Petri Nets

1

Petri Nets

Marco Sgroi ([email protected])

EE249 - Fall 2002

2

Outline

• Petri nets– Introduction– Examples– Properties– Analysis techniques

3

Petri Nets (PNs)

• Model introduced by C.A. Petri in 1962– Ph.D. Thesis: “Communication with Automata”

• Applications: distributed computing, manufacturing, control, communication networks, transportation…

• PNs describe explicitly and graphically:– sequencing/causality

– conflict/non-deterministic choice

– concurrency

• Asynchronous model (partial ordering)• Main drawback: no hierarchy

4

Petri Net Graph• Bipartite weighted directed graph:

– Places: circles– Transitions: bars or boxes– Arcs: arrows labeled with weights

• Tokens: black dots

t1p1

p2

t2

p4

t3

p3

2

3

5

Petri Net• A PN (N,M0) is a Petri Net Graph N

– places: represent distributed state by holding tokens• marking (state) M is an n-vector (m1,m2,m3…), where mi is the non-

negative number of tokens in place pi.

• initial marking (M0) is initial state

– transitions: represent actions/events• enabled transition: enough tokens in predecessors

• firing transition: modifies marking

• …and an initial marking M0. t1p1

p2 t2

p4

t3p3

2

3Places/Transition: conditions/events

6

Transition firing rule• A marking is changed according to the following rules:

– A transition is enabled if there are enough tokens in each input place

– An enabled transition may or may not fire

– The firing of a transition modifies marking by consuming tokens from the input places and producing tokens in the output places

22

3

2 2

3

7

Concurrency, causality, choice

t1

t

2

t3 t4

t5

t6

8


Concurrency

t1

t

2

t3 t4

t5

t6

9


Causality, sequencing

t1

t

2

t3 t4

t5

t6

10


Choice,

conflict

t1

t

2

t3 t4

t5

t6

11


Choice,

conflict

t1

t

2

t3 t4

t5

t6

12

Communication Protocol

P1

Send msg

Receive Ack

Send Ack

Receive msg

P2

13


P1

Send msg

Receive Ack

Send Ack

Receive msg

P2

14


P1

Send msg

Receive Ack

Send Ack

Receive msg

P2

15


P1

Send msg

Receive Ack

Send Ack

Receive msg

P2

16


P1

Send msg

Receive Ack

Send Ack

Receive msg

P2

17


P1

Send msg

Receive Ack

Send Ack

Receive msg

P2

18

Producer-Consumer Problem

Produce

Consume

Buffer

19


Produce

Consume

Buffer

20


Produce

Consume

Buffer

21


Produce

Consume

Buffer

22


Produce

Consume

Buffer

23


Produce

Consume

Buffer

24


Produce

Consume

Buffer

25


Produce

Consume

Buffer

26


Produce

Consume

Buffer

27


Produce

Consume

Buffer

28


Produce

Consume

Buffer

29


Produce

Consume

Buffer

30


Produce

Consume

Buffer

31


Produce

Consume

Buffer

32

Producer-Consumer with priority

Consumer B can

consume only if

buffer A is empty

Inhibitor arcs

A

B

33

PN properties

• Behavioral: depend on the initial marking (most interesting)– Reachability

– Boundedness

– Schedulability

– Liveness

– Conservation

• Structural: do not depend on the initial marking (often too restrictive)– Consistency

– Structural boundedness

34

Reachability

• Marking M is reachable from marking M0 if there exists a sequence of firings M0 t1 M1 t2 M2… M that transforms M0 to M.

