Fluidization of�Discrete Event Modelsor a�marriage between the discrete

and�the continuous

Manuel SilvaUniversidad�de�Zaragoza

1. As�an�appetizer…�on�models,�systems�and�fluidization…

2.�(Discrete)�Petri�Nets:�elements�to�be�used

3.�Fluidization�of�autonomous�discrete�models

4.�Fluid�timed�models:�server�semantics…

Informal, trying to provide intuitionUsing examples as simple as possible

Well known,�model is:*�in�Science &�Technology: the representation*�in�Fine�Arts:�the reference…

…�just the contrary.

La�trahison des�images,René�Magritte,�1928Ͳ29

…�but if model is a representation,�it is only “an image”, “a view”

constructed with a�purpose,�not “the reality”:a

A MODEL vs THE (REAL)�SYSTEM

how representative is a�model?�ͲͲͲ Fidelity vs complexity?

This system is discrete or continuous?

1)Continuous?2)Discrete?3)…???

The system &�the model:

1) Purpose2) Fidelity3) Complexity

Models are “views”

…�sometimes in�manufacturing:• a�production engineer?• or and�hydraulic engineer?

• Flows• Levels (deposits…)• 3Ͳways�valves(mixing flows)

• …

In�manufacturing,��������in�informatics…������most frequently:rendez-vous lead�to�a�minimum operator

In�population dynamics,�in�chemistry…�are�mostfrequent other functions

The state explosion problem…�(�a�very naive view…)�H

omot

ecy

Increasing the population (or initial marking):����������������������• the approximation improves• much bigger computational savings

MarkovianT-time

interpretation

SteadyState

FLUIDThe bigger the marking,�the better theaproximation &�the computational savings

If we fluidify (from integer to�reals)��the model&�use�infinite server�semantics (min�operator)…infinite

From individuals to�populations:decoulourization

A�classical &�basic predatorͲprey system (revisited)

If bounded populations

Coloured PN Place/Transition net (PN)

VolterraͲLotka

If we fluidify (from integer to�reals)��the previous model&�use��product server�semantics…

The timed continuousPN�is Live &�Bounded

The timed discrete PN�isNonͲlive &�Unbounded

LotkaͲVolterra orbit

but…

product

If populations are�as�in�the LotkaͲVolterra model:

?

Fluidization of DEDS (also “views”…)*�Queuing�Networks�(Newell,�1971)*�Petri�Nets�(David&Alla:�Continuous�PNs,�1987�//��

Silva&Colom,�PNs�&�LPP,�1987+1999)*�Process�Algebra�(Hillston,�2005)

Fluid “views” of DEDS*�Forrester (or�Stock�&�Flow)�diagrams (1961)*�Stochastic Flow (or Fluid) Systems (Cassandras,���

2001) “We�use�Stochastic�Fluid�Models�(SFM)�for�control�andoptimization�of�communication�networks�in�whichdetailed�discrete�event�models�become�impractical.”

Fluidization and�fluid�“views”�(to�overcome the state explosion problem) With a�preliminary character…

If populations (i.e.,�initial markings)�are:• “veryͲvery big”:�fluid�views are�most surely very interesting• “big”:���fluid�views may usually be�very interesting• “not so�big”:��fluid�views may or may not be�

(very)�interesting

* k >>> 1* k >> 1* k >~ 1

With fluidization (total�or partial):• What we keep?• What we lost?• What we gain?

The balance?¾ Like in�differential equations:�NOT�all can�be�

approximated by linearization (e.g.�Lorentz)¾ In�PNs:�NOT�all can�be�approximated by

(partial)�fluidization

How to��improve the (partially)�fluid�relaxations?

1. As�an�appetizer2.�(Discrete)�Petri�Nets:�elements�to�be�used¾ A�quick�global�perspective�of�place/transition�nets�(P/T)¾ Fundamental�equation�and�mathematical�programming¾ Components,�invariants�&�stable�predicates¾ Removing�spurious�solutions�&�implicit�places¾ Rank�theorems:�a�proof�of�synergy�between�the�

performance�and�logic�analysis

3.�Fluidization�of�autonomous�discrete�models

4.�Fluid�timed�models:�server�semantics…

Place/Transition nets�(P/T,�PNs)

PNs as�directedbipartitemultigraph

PNs and�incidencematrices

Dynamicsystem

Marking:*�Distributed*�Numerical

Marking (numerical distributed state)�& evolution

Thenegation

is notpresent!

Logic:consumption/ production……….…………..

