1 Quasi-Static Scheduling of Embedded Software Using Free-Choice Petri Nets Marco Sgroi, Alberto Sangiovanni-Vincentelli Luciano Lavagno University of

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Quasi-Static Scheduling of Embedded 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 - Fall2001

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Outline

• Motivation

• Scheduling Free-Choice Petri Nets

• Algorithm

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Embedded Software Synthesis

• System specification: set of concurrent functional blocks (DF actors, CFSMs, PNs, …)

• Software implementation: set of concurrent software tasks

• Two sub-problems:

– Generate code for each task (software synthesis)

– Schedule tasks dynamically (dynamic scheduling)

• 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

t12 t13t16

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Petri Nets Model

Shared Processor+ RTOS

Task 1

Task 2

Task 3

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Classes of Scheduling

• Static: schedule completely determined at compile time

• Dynamic: schedule determined at run-time

• Quasi-Static: most of the schedule computed at compile time, some scheduling decisions made at run-time (but only when necessary)

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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

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

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

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

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Data computation (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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Existing approaches• Lee - Messerschmitt ‘86

– Static Data Flow: cannot specify data-dependent control

• Buck - Lee ‘94

– Boolean Data Flow: scheduling problem is undecidable

• Thoen - Goossens - De Man ‘96

– Event graph: no schedulability check, no minimization of number of tasks

• Lin ‘97

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

• Thiele - Teich ‘99

– Bounded Petri Net: partial schedulability check, complex (reachability-based) algorithm

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PNs and BDF

BDF network

F

T

F

T

>0

Petri Net

t1

t2

t3

t4

t5

<=0

<=0

>0

>0

Switch/Select vs. choice/merge

PNs: No correlation between different choices

TFF

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PNs and BDF

BDF network

F

T

F

T

>0

Petri Net

PNs are not-determinate

F

t1t2 t4 t6

t8

t7t5t3

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PNs and BDF

BDF network

F

T

F

T

>0

Petri Net


TF

t1t2 t4 t6

t8

t7t5t3

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PNs and BDF

BDF network

F

T

F

T

>0

Petri Net


TTF

t1t2 t4 t6

t8

t7t5t3

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Existing approaches• Lee - Messerschmitt ‘86

– Static Data Flow: cannot specify data-dependent control

• Buck - Lee ‘94

– Boolean Data Flow: scheduling problem is undecidable

• Thoen - Goossens - De Man ‘96

– Event graph: no schedulability check, no minimization of number of tasks

• Lin ‘97

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

• Thiele - Teich ‘99

– Bounded Petri Net: partial schedulability check, complex (reachability-based) algorithm

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Scheduling Petri Nets

• Petri Nets provide a unified model for mixed control and dataflow specification

• Most properties are decidable

• A lot of theory available

• Abstract Dataflow networks by representing if-then-else structures as non-deterministic choices

• Non-deterministic actors (choice and merge) make the network non-determinate according to Kahn’s definition

• Free-Choice: 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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Bounded scheduling (Marked Graphs)

• A finite complete cycle is a finite sequence of transition firings that returns the net to its initial state

• Bounded memory

• Infinite execution

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

t1 t2 t3

T-invariant f() = (4,2,1)

2 22

t1t2

t3

No schedule

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

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Bounded scheduling (Marked Graphs)

• Existence of a T-invariant is only a necessary condition

• 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

= t1t1t1t1t2t2t4

Not enough initial tokens

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Free-Choice Petri Nets (FCPN)

Marked Graph (MG)

Free-Choice Confusion (not-Free-Choice)

Free-Choice: choice depends on token value rather than arrival timeeasy 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

• 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


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t1 t2t3

t4

t5t7

t6


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Schedulability (FCPN)

• Quasi-Static Scheduling • at compile time find one schedule for every

conditional branch

• at run-time choose one of these schedules according to the actual value of the data.

={(t1 t2 t4),(t1 t3 t5)}

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• Valid schedule• is a set of finite firing sequences that return the net to

its initial state

• contains one firing sequence for every combination of outcomes of the free choices

t3

t2t1

t5

t4

Schedulable={(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

• Basic intuition: every resolution of data-

dependent choices must be schedulable

• Algorithm:

– Decompose (by applying the Reduction Algorithm) the

given Equal Conflict Net into as many Conflict-Free

components as the number of possible resolutions of

the non-deterministic choices.

– Check if every component is statically schedulable

– 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)

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

• A Reduction is the Conflict Free Net generated from one Allocation by applying the Reduction Algorithm.

• A ECN is allocatable if every Reduction generated from an allocation is consistent.

• Theorem: A ECN is schedulable iff

– it is allocatable and

– 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)

1086531

1086421

5976531

5976421

tttttt

tttttt

ttttttt

ttttttt

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C code implementation

={(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

• No static schedule due to:– Inputs with independent rates

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

(can use Quasi-Static Scheduling)

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Petri Nets Model

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Decomposition with min # of tasks

2 Tasks

Input cell processing

Output cell processing

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Real-time scheduling of independent tasks

+ RTOS

Shared Processor

Task 1

Task 2

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Functional decomposition

4 Tasks

Accept/discard cell

Output time selector

Output cell enablerClock divider

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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

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

Scheduling by

Automatic partitioning of the system functions into a minimum number of concurrent tasks

The underlying model is FCPN: can check schedulability before code generation

• Future work– Larger PN classes

– Code optimizations

Documents

1 Quasi-Static Scheduling of Embedded Software Using Free-Choice Petri Nets Marco Sgroi, Alberto Sangiovanni-Vincentelli Luciano Lavagno University of