• The reachability problem is decidable.

t1p1

p2t2

p4

t3p3

= (1,0,1,0)M = (1,1,0,0)

= (1,0,1,0) t3

M1 = (1,0,0,1) t2

M = (1,1,0,0)

35

• Liveness: from any marking any transition can become fireable– Liveness implies deadlock freedom, not viceversa

Liveness

Not live

36


Liveness

Not live

37


Liveness

Deadlock-free

38


Liveness

Deadlock-free

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• Boundedness: the number of tokens in any place cannot grow indefinitely– (1-bounded also called safe)

– Application: places represent buffers and registers (check there is no overflow)

Boundedness

Unbounded

40



Boundedness

Unbounded

41



Boundedness

Unbounded

42



Boundedness

Unbounded

43



Boundedness

Unbounded

44

Conservation

• Conservation: the total number of tokens in the net is constant

Not conservative

45

Conservation


Not conservative

46

Conservation


Conservative

2

2

47

Analysis techniques

• Structural analysis techniques– Incidence matrix

– T- and S- Invariants

• State Space Analysis techniques– Coverability Tree

– Reachability Graph

48

Incidence Matrix

• Necessary condition for marking M to be reachable from initial marking M0:

there exists firing vector v s.t.:

M = M0 + A v

p1 p2 p3t1t2

t3A=A=

-1-1 00 0011 11 -1-100 -1-1 11

t1 t2 t3

p1

p2

p3

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State equations

p1 p2 p3t1

t3

A=A=-1-1 00 0011 11 -1-100 -1-1 11

• E.g. reachability of M =|0 0 1|T from M0 = |1 0 0|T

but also vbut also v2 2 = | 1 1 2 |= | 1 1 2 |TT or any v or any vk k = | 1 (k) = | 1 (k) (k+1) |(k+1) |TT

t2

110000

++-1-1 00 0011 11 -1-100 -1-1 11

000011

==110011

vv1 1 ==110011

50

Necessary Condition only

22

Firing vector:

(1,2,2)

t1

t2

t3

Deadlock!!

51

State equations and invariants

• Solutions of Ax = 0 (in M = M0 + Ax, M = M0)

T-invariants– sequences of transitions that (if fireable) bring back to original marking

– periodic schedule in SDF

– e.g. x =| 0 1 1 |T

p1 p2 p3t1

t3

A=A=-1-1 00 0011 11 -1-100 -1-1 11

t2

52

Application of T-invariants

• Scheduling– Cyclic schedules: need to return to the initial state

i *k2 +

*k1

Schedule: i *k2 *k1 + o

T-invariant: (1,1,1,1,1)

o

53

State equations and invariants

• Solutions of yA = 0

S-invariants– sets of places whose weighted total token count does not

change after the firing of any transition (y M = y M’)

– e.g. y =| 1 1 1 |T

p1 p2 p3t1

t3

AATT==-1-1 11 0000 11 -1-100 -1-1 11

t2

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Application of S-invariants

• Structural Boundedness: bounded for any finite initial marking M0

• Existence of a positive S-invariant is CS for structural boundedness – initial marking is finite

– weighted token count does not change

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Summary of algebraic methods

• Extremely efficient (polynomial in the size of the net)

• Generally provide only necessary or sufficient information

• Excellent for ruling out some deadlocks or otherwise dangerous conditions

• Can be used to infer structural boundedness

56

Coverability Tree

• Build a (finite) tree representation of the markings

Karp-Miller algorithm• Label initial marking M0 as the root of the tree and tag it as new

• While new markings exist do:– select a new marking M

– if M is identical to a marking on the path from the root to M, then tag M as old and go to another new marking

– if no transitions are enabled at M, tag M dead-end

– while there exist enabled transitions at M do:• obtain the marking M’ that results from firing t at M

• on the path from the root to M if there exists a marking M’’ such that M’(p)>=M’’(p) for each place p and M’ is different from M’’, then replace M’(p) by for each p such that M’(p) >M’’(p)

• introduce M’ as a node, draw an arc with label t from M to M’ and tag M’ as new.