No mono-tonicity

OR constructions AND constructions

Marking (numerical distributed state)�& evolution

Synchronizations (two mechanisms…):*�Joins /�RendezͲVous*�Weigths on arcs

Fundamental�or state (transition)�equationWithm�=�m0 +�C�V,�two main computationalproblems:1)�in�the naturals (computational cost)�&2)�with spurious (validity of�computations…)

Our focus is today on:1)�Fluidization:�relaxation into the nonͲnegative reals2) How to�remove spurious solutions on:

2.1)�The discretemodel:�untimed &�timed2.2)�The fluidmodel:�untimed &�timed

…�in�the PN�paradigm

Fundamental�equation &�fluid�relaxation

Decomposed views…�(I):�Invariants

From the fundamental equation (structure):�1.�Annuler vectors of�the incidence matrix,�C:* Left &�nonͲnegative:�PͲsemiflows (y):��y C = 0, y �0*�Rigth &�nonͲnegative:�TͲsemiflows (x):�C x = 0, x �02.�Invariants (laws…):* y m0 =�y m�ͲͲͲ token conservation law (PͲinv.)* Firing &�marking repetitive sequences law (TͲinv.)3.�Subnets:�*�Conservative component (PͲsubnet)*�Repetitive component (TͲsubnet) Duality

The semiflows allows to�“see” relevantcomponents of�themodel (some “intrinsic”�descompositions)

y C = 0y �0

� A single�(elem.)�TͲinvariant:�X�=�(1�1�1�1�1�1�1�1�1�1)� In�this case,�the TͲsubnet is the full�net�(i.e.,�no�decomposition)

Decomposed views…�(I):�Invariants

The 4�elementary components (3�FS&DA�+�1�store/SM)*�Proof of�properties*�Synthesis of�controllers*�Implementation (programmed,�PͲprogrammed,�hardwired…)

Decompose to�analyze,�synthetize &�implement

Generators of elementary traps (also siphons) can be computed as P-semiflows of a transformed net (join work MASI/LIP6-UNIZAR, 93)

Decomposed views…�(II):�stable predicatesLooking the net�as�a�directed &�bipartite graph (structure):1.�Inclusions concerning subsets of�places (here we do�notconsider weigths on arcs…):* Trap: subset of�places�s.t.�the subset of�output�transitionsare�included in�their input�subset of�transitions*�Siphon: the inverse of�a�trap2.�Stable predicates (laws…):* Trap:�if marked,�cannot be�fully unmarked (tokens are�trapped)* Siphon:�if emptied,�cannot be�marked again3.�Subnets:�*�Trap component (PͲsubnet)*�Siphon component (PͲsubnet)

Reverseconcepts

Decomposed views…�(II):�stable predicates� p1 is a trap &�a�siphon,�but not a�conservative

component

Conservative components are�trap &�siphon components� y =�(1�0�1)�is a�PͲsemiflow,�thus:�m1+m3 =10� y =�(0�1�1)�is a�PͲsemiflow,�thus:�m2+m3 =11

How�to�cut�the�spurious�deadlock�m�=�(0�1�10)?

� m3+q3 =�9,�

q3:�complement�of�m3

� p1 is�an�initially�marked�trap,�

thus:�m1 ш�1

� but:�m1+m3 =�10,�thus

m2+m3 =�11�,�thus�m1 ш�2��(p2 has�2�frozen�tokens)

(m1�

Removing aspurioussolution thatunmark a�trapis a�polynoͲmial time�problem

2�frozentokens

m3+q3 =�9

m3 ч�9

q3 makes�p2 implicit

Implicit placesPlaces�comptabilize theresources:�constraint thefiring of�transitions

Sufficient condition in�polynomial time�(PPL)

Implicit place:�never is theunique to�constraint…�but:¾ Interleaving semantic?:�

Seq.�IP¾ Step semantics?:�Con.�IP

• Elimination:�to�simplify the implementations;�to�reduce�in�analysis(boundedness,�liveness,�reversibility,�etc.;…

• Addition:�to�create new�traps (&�PͲsemiflows);�to�remove spurioussolutions;�to�allow a�nonͲintrinsic decomposition;�to�increasethe Hamming distance;…

Implicit places,�spurious solutions &�traps

� The addition of�implicit places�p,�r &�s remove all the natural�spurious solutions.

� In�general:�Alleviate,�but do�not dress (always)

Removing natural�solutions is beyond the integer hulldefined by the fundamental�equation (…�Gomory’s cuts)

m1+m2+

+m4+ms=�1

� Did siphons also help to�cut spurious deadlocks?�:�YES!

