liveness and boundedness of synchronous data flow graphs

technische universiteit eindhoven

Department of Electrical EngineeringElectronic Systems

Liveness and Boundedness of Synchronous Data Flow Graphs

A.H. Ghamarian, M.C.W Geilen, T. Basten, B. Theelen, M. Mousavi, and S. Stuijk

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Electronic Systems

Outline

Introduction Synchronous Data Flow Graphs (SDFGs) Definition of Liveness and Boundedness Live and Bounded SDFGs Live and Strictly Bounded SDFGs Live and Self-timed Bounded SDFGs Conclusions

intro sdf definitions boundedness strict bndss self-tmd bndss conclusions

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Introduction

Streaming Multimedia and DSP Applications Throughput Buffer sizes …

SDFG Modeling and analysis Single and multiprocessor platforms

Given an SDFG,is it realizable within bounded memory while running forever ?


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Electronic Systems

C,1

Structure of a (Timed) SDFG

channel (unbounded)

rate

token

execution time

actor

A,2 B,1

3

3

2

211

1

3

Self-loop channel

1 1


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D,1

Behavior: Firing an Actor

C, 1

B, 1

A,1 1

1

1

2

2

3

4

5

2

23

2

1


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Behavior: Execution and Self-timed Execution

An execution is a finite or infinite sequence of actor firings. An execution is maximal iff it is finite with no actor enabled in the

final state or if it is infinite. A self-timed execution is an execution in which each actor fires as

soon as it can fire.


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Consistency and Repetition Vector

A,2 B,1 C,13

3

2

211

1

3

1.q(A)=1.q(A)

1.q(A)=1.q(B)

2.q(B)=3.q(C)

3.q(C)=2.q(B)

An SDFG is consistent iff q(A) > 0 for all actors A

Repetition vector q={(A,3), (B,3), (C,2)}

1 1


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Liveness and Deadlock

An SDFG is live iff it has an execution in which all actors fire infinitely often.

An SDFG has a deadlock iff it has a maximal execution of finite length.

Consistent, but neither live nor deadlocked !

A,2 B,1 C,1

33

3

2

211

Cannot fire after one firing of B and C

A can fire infinitely often

1 1


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Boundedness

An execution is bounded iff for all states of the execution the number of tokens in all channels is at most some bound.

SDFG bounded iff it has a bounded execution. ((un-)timed) Strictly bounded iff all possible executions are bounded. ((un-)timed) Self-timed bounded iff self-timed execution is bounded. (timed)

Necessary and sufficient conditions for an SDFG to be live in combination with each of the

three definitions of boundedness.


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Liveness and Boundedness

A live SDFG is bounded iff it is consistent. A strongly connected SDFG is live iff it is deadlock free. If one strongly connected component (SCC) in an SDFG G

deadlocks then either G deadlocks or it is unbounded.

A,2 B,1 C,1

33

3

2

211

1

{(A,3), (B, 3), (C, 2)}

An SDFG is live and bounded iff it is consistent and all its SCCs are deadlock free.

1 1


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Liveness and Strict Boundedness

A live SDFG is strictly bounded iff it is consistent and strongly connected.

An SDFG is live and strictly bounded iff it is deadlock free, consistent and strongly connected.

B,1 C,1

33

3

2

2

A,211

15

1 1


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Throughput Definition

Actor throughput:The average number of firings of one actor per time unit in the self-

timed execution.

(Normalized) graph throughput (if SDFG is consistent):

firings of ( ) lim .

end time of these firingsk

k aTh a

( )min .

( )actorsa

Th a

q a


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Local and Normalized Throughput

pa b

c Each firing of a provides tokens for p/c

firings of b.

( , ) ( / ) ( )NTh a b p c Th aNormalized actor throughput of a for b :

a,E

Local actor throughput : ( ) /LTh a x E

x


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Throughput Relation between Actors

Throughput of each actor equals the minimum of its local throughput and normalized throughput of its predecessors.

b

c

d

a,1

14

2 3

2

2

1/2

4/3

1/4

3

Th(b)=1Th(c)=2Th(d)=3

NTh(b,a)=1/4NTh(c,a)=4/3NTh(d,a)=3

LTh(a)=1/2

Th(a)=1/4


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A channel between a and b is self-timed bounded iff

Main Rule for Self-timed Boundedness

pa b

c

Golden Rule


).())()/()(,( bThaThcpbaNTh

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Liveness and Self-timed Boundedness for Acyclic SDFGs

Liveness and self-timed boundedness for an acyclic SDFG: Acyclic SDFGs are live Checking self-timed boundedness for acyclic SDFGs

Calculate the throughput for all actors Sort the graph topologically Calculate the throughput for the source actors (actors without any

predecessors) and propagate the calculation

For each channel between actors a and b check the golden rule.


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Example for Acyclic SDFGs

b

C

d

a,1

14

2 3

2

2

1/4

4/3

1/4

3

Self-timed unbounded !!!


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doesitif

deadlocktdoesnitifKLThxE ii

')(/1)(

Reduction

K1

K2

K3

K4

p4

c4

p3p2

c3

c2

c1p1

x1 x3

x2x4


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Properties of the Reduced SDFG

The reduction preserves Throughput Self-timed boundedness Liveness


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Example for a General SDFG

A,2 B,1 C,1

33

3

2

211

1

x1,6 x2,4

Self-timed bounded

1/6 < 1/4


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Conclusions

Necessary and sufficient conditions for checking Liveness and boundedness of (timed) SDFGs Liveness and strict boundedness of (timed) SDFGs Liveness and self-timed boundedness of timed SDFGs

Throughput calculations for general SDFGs (not necessarily strongly connected!)


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Electronic Systemsintro sdf definitions boundedness strict bndss self-tmd bndss conclusions

Questions

liveness and boundedness of synchronous data flow graphs

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