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Coverability Tree• Boundedness is decidable

with coverability tree

p1 p2 p3

p4

t1t2

t3

1000

58



p1 p2 p3

p4

t1t2

t3

1000

0100t1

59



p1 p2 p3

p4

t1t2

t3

1000

0100

0011

t1

t3

60



p1 p2 p3

p4

t1t2

t3

1000

0100

0011

t1

t3

0101

t2

61



p1 p2 p3

p4

t1t2

t3

1000

0100

0011

t1

t3

t2010

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Coverability Tree• Is (1) reachable from (0)?

t1 p1 t2

t1

0

t1

0

t1

p1 t22 2

63


t1 p1 t2

t1

0

t1

0

t1

p1 t22 2

(0) (1) (2)…

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• Cannot solve the reachability problem

t1 p1 t2

t1

0

t1

0

t1

p1 t22 2

(0) (1) (2)…

(0) (2) (0)…

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Reachability graph

p1 p2 p3t1t2

t3

100

• For bounded nets the Coverability Tree is called Reachability Tree since it contains all possible reachable markings

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Reachability graph

p1 p2 p3t1t2

t3

100

010t1


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Reachability graph

p1 p2 p3t1t2

t3

100

010

001

t1

t3


68

Reachability graph

p1 p2 p3t1t2

t3

100

010

001

t1

t3t2


69

Subclasses of Petri nets• Reachability analysis is too expensive• State equations give only partial information • Some properties are preserved by reduction rules

e.g. for liveness and safeness

• Even reduction rules only work in some cases• Must restrict class in order to prove stronger results

70

PNs Summary

• PN Graph: places (buffers), transitions (actions), tokens (data)

• Firing rule: transition enabled if there are enough tokens in each input place

• Properties– Structural (consistency, structural boundedness…)

– Behavioral (reachability, boundedness, liveness…)

• Analysis techniques– Structural (only CN or CS): State equations, Invariants

– Behavioral: coverability tree

• Reachability

• Subclasses: Marked Graphs, State Machines, Free-Choice PNs

2

71

Subclasses of Petri nets: MGs• Marked Graph: every place has at most 1 predecessor

and 1 successor transition• Models only causality and concurrency (no conflict)

NO

YES

72

Subclasses of Petri nets: SMs

• State machine: every transition has at most 1 predecessor and 1 successor place

• Models only causality and conflict – (no concurrency, no synchronization of parallel activities)

YES NO

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Free-Choice Petri Nets (FCPN)

Free-Choice (FC)

Extended Free-Choice Confusion (not-Free-Choice)

t1

t2

Free-Choice: the outcome of a choice depends on the value of a token (abstracted non-deterministically) rather than on its arrival time.

every transition after choice

has exactly 1 predecessor place

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Free-Choice nets

• Introduced by Hack (‘72)• Extensively studied by Best (‘86) and Desel and

Esparza (‘95)• Can express concurrency, causality and choice without

confusion• Very strong structural theory

– necessary and sufficient conditions for liveness and safeness, based on decomposition

– exploits duality between MG and SM

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MG (& SM) decomposition• An Allocation is a control function that chooses which

transition fires among several conflicting ones ( A: P T).

• Eliminate the subnet that would be inactive if we were to use the allocation...

• Reduction Algorithm– Delete all unallocated transitions

– Delete all places that have all input transitions already deleted

– Delete all transitions that have at least one input place already deleted

• Obtain a Reduction (one for each allocation) that is a conflict free subnet

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• Choose one successor for each conflicting place:

MG reduction and cover

77



78



79



80



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MG reductions• The set of all reductions yields a cover of MG

components (T-invariants)

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MG reductions• The set of all reductions yields a cover of MG

components (T-invariants)

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Hack’s theorem (‘72)

• Let N be a Free-Choice PN:– N has a live and safe initial marking (well-formed)

if and only if• every MG reduction is strongly connected and not empty, and

the set of all reductions covers the net

• every SM reduction is strongly connected and not empty, and

the set of all reductions covers the net

84

Hack’s theorem

• Example of non-live (but safe) FCN

85

Hack’s theorem


86

Hack’s theorem


87

Hack’s theorem


88

Hack’s theorem


89

Hack’s theorem


90

Hack’s theorem


91

Hack’s theorem


92

Hack’s theorem


93

Hack’s theorem


94

Hack’s theorem


95

Hack’s theorem


96

Hack’s theorem


97

Hack’s theorem


98

Hack’s theorem


99

Hack’s theorem


Deadlock

100

Summary of LSFC nets

• Largest class for which structural theory really helps

• Structural component analysis may be expensive (exponential number of MG and SM components in the

worst case)