A GMEC

• A�live S3PR�model• S=�{r2,�r3,�q1,�q4,�p1,�p4}�

is a�siphon• m,�is a�solution of�the

fundamental�equation:¾ m[r1]�=�m[q3]=�

m[p3]�=�1,¾ &�zero for the rest

• m empties S.• Thusm is a�killing

spurious solution• The addition of�place�u

remove the killingspurious solution

Avoiding siphons to�be�emptied,�killing (spurious)�solutions can�be�removed�

Remark:�The added place�u is implicit in�the discretePN�system (makes the system DF�as�continuous)

Implicit places,�spurious solutions &�siphons

A GMEC:m[u]+m[q3]+m[p3]=1

Logical &�performance�properties…

Improvement1: Adding the implicit place p6, a new P-component is createdImprovement2: embeded QNs with product form (GNQN…)

* Descomposition in�PͲcomponents* Solvable�by�a�LPP (polynomial�time)* Idea:�detection�of�bottlenecks 2:�Throughput bound of�t5?

1:�Are�p2 &�p3 in�mutex?

(adding p6)

From performance�evaluation:�a�family of�rank theorems

… found when computing the visit ratios:

• Necessary condition for structural boundedness &�liveness• N&S�cond. for SB&�SL�in�net�subclasses (e.g.,�equal conflict,�DSSP…)• Sufficient condition for structural boundedness &�liveness

Polyn.�time+�Synergy

Str.�non�live ¿?

SB & SL: ¾Consistent:

C x = 0, x � 1¾Conservative:

y C = 0, y � 1¾ rank(C)ч�/ECS/Ͳ1

rank (C)�=�4rank(C)ч�4Ͳ1�=�3?

� Coherence� Economy� Synergy

The Petri�nets�paradigm

+�Fluid�PNs+�Hybrid PNs+�…

1. As�an�appetizer2.�(Discrete)�Petri�Nets:�elements�to�be�used3.�Fluidization�of�autonomous�discrete�models

3.1�Fluidization of�DES�&�fluidizability3.2.�Basic�properties�of�autonomous�continuous�

models3.3.�Improving�the�approximation�(I):�

removing�spurious�solutions

4.�Fluid�timed�models:�server�semantics…

3.1�Fluidization of�DES:�concept�&�fluidizability

Definition:��A�continuous�transition*�Enabling degree:

*�Firing:����

Neighbour places�become�continuous

¿¾½

¯®­

0>][][

][min)enab(t, p,tp,tp Pre

Premm

m mD t� o� '

Continuous net:��all transitions are�continuousHybrid net:��onlysome transitionsare�continuous

Reachability (under total�continuity)

*�m�is reachable fromm0 iff a�finite sequence

exists,�such that*�Reachability space,�����������������=��set�of��reachable

markings

–

kktt DD V �11

mm0 ��� o���� o�DD kktt

11 �

), RS(N 0m

),(NRS ), (NRS CD 00 mm �

00.2

0.40.6

0.81

00.2

0.40.6

0.81

0

0.5

1

m(p1)m(p2)

m(p

3)

Same reachability space

All (discrete)�PN�models can�be�fluidified?

ZENO

nonͲlive as�discrete &live as�continuous

live as�discrete &�nonͲlive as�continuous

¾ At�the limit:�a�trap is emptied¾ So�traps are�no�more�traps…!�

DeadlockͲfreeness:�discrete vs continuous (!!!)

Live:�yes�or not?�

New�concept: limͲreachability

A�main question:�When a�given property is preserved by fluidization?Fluidization:�can�be�approximated by the scaling of�the initial marking¾ homotheticmarkings!¾ amonotonicity property

Monotonicities

DeadlockͲfreefor the marking,�and

Bigger markingand�deadlock!

Homothetic monotonous,�…but not monotonous

x k

3.2. Basic�properties�of�autonomous�continuous�models

Let�us�assume�in�the�sequel�that:1)�The�net�N�is�consistent�(��x�>�0,�Cͼx =�0 )�and2)�all�the�transitions�can�be�fired�“a�little”�at�m0

(otherwise�stated:�no�siphon�is�empty�at�m0),

RS(N,�m0)�=�{m|m =�m0�+�CͼV,��V t 0}* Thus RS(N,�m0) is linearly delimited,�it is a�convex set* Spurious�solutions?:�

• NOT�on�the�continuousmodel;• But�spurious�solutions�in�the�originally�discrete net�model�are�here�integrated.�¾ This�raise�the�question�of�how�to�remove�it!�

