Petri Nets

Marco Sgroi (sgroi@eecs.berkeley.edu)

EE249 - Fall 2002

Outline

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

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

Petri Net Graph• Bipartite weighted directed graph:

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

• Tokens: black dots

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

3Places/Transition: conditions/events

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

Concurrency, causality, choice

Concurrency

Causality, sequencing

Choice,

conflict

Choice,

conflict

Communication Protocol

Send msg

Receive Ack

Send Ack

Receive msg

Send msg

Receive Ack

Send Ack

Receive msg

Send msg

Receive Ack

Send Ack

Receive msg

Send msg

Receive Ack

Send Ack

Receive msg

Send msg

Receive Ack

Send Ack

Receive msg

Send msg

Receive Ack

Send Ack

Receive msg

Producer-Consumer Problem

Produce

Consume

Buffer

Produce

Consume

Buffer

Produce

Consume

Buffer

Produce

Consume

Buffer

Produce

Consume

Buffer

Produce

Consume

Buffer

Produce

Consume

Buffer

Produce

Consume

Buffer

Produce

Consume

Buffer

Produce

Consume

Buffer

Produce

Consume

Buffer

Produce

Consume

Buffer

Produce

Consume

Buffer

Produce

Consume

Buffer

Producer-Consumer with priority

Consumer B can

consume only if

buffer A is empty

Inhibitor arcs

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

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.

= (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)

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

Liveness

Not live

Liveness

Not live

Liveness

Deadlock-free

Liveness

Deadlock-free

• 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

Boundedness

Unbounded

Boundedness

Unbounded

Boundedness

Unbounded

Boundedness

Unbounded

Conservation

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

Not conservative

Conservation

Not conservative

Conservation

Conservative

Analysis techniques

• Structural analysis techniques– Incidence matrix

– T- and S- Invariants

• State Space Analysis techniques– Coverability Tree

– Reachability Graph

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

State equations

p1 p2 p3t1

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

110000

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

000011

==110011

vv1 1 ==110011

Necessary Condition only

Firing vector:

(1,2,2)

Deadlock!!

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

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

Application of T-invariants

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

i *k2 +

Schedule: i *k2 *k1 + o

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

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

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

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

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

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.

Coverability Tree• Boundedness is decidable

with coverability tree

p1 p2 p3

0100t1

p1 p2 p3

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

t1 p1 t2

p1 t22 2

t1 p1 t2

p1 t22 2

(0) (1) (2)…

• Cannot solve the reachability problem

t1 p1 t2

p1 t22 2

(0) (1) (2)…

(0) (2) (0)…

Reachability graph

p1 p2 p3t1t2

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

Reachability graph

p1 p2 p3t1t2

Reachability graph

p1 p2 p3t1t2

Reachability graph

p1 p2 p3t1t2

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

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

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)

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

Free-Choice Petri Nets (FCPN)

Free-Choice (FC)

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

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

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

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

• Choose one successor for each conflicting place:

MG reduction and cover

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

components (T-invariants)

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

components (T-invariants)

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

Hack’s theorem

• Example of non-live (but safe) FCN

Hack’s theorem

Deadlock

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

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

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

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

Outline

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

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

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)

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

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();

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…

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

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

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

Filter example

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

i * c2

Filter example

i * c2

Filter example

i * c2

Filter example

i * c2

i i *c1

Filter example

i * c2

i i *c1 *c2

Filter example

i * c2

i i *c1 *c2 *c1

Filter example

i * c2

i i *c1 *c2 *c1 +

Filter example

i * c2

i i *c1 *c2 *c1 + *c2

Filter example

i * c2

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

Filter example

i * c2

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

Filter example

i * c2

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

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

FFT1024 1024 10 1

select switchT FFT

TTTF TTTF

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

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

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

Balance equations

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

– vS(B) - vS(C) = 0

– 2 vS(A) - vS(C) = 0

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

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

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

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...

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)

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)

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)

• 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

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)

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)

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

Outline

• Motivation

• Scheduling Free-Choice Petri Nets

• Algorithm

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

Petri Nets Model

Schedule: t12, t13, t16...

a = 5c = a + b

t12 t13t16

Petri Nets Model

Shared Processor+ RTOS

Task 1

Task 2

Task 3

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)

Embedded Systems Specifications

Static

Quasi-Static

Dynamic

Specification Scheduling

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

Real-time Control(preemption, suspension)

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

An example

i k2 + o

Static Data Flow network

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

Data Processing

i *k2 + o

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

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

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

Data computation (Multirate)

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

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

Data-dependent Control

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

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

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

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)

No schedule

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

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

t1 t22 3

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

= t1t1t1t1t2t2t4

Not enough initial tokens

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)

t1 t2 t3 t5 t6

Bounded scheduling (Free-Choice Petri Nets)

t1 t2t3

t1 t2 t3 t5 t6

• Can the “adversary” ever force token overflow?

t1 t2 t4 t7

t1 t2t3

t1 t2 t4 t7

t1 t2t3

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)}

• 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

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

t4(t1 t2 t4)

(t1 t3 t5)

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

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)

Reduction Algorithm

t4t2t6t1

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

How to find a valid schedule

t8 t10

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

Valid schedulet1 t2 t4

t1 t2 t4

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

ttttttt

C code implementation

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

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;}

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)

Petri Nets Model

Functional decomposition

4 Tasks

Accept/discard cell

Output time selector

Output cell enablerClock divider

Decomposition with min # of tasks

2 Tasks

Input cell processing

Output cell processing

Real-time scheduling of independent tasks

+ RTOS

Shared Processor

Task 1

Task 2

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)

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

Documents

Transcript of Petri Nets

APPLICATIONS OF PETRI NETS

Timed Petri Nets - IntechOpen

Coloured Petri Nets - artamonoviv.ru · This textbook presents Coloured Petri Nets (also known as CP-nets or CPNs). Coloured Petri Nets is a language for the modelling and validation

1 Petri nets Classical Petri nets: The basic model.

Petri Nets Peterson

Petri Nets 2000 - uni-hamburg.de · Petri Nets 2000 21st International Conference on Application and Theory of Petri Nets Aarhus, Denmark, June 26-30, 2000 ... Marked place/transition-nets

Petri nets - TAUnachumd/models/Nets.pdf · Petri nets Petri nets are a basic model of parallel and distributed systems (named after Carl Adam Petri). The basic idea is to describe

Open Petri Nets - math.ucr.edu

1 Petri Nets Ina Koch and Monika Heiner. 2 Petri Nets(1962) Carl Adam Petri.

Implementation of Petri Nets Based Controller using SFC · Implementation of Petri Nets Based Controller using SFC Abbas Dideban, ... Implementation of Petri Nets Based Controller

Petri Nets 2000

Coloured Petri Nets - HVLhome.hvl.no/ansatte/lmkr/acpn2010/ColouredPetriNets.pdf · 2010. 9. 15. · Advanced Course on Petri Nets 2010 - 2. Coloured Petri Nets (CPNs) Petri Nets

Introduction to Petri Nets

Stochastic Petri nets

Petri nets - LSV

Coloured Petri Nets

24 Petri Nets - edisciplinas.usp.br

Generalized Stochastic Petri Nets

Petri Nets 2004

SMS - Stochastic Petri Nets