• But… – number of MG components is generally small

– FC restriction simplifies characterization of behavior

101

Petri Net extensions

• Add interpretation to tokens and transitions– Colored nets (tokens have value)

• Add time– Time/timed Petri Nets (deterministic delay)

• type (duration, delay)• where (place, transition)

– Stochastic PNs (probabilistic delay)– Generalized Stochastic PNs (timed and immediate

transitions)

• Add hierarchy– Place Charts Nets

102

Summary of Petri Nets

• Graphical formalism• Distributed state (including buffering)• Concurrency, sequencing and choice made

explicit• Structural and behavioral properties• Analysis techniques based on

– linear algebra – structural analysis (necessary and sufficient only for

FC)

103

References

• T. Murata Petri Nets: Properties, Analysis and Applications

• http://www.daimi.au.dk/PetriNets/• Cortadella, J.; Kishinevsky, M.; Lavagno, L.; Yakovlev, A.

Synthesizing Petri nets from state-based models. 1995 IEEE/ACM International Conference on Computer-Aided Design. Digest of Technical Papers

104

Outline

• Part 3: Models of Computation– FSMs– Discrete Event Systems – CFSMs– Petri Nets – Data Flow Models– The Tagged Signal Model

105

Data-flow networks

• Kahn Process Networks• Dataflow Networks

– actors, tokens and firings

• Static Data-flow– static scheduling

– code generation

– buffer sizing

• Other Data-flow models– Boolean Data-flow

– Dynamic Data-flow

106

Data-flow networks

• Powerful formalism for data-dominated system specification

• Partially-ordered model (no over-specification)• Deterministic execution independent of

scheduling• Used for

– simulation

– scheduling

– memory allocation

– code generation

for Digital Signal Processors (HW and SW)

107

A bit of history

• Karp computation graphs (‘66): seminal work • Kahn process networks (‘58): formal model• Dennis Data-flow networks (‘75): programming

language for MIT DF machine• Several recent implementations

– graphical:• Ptolemy (UCB), Khoros (U. New Mexico), Grape (U. Leuven)• SPW (Cadence), COSSAP (Synopsys)

– textual:• Silage (UCB, Mentor)• Lucid, Haskell

108

Kahn Process networks

• Network of sequential processes running concurrently and communicating through single-sender single-receiver unbounded FIFOs.

• Process: mapping from input sequences to output sequences (streams)

• Blocking read: process attempting to read from empty channel stalls until the buffer has enough tokens

• Determinacy: the order in which processes are fired does not affect the final result

• Difficult to schedule– Dataflow Networks

…

Send();

…

…

Wait();

…

109

Determinacy

• Process: “continuous mapping” of input sequence to output sequences

• Continuity: process uses prefix of input sequences to produce prefix of output sequences. Adding more tokens does not change the tokens already produced

• The state of each process depends on token values rather than their arrival time

• Unbounded FIFO: the speed of the two processes does not affect the sequence of data values

Fx1,x2,x3… y1,y2,y3…

110

Scheduling

• Multiple choices for ordering process execution• Dynamic scheduling• Context switching overhead• Data-driven scheduling

– run processes as soon as data is available– boundedness

• Demand-driven scheduling– run a process when its output is needed as input by another process

• Bounded scheduling [Parks96]– define bounds on buffers, if program deadlocks extend capacity

A C

B

A

B

C

…

111

Data-flow networks

• A Data-flow network is a collection of actors which are connected and communicate over unbounded FIFO queues

• Actors firing follows firing rules– Firing rule: number of required tokens on inputs

– Function: number of consumed and produced tokens

• Breaking processes of KPNs down into smaller units of computation makes implementation easier (scheduling)