Marking homotecy and�DeadlockͲfreeness

(y C = 0, y >0)

Sufficient condition for DF…

Marking homotecy and�properties preservation

23 = 8 LS

Approaching DF�(N is consistent & all siphons initially marked)

Needed?

k =�1

Approaching DF�(N is consistent & all siphons initially marked)� Structural (upper)�bound of�p:�SB(p)�=�{maxm(p) /�m�=�m0 +�C�V`� Structural lower bound (frozen tokens)�of�p:�SF(p)�=�{maxm(p)�/�

/�m�=�m0 +�C�V`

1 LS+�4�var.�+�2�equ.

¾ Firing languages are�identical if the samem0 and�tp &�tq are�“silent”�transitions¾ No�system has�a�a solution,�thus they are�deadlock-free systems

k =�1

8 LS

1 LS

3.3.�Improving�the�autonomous (thus,�in�general,�all�timed)�approximations�:�cutting�spurious�solutions

Complementary (efficient)�techniques:

� Enforce the bound to�the bigger integer of�¾ the marking of�places�(addition of�complementary places)¾ the enabling of�transitions (addition of�selfͲloop places)

� …� … below Gomory cutting planes

Truncating

� Spurious solutions that empty a�trap (like in�discretemodels;�reachable in�the limit)� Spurious solutions that empty a�siphon (feasible withfinite sequences)

if integer … beyond Gomory cutting planes

If the initial marking is “not too big”,�any SB(p)�&�SF(p)�should be�truncated to�natural�values:�floor functionsSB(p3)�=�1,5:�m(p3)+m(q3)�=�1¾ Adding q3 avoids siphonS�=�{p1,�p4}�to�be�emptied in�the continuous approximation

Adding q4 does not prevent the non�integer (deadlock)�solution:�

m =�(0�0�0�1,5�1,5)Thus the integer hull is not obtained(too expensive for the benefits…)

Decreasing�improvement�with�increasing�markings...

Remarks:�*�Non�homotheticvalue:�NOT�9k!*�Relative changedecrease with k

m�=�(0�0�10)�is�a��spuriousdeadlock�

Removing a�knownspurious solutionbecause a�trap isemptied is a

polynomial problem

1. As�an�appetizer2.�(Discrete)�Petri�Nets:�elements�to�be�used3.�Fluidization�of�autonomous�discrete�models4.�Fluid�timed�models:�semantics,�properties

and�improvements4.1�(Deterministic)�Infinite�server�(&�product)�semantics4.2�Patterns�of�behaviour4.3�Improving�the�timed�fluidization�(II)

4.3.1�Bound�reaching�problem�and�roͲsemantics4.3.2�Stochastic�approximation

x

m�=�m0 +�C�V (autonomous:�state transition eq.)

… deriving

m�W��=�C�f�W�� and��m(0)�=�m0,�with f�W��=V�W��x

… if the firing is a function of time

¾ How is f defined?�o firing semantics? Exist several…¾ An interesting source of�inspiration isMarkovian PNs

(inheritance of�results,�coherence with discrete perforͲmance models):�lead�to�infinite server semantics

4.�Timed�continuous�models:�semantics,properties�and�improvements

m�W��=�m0 +�C�V�W�

Infinite server�and�product (population) semantics

In�the sequel wejust concentrate

on infiniteserver�semantics

ʄ�=�[0.5�0.1�1.0�0.3]

From configuration to�operation mode

From the graph (structure):�1.�Configuration (allocation):�A�set�of�pͲt arcs defining a�1Ͳcover of�all transitions:�those that constraint the firing…2.�Region:�the subͲreachability space defined by a�configuration (are�disjoint except on the boundary)3.�Operation mode:�the dynamic linear�system corresponͲding to�a configuration

4.1.�(Deterministic)�Infinite server�semantics

3�configurations out of�4�possible for ʄ�=�[0.5 0.1 1.0�0.3] &m0 =�(1�1�0�0)

1)�Configuration,2)�Partition (except

on the borders)�of�thepolytope &��������3)�Phase portrait

Region 4�is notreached in�the

trajectory

m3 ч�m4 &�m2 чm3

m3 шm4 &m2 чm3

m3 ч�m4 &m2 ш�m3

m3 шm4 &m2 ш�m3

About “infinite servers�semantics”1. The�firing�flow�is�proportional�to�the�input�“level”2. The�differential�system�is:

1. Ordinary2. Piecewise�(the�minimum operator)�linear�with�constant�

coefficients�(idea�of�configuration)3. BUT:�pͲflows�(the�marking�of�some�places�are�NOT�state�

variables,�but�constraints)�lead�to�PIECEWISE�AFFINE3. Flows are bilinear on firing speeds and�markings:�it reflects

the duality of transitions & places.4. Continuous�PNs�are�technically�hybrid�(the�discrete�state,�

the�configuration,�is�implicit�in�the�continuous)5. Under�infinite�server�semantics:�Time�differentiable6. BUT:�able�to�simulate�TURING�MACHINES!!!�(ENS�Cachan /�

UNIZAR,�2007)

Infinite servers�semantics.�Properties

� Positive�systems:�f(W)�t 0,�m(W)�t 0���Marking &�time�homothecies (duality):� If m’(0)�=�k ͼm(0)�thenm’(W)�=�k ͼm(W)

&�f�’(W)�=�k ͼf(W)� If O’ =�k ͼO thenm’(W)�=�m(k ͼW)

&�f�’(W)�=�k ͼf(k ͼW)

But no�superposition,�what lead�to�many counterͲintuitive behaviors

Counterintuitive�properties (I)

• The�steadyͲstate of�the�throughput�of�the�continuous�PN�relaxation�may�not be�an�upper�bound�of�corresponding�discrete�models

2

p3

2

5

4

p1

p4

p5p

2

t1 t2

t3 t4

Discrete: 0.801 f/t.u.

Continuous: 0.535 f/t.u.

To fluidify does not improve!

Counterintuitive�properties (II)

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

p3

2

5p1

p4

p5p

2

t1 t2

t3 t4

• No�performance�monotonicities with respect toresources

m[p5]=3 � F[t3]=1.071

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Counterintuitive�properties (II)

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

p3

2

5p1

p4

p5p

2

t1 t2

t3 t4 m[p5]=4 � F[t3]=0.535

• No�performance�monotonicities with respect toresources

m[p5]=3 � F[t3]=1.071

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

.Discretemodel

Counterintuitive�properties (II)

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

p3

2

5p1

p4

p5p

2

t1 t2

t3 t4 m[p5]=4 � F[t3]=0.535

m[p5]=5 � Deadlock

• No�performance�monotonicities with respect toresources

m[p5]=3 � F[t3]=1.071

Counterintuitive�properties (III):�steadyͲstate� No performance monotonicities with respect to firing speeds� Discontinuity / Bifurcation (loss of hyperbolicity)

Discrete model

Performance�monotonicity� MonoͲTͲsemiflow reducible (MͲTͲSͲred)�nets�are�conserͲvative with a�single�TͲsemiflow if Homothetic EqualConflicts [Pre�(t1)�=�q�Pre�(t2)�]�are�reduced.

� Let N be�a�MͲTͲSͲred�net.�Assume that steadyͲstates are�reached for given firing speeds and�initial markings.�If thestͲst configurations contains the support of�a�PͲsemiflow,� <�N,�O,�m1>�&�<N,�O,�m2>,�with m1�ч�m2�,��or� <�N,�O1,�m0>�&�<N,�O2,�m0>,�with O1�ч�O2

then f1 ч�f2��(i.e.,�system 2�never slower!)

Duality

� Corollary:�SL&SB�EQ�net�systems are�throughputmonotonous wrt themarking and�the firing speeds. m(p1)�+�m(p3)�=�10 m(p1)�+�m(p3)�=�10,

only if m(p1)�=�5!No�PͲcomponent“covered by theconfiguration”

Monotonousoperation mode

Non�monotonousoperation mode

M-T-S net 3 of 4 conf.

4.2.�Patterns�of�behaviours

(1) Equilibrium (point attractor)

p4

p3

p2

p1

p6

p5

p7

t1

t2

t3

t4

t5t6

32

4

2

2

2)�Basic�oscillatory (linear)

H : hiringF : firing

E : employeesP : productionS : sales

St : stock

Patterns�of�behaviours

3)�Oscillatory (nonͲlinear)�&�timing depending collapse:�Siphons Flows and�phaseͲportraits:�collapse if O2 >�10

O2 >�10

O2 <�1

Two discontinuities wrt O2 (�at�1.0�&�10.0) High�&�low frequencies and�collapse

+++ Small�variations in�parameters:¾ Orbits (like in�LotkaͲVolterra predatorͲprey model),¾ Limit cycles¾ …�

2

k

t1 t2

t3 D

~r ~f

r f

20

2020D��

4.3�Improving�the�timed fluidization�(II)With a�preliminary character:��use�all improveͲ

ments for untimed models that remove spurioussolutions!

