9,t10}

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Valid schedulet1 t2 t4

t5

t6

t7t9

t1

t3 t5

t6

t7t9

t1 t2 t4

t6t8 t10

t1

t3t5

t6t8 t10

(t1 t2 t4 t6 t7 t9 t5) (t1 t3 t5 t6 t7 t9 t5)(t1 t2 t4 t6 t8 t10) (t1 t3 t5 t6 t8 t10)

=Σ

1086531

1086421

5976531

5976421

tttttt

tttttt

ttttttt

ttttttt

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C code implementation

Σ={(t1 t2 t1 t2 t4 t6 t7 t5) (t1 t3 t5 t6 t7 t5)}

t1

t3 t5

t4t22

t6 t7

Task 1:{ t1; if (p1) then{ t2; count(p2)++; if (count(p2) = 2) then{ t4; count(p2) = count(p2) - 2;} else{ t3; t5;} }}

Task 2:{ t6; t7; t5;}

p1

p3

p4

p2

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Application example:ATM Switch

Input cells: accept?

Output cells: emit?

Internal buffer

Clock (periodic)

Incoming cells (non-periodic)

Outgoing cells

n No static schedule due to:u Inputs with independent rates

(need Real-Time dynamic scheduling)u Data-dependent control

(can use Quasi-Static Scheduling)

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Petri Nets Model

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Functional decomposition

4 Tasks

Accept/discard cell

Output time selector

Output cell enablerClock divider

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Decomposition with min # of tasks

2 Tasks

Input cell processing

Output cell processing

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Real-time scheduling ofindependent tasks

+ RTOS

Shared Processor

Task 1

Task 2

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ATM: experimental results

Sw Implementation QSS Functional partitioning

Number of tasks 2 5

Lines of C code 1664 2187

Clock cycles 197526 249726

Functional partitioning (4+1 tasks) QSS (2 tasks)

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Conclusion

n Advantages of Quasi-Static Scheduling:QSS minimizes run-time overhead with respect to Dynamic

Scheduling byAutomatic partitioning of the system functions into a

minimum number of concurrent tasksThe underlying model is FCPN: can check schedulability

before code generation

n Future worku Larger PN classes (synchronization-dependent choice)u Code optimizations

quasi-static scheduling of embedded software using free...

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