• Tokens carry values – integer, float, audio samples, image of pixels

• Network state: number of tokens in FIFOs

+

112

Intuitive semantics

• At each time, one actor is fired

• When firing, actors consume input tokens and produce output tokens

• Actors can be fired only if there are enough tokens in the input queues

113

Filter example

• Example: FIR filter– single input sequence i(n)– single output sequence o(n)– o(n) = c1 i(n) + c2 i(n-1)

* c1

+ o

i * c2

i(-1)

114

Filter example


* c1

+ o

i * c2

i(-1)

i

115

Filter example


* c1

+ o

i * c2

i(-1)

i i

116

Filter example


* c1

+ o

i * c2

i(-1)

i i *c1

117

Filter example


* c1

+ o

i * c2

i(-1)

i i *c1 *c2

118

Filter example


* c1

+ o

i * c2

i(-1)

i i *c1 *c2 *c1

119

Filter example


* c1

+ o

i * c2

i i *c1 *c2 *c1 +

120

Filter example


* c1

+ o

i * c2

i i *c1 *c2 *c1 + *c2

121

Filter example


* c1

+ o

i * c2

i i *c1 *c2 *c1 + *c2 +

122

Filter example


* c1

+ o

i * c2

i i *c1 *c2 *c1 + *c2 + o

123

Filter example


* c1

+ o

i * c2

i i *c1 *c2 *c1 + *c2 + o o

124

Examples of Data Flow actors

• SDF: Synchronous (or, better, Static) Data Flow

• fixed number of input and output tokens per invocation

• BDF: Boolean Data Flow• control token determines consumed and produced tokens

+

1

11

FFT1024 1024 10 1

select switchT FFT

TTTF TTTF

125

Static scheduling of DF

• Number of tokens produced and consumed in each firing is fixed

• SDF networks can be statically scheduled at compile-time – no overhead due to sequencing of concurrency– static buffer sizing

• Different schedules yield different – code size– buffer size– pipeline utilization

126

Static scheduling of SDF

• Based only on process graph (ignores functionality)• Objective: find schedule that is valid, i.e.:

– admissible

(only fires actors when fireable)

– periodic (cyclic schedule)

(brings network back to initial state firing each actor at least once)

• Optimize cost function over admissible schedules

127

Balance equations

• Number of produced tokens must equal number of consumed tokens on every edge

• Repetitions (or firing) vector vS of schedule S: number of firings of each actor in S

• vS(A) np = vS(B) nc

must be satisfied for each edge

np nc

A B

128

Balance equations

B C

A3

1

1

1

22

11

• Balance for each edge:– 3 vS(A) - vS(B) = 0

– vS(B) - vS(C) = 0

– 2 vS(A) - vS(C) = 0

– 2 vS(A) - vS(C) = 0

129

Balance equations

• M vS = 0

iff S is periodic• Full rank (as in this case)

• no non-zero solution

• no periodic schedule

(too many tokens accumulate on A->B or B->C)

3 -1 00 1 -12 0 -12 0 -1

M =

B C

A3

1

1

1

22

11

130

Balance equations

• Non-full rank• infinite solutions exist (linear space of dimension 1)

• Any multiple of q = |1 2 2|T satisfies the balance equations

• ABCBC and ABBCC are minimal valid schedules• ABABBCBCCC is non-minimal valid schedule

2 -1 00 1 -12 0 -12 0 -1

M =

B C

A2

1

1

1

22

11

131

Static SDF scheduling

• Main SDF scheduling theorem (Lee ‘86):– A connected SDF graph with n actors has a periodic

schedule iff its topology matrix M has rank n-1– If M has rank n-1 then there exists a unique smallest

integer solution q to

M q = 0• Rank must be at least n-1 because we need at least

n-1 edges (connected-ness), providing each a linearly independent row

• Admissibility is not guaranteed, and depends on initial tokens on cycles

132

Admissibility of schedules

• No admissible schedule:BACBA, then deadlock…

• Adding one token on A->C makesBACBACBA valid

• Making a periodic schedule admissible is always possible, but changes specification...