In�particular,�those that :1)�empty a�marked trap2)�empty a�siphon (and�are�known to�be�spurious)3)�add complementary places�(qi)�that truncate the structural boundof�places

Also:�4)�add selfͲloop places�that truncate the enabling boundof�transitions

Reuse…:Interleavingqualitative & quantitative

sf = spurious-free(i.e., m[p2]=9

(k = 1)

Removing the spuriousdeadlockm=(0�1�10)T

O� O� O� SPN TCPN TCPN+q3 TCPN+f11 3,00 1,00 1,00 0,480 0,0062 0,500 0,00632 0,40 1,00 10,00 0,346 0,476 0,385 0,400

If O1 (vs O2)�is relatively very fast:1)�The basic TCPN�approach thespurious deadlock on the discretemodel.2)�The addition of�q3 (complementof�p3)�remove the spuriousdeadlock,�leading to�theremarkable improvement in�theTCPN+q3 approximation.

f1 may help to�improve,�but hereit is improbable…

Adding marking and�flow truncating placesBecause close to�a�spurious deadlock

4.3.1 Bound reaching problem & U-semantics

� What to�do�whenm0 is “not that big”,�and�exist transiͲtions with enabling degree =�1�(particularly if a�synchroͲnization appears as�weight on the arc from p�to�t)???

� One possibility:�to�mantain the transition t1 as�discrete(problem similar�to�responseͲtime in�RC�circuits).

� Alternative IDEA:�To�divide�the load�of�tokens in�such a�way that the firing of�the transition start with a�treshold value:�k–U¾ Question:�a�“good”�value for U ?�(heuristic?)

Infinite server�semantics Hybrid net UͲsemantics

The enabling degree of�t1

Infinite server�semantics

Hybrid approximation

With anheuristic

technique to�choose thevalue for U

4.3.2 Stochastic approximation

�If the discrete PN�model to�be�approximated is providedwith exponential timing (Markovian PN,�analyzable bymeans of�a�Markov Chain that is isomorphous to�thereachability graph),�then:

The fluid�PN�approximation lead�to�a�system ofstochastic piecewise affine differencial equations:�like the deterministic +�white gaussian noise

� m0 “very big”:�applicable a�kind of�functional law of�numbers or deterministic fluid�limit

� m0 “not that big”: applicable a�kind of�functionalcentral�limit theorem (in�the line�of�Donsker theorem)

Stochastic Continuous Petri�Nets�(STCPN):�adding Gaussian White�Noise to�the firing flow

Noise:independentnormallydistributed RVswith average

and�covariancematrix

STCPN: obtainedadding thenoise to�thefiring flows

STCPNs:�particularly interesting when “moving”�among two or more�configurations

Exp {max}�ш�max {Exp}�

Improvement when the discrete system in�the steadyͲstate switch among different regions

Transient for m0

m�=�m0 +�C�V is:1)�in�the naturals (computational cost)�&2)�with spurious (care with computations …)

Our focus today,�was on:1)�Fluidization:�relaxation into the nonͲnegative reals2)�How to�remove spurious solutions on:

The discrete and�corresponding fluidmodels(untimed &�timed)

Integratedview in�the PN�paradigm

Concluding remarks

GOAL:��To�overcome the state explosion problem…�in�certain cases�(the constraint:�the existence of�nonͲfluidizable PN�models)¾ The bigger the marking: the better the approximation &��������

the bigger the saving on computations.¾ Useful with untimed &�timed net�models¾ Reuse /�adaptation of�results from discrete PNs theory:�

Integration in�the PN�paradigm (economy;�coherence;�synergy)¾ Partial fluid�relaxations:�(a�kind of)�Hybrid Petri�Nets

Concluding remarks

The use�of�“good”�timed fluid�(technically hybrid)�models:� Parametric optimization (�where to�go?,�dim.�of�buffers?...�� Sensibility analysis

� Observers

� Controllers (min.�time;�…)� Diagnosis�under faulty behaviours� …

“Good”�is related to:�� the relative absence of�spurious solutions� the level of�fluidization:�Hybrid vs continuous nets?� In�timed models:�an appropriate server�semantics

(deterministic or stochastic)?

StructuralGenericPunctual

Fluidization of�Discrete Event Modelsor a�marriage between the discrete

and�the continuous

Manuel SilvaUniversidad�de�Zaragoza