Overview and Recent Trends of Petri Net Research

1

Overview and Recent Trends of Petri Net Research

Tadao Murata

University of Illinois at Chicago

[email protected]

Romanian Academy of Science

Bucharest, Romania

March 24, 2005

University of Illinois at Chicago C S Dept.

Overview of Petri Net Research Our Recenet Work: Fast Performance Evaluation

Using Fuzzy Logic and Petri Nets Fuzzy Logic and Soft Computing (SC) Examples of Possibility Distributions Probability vs. Possibility Simple Examples of Performance and Possibility

Evaluation by Using Fuzzy Logic and Petri Nets Degrees of Possibilities for Satisfying Given Specifications Concluding Remarks

Plan of Talk


What is a Petri Net?Petri Nets are a graphical and mathematical modeling tool, and good for describing and studying information processing systems that are characterized as being:

1. Concurrent2. Parallel3. Asynchronous4. Distributed5. Non-deterministic6. and/or Stochastic


"Three-In-One" Characteristics of Petri Nets

1) Graphical or Intuitive Model,

2) Mathematical or Formal Model, and

3) Can be used as Simulation Tool

An Analogy: A Vehicle that can travel

On Land like a car,

On Water like a boat, and

On Air like an airplane.


Application Areas

Successful application examples are often found in the areas of Communication protocols and networks, Performance evaluation of time-critical systems, Flexible manufacturing systems, Discrete event control systems, Business and other work-flow management systems, System and Computational Biology, etc.

For actual (non-toy) examples of applications, visit, e,g., http://www.daimi.au.dk/PetriNets/applications/, and http://www.daimi.au.dk/CPnets/intro/example_indu.html


Analysis Methods1) State Equation and Invariants

2) Reduction Techniques (Expansion for Synthesis)

3) Use of subgraphs: Siphons, Traps, Handles, Bridges, SM- & MG components, etc.

4) Reachability (Coverability) Graphs

The first three are applicable to subclasses or with certain conditions, and the forth has the state space explosion problem.

7

Our Recent Work:Fast Performance Evaluation Using

Fuzzy Logic and Petri Nets


Fuzzy Logic is a Main Component of Soft Computing (SC), which is:

A set of methodologies that function effectively in an environment of imprecision and/or uncertainty;

Aims at exploiting the tolerance for “fuzziness” (imprecision, uncertainty, and partial truth) to achieve tractability (or scalability), and low-solution cost.

Methodologies in SC include Fuzzy Logic, Computing with Words, Neurocomputing, Probabilistic Reasoning, etc.

*[Zad94] Lotfi A. Zadeh, "Fuzzy Logic, Neural Networks, and Soft Computing," Comm. of ACM, vol.37, pp.77-84, 1994


Fuzzy Set is a Generalization of Crisp Set

Any crisp theory can be generalized to the concept

of a fuzzy set (from a set):

Membership grade: =0 or 1 v.s, 01

Crisp, Non-Fuzzy Fuzzy

Linear Nonlinear

Deterministic Non-Deterministic


Example of a crisp setThe set of people who are 20 years old or younger

The set of “younger people”

=1=1 inside

=0 outside the set

=1 for 15 years old=0.5 for 30 years old=0.1 for 40 years old =1

=0.1 =0.5


Fuzzy Timing is a Generalization of Deterministic TimingWithout loss of generality, we can use trapezoidal fuzzy time functions or possibility distributions*, using 4 parameters, () = (1, 2, 3, 4).

Note: (probabilities) = 1, but (possibilities) ≠ 1 Special Cases:

1. Deterministic Timing if 1= 2= 3= 4 (=)

2. Deterministic Time Interval if 1= 2 and 3= 4

3. Triangular Distributions (Fuzzy Numbers) if 2= 3

* This is not restriction. Any possibility distributions can be used in this method.

0

1

1 0

1

2 3 4 1 2 = 3 4

(a) (b)


Typical Building Blocks of Possibility Distributions

Normal Possibility Distribution by Triangular

(Trapezoidal) or Exponential Functions

0

1

20 pointsC=60 100

(x)=e-1/2((x-c)/)2

Special Cases Special Cases


Example 1: Possibility distribution of a typical exam in my classIt is a normal probability distribution which can be approximated by the triangular possibility distribution: (1, 2, 3, 4) = (20, 60, 60, 100) points.

0

1

20 points60 100


Example 2: Possibility distribution of driving time from my home to work (20 miles). Note that arbitrary possibility distributions can be decomposed into a set of trapezoidal distributions.

0

1

20 minutes40 12030

0

1

20 minutes40 12030

1 2 3 4 5 6

(1, 2, 3, 4) = (20, 30, 40, 120) minutes

Approximation using 6 trapezoidals


Example 3: Possibility distribution of time to download a “big” file (of 1Mb to 1Gb)

(1, 2, 3, 4) = (1, 5, 10, 100) sec.

0

1

1 sec10 100

5


Example 4: Possibility distribution of the total # of hours spent by a student on HWs

HW.course1 HW.course2 HW.course3 = (1, 2, 3, 4) (2, 3, 3, 4) (2, 2, 3, 3) = (5, 7, 9, 11) hours.

0

1

1 2 3 4

0

1

5 7 9 11 hours

0

1

2 3 4 0

1

2 3

=


Fuzzy vs. VagueA proposition is fuzzy if it contains terms that are labels of fuzzy sets, such as possibility distributions: e.g.,

"I will be back in a few minutes.“The possibility distribution of "a few minutes" is shown below.

But “I will be back sometime” is vague, unless the possibility distribution of ”sometime” is given.

0

1

2 minutes101 3


Negation of a Proposition and its Possibility DistributionThe possibility distribution of "young":

The possibility distribution of "not young":

0

1

yr. old4020

0

1

yr. old4020


Computing with WordsIn a broader sense, computing with words is a computational

theory of perceptions.

It is a methodology in which the objects of computation are words such as:

a few days, young, rich, not very likely, …, and propositions in natural languages such as:

It takes a few days,I'll do it in the near future, The stock price will go up eventually, etc.

In this talk we restrict the perception related to time or delay (performance).


Computing Over-all Possibilities: ExampleWe have a project which consists of three steps to do in sequence. Each step takes a few days to complete. What is the possibility to finish this project within the deadline of 9 days?Solution:

Suppose that the possibility distribution of a few days is given by (1,2,3,5) days. Then 3 steps take 3(1, 2, 3, 5) = (3, 6, 9, 15) days. Thus the possibility distribution to finish this project is:

0

1

days63 9 15


(Continued from the preceding page)

The possibility to finish this project in 9 days is computed by the radio of the areas:

= Area A (the part of the trapezoidal area before 9 days) / Area B (the entire trapezoidal area) = (1.5 + 3)/(1.5 + 3 + 3) = 4.5/7.5 = 0.6 or 60 %.

Step 1 Step 2 Step 3

Deterministic: 3 days + 3 days + 3 days = 9 days


Computing Possibilities of Satisfying Maximum Tolerable Skew in Multimedia Synchronization

Given the following Dynamic Parameters for a

Multimedia (Audio and Video) Application: Throughput: 10 images per sec; Max. Tolerable jitter on audio or video: 10ms; Max. tolerable skew between audio and video 50ms.


(Continued from the preceding page)Normal playout duration per image=100ms;

Possibility distribution = (90, 100, 100, 110)ms.

Synchronizing every 4 audio-video unit gives the

playout duration for 4 units = 4X(90,100,100,110)=(360, 400, 400, 440)

0

1

90 ms100 110 0

1

360 ms400 440


(Continued from the preceding page)The max possible skew=440-360=80 ms or possibility

distribution is (-80, 0, 0, 80) ms

The degree of possibility that the max skew requirement 50ms is satisfied.

The shaded Area between and Area of the whole triangular = 0.859375

0

1

-80 0 8050-50


(Continued from the preceding page)

Thus, synchronizing every 4 units yields the 85.9% possibility that the skew between video and audio will not exceed 50 ms. Thus the requirement is satisfied 85.9% of time.

Synchronizing every 2 units yields the possibility distribution of the skew = 2x(90,100,100,110) =(180,200,200,220)ms: Max. possible skew is 220-180 = 40ms < 50ms limit. Thus the requirement is satisfied 100% this time.


Probability vs. PossibilityProbabilities are normalized:

(probabilities) = 1, but (possibilities) 1.

Probability theory offers no techniques for dealing with fuzzy quantifiers like few, many, most, several, ….

Probability theory does not provide a system for computing fuzzy probabilities expressed as likely, unlikely, not very likely, etc.


Probability theory is much less effective than fuzzy logic in those fields where:

1) The knowledge of probability is imprecise and/or

incomplete;

2) The systems are not mechanistic (have no equations governing system behaviors); and

3) Human reasoning, perceptions and emotion do play an important role.

This is the case, in varying degree, in expert systems,

economics, speech recognition, analysis of evidence,

etc.


Petri Net Model of a Job-Shop

A job shop has a machine (Pfree) which can process two types of job a or b.

a

b

Pfree

e1a

e1b

e2a

e2b

P1a

P1b Pout-b

Pout-a


Meanings of Places and TransitionsPlace a gets a token when the request for job a arrives.

Place b gets a token when the request for job b arrives.

Place Pfree gets a token when the machine is available.

Transition e1a or e1b represents job a or b gets the machine;

Transition e2a or e2b represents job a or b performs the job and release the machine, respectively; and

Place Pout-a (or Pout-b) gets a token when Job a (or b) completed its job, respectively.


Fuzzy-Timing Petri Net (FTPN) Model of a simple resource sharing system

Pfree

t1a

t1b

t2a

t2b

P1a

P1b Pout-b

Pout-a

Pa

Pb

d1a() d2a()

d1b()

d2a()

d2b()

d2b()

(0,0,0,0)

(0,0,0,0)

(4,5,7,9)

(4,5,7,9)

(4,5,7,9)

(4,5,7,9)


Mutual Exclusion ModelThis Petri net model also represents a mutual

exclusion in which a common resource Pfree is shared by two processes a and b, where:

Pa (or Pb): process a (or b) is waiting;

P1a (or P1b): process a (or b) is using the resource

Pout-a (or Pout-b): Process a (or b) finishes using the resource Pr;

e1a (or e1b): process a (or b) gets the resource;

e2a (or e2b): process a (or b) releases the resource.


Non-Fuzzy CaseSuppose job a