B C

A1

2

1

3

2

3

133

From repetition vector to schedule

• Repeatedly schedule fireable actors up to number of times in repetition vector q = |1 2 2|T

• Can find either ABCBC or ABBCC • If deadlock before original state, no valid schedule

exists (Lee ‘86)

B C

A2

1

1

1

22

11

134

From schedule to implementation

• Static scheduling used for:– behavioral simulation of DF (extremely efficient)– code generation for DSP – HW synthesis (Cathedral by IMEC, Lager by UCB, …)

• Code generation by code stitching(chaining custom code for each actor)

• Issues in code generation– execution speed (pipelining, vectorization)– code size minimization– data memory size minimization (allocation to FIFOs)

135

Code size minimization

• Assumptions (based on DSP architecture):– subroutine calls expensive

– fixed iteration loops are cheap (“zero-overhead loops”)

• Absolute optimum: single appearance schedulee.g. ABCBC -> A (2BC), ABBCC -> A (2B) (2C)

• may or may not exist for an SDF graph…

• buffer minimization relative to single appearance schedules

(Bhattacharyya ‘94, Lauwereins ‘96, Murthy ‘97)

136

• Assumption: no buffer sharing• Example:

q = | 100 100 10 1|T

• Valid SAS: (100 A) (100 B) (10 C) D• requires 210 units of buffer area

• Better (factored) SAS: (10 (10 A) (10 B) C) D• requires 30 units of buffer areas, but…• requires 21 loop initiations per period (instead of 3)

Buffer size minimization

C D1 10

A

B10

10

1

1

137

Dynamic scheduling of DF

• SDF is limited in modeling power – no run-time choice

– cannot implement Gaussian elimination with pivoting

• More general DF is too powerful– non-Static DF is Turing-complete (Buck ‘93)

– bounded-memory scheduling is not always possible

• General case: thread-based dynamic scheduling (Parks ‘96: may not terminate, but never fails if feasible)

138

Summary of DF networks

• Advantages:– Easy to use (graphical languages)– Powerful algorithms for

• verification (fast behavioral simulation)• synthesis (scheduling and allocation)

– Explicit concurrency

• Disadvantages:– Efficient synthesis only for restricted models

• (no input or output choice)

– Cannot describe reactive control (blocking read)

139

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 Berkeley Labs

140

Outline

• Motivation

• Scheduling Free-Choice Petri Nets

• Algorithm

141

Embedded Software Synthesis

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

• Software implementation: (smaller) 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

142

Petri Nets Model

143

Petri Nets Model

Schedule: t12, t13, t16...

a = 5c = a + b

t12 t13t16

144

Petri Nets Model

Shared Processor+ RTOS

Task 1

Task 2

Task 3

145

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)

146

Embedded Systems Specifications

Static

Quasi-Static

Dynamic

Specification Scheduling

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

Real-time Control(preemption, suspension)

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

147

An example

i k2 + o

k1

Static Data Flow network

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

148

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

149

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)

150

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

151

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

152

Scheduling FC 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

153

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 solvef() 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

154

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

155

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)

156

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?

157

t1 t2 t4 t7


t1 t2t3

t4

t5 t6

t7

t1 t2 t4 t7


158


t1 t2t3

t4

t5t7

t6


159


t1 t2t3

t4

t5t7

t6


160


t1 t2t3

t4

t5t7

t6


161

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)}

162


• 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)

163

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 (FCPN with weights) 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

164

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 FCPN is allocatable if every Reduction generated from an allocation is consistent.

• Theorem: A FCPN is schedulable iff

– it is allocatable and

– every Reduction is schedulable (following Lee)

165

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}

166

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

}

167

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

168

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

169

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)

170

Petri Nets Model

171

Functional decomposition

4 Tasks

Accept/discard cell

Output time selector

Output cell enablerClock divider

172

Decomposition with min # of tasks

2 Tasks

Input cell processing

Output cell processing

173

Real-time scheduling of independent tasks

+ RTOS

Shared Processor

Task 1

Task 2

174

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)

175

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

• Related work– Larger PN classes

– Code optimizations

Petri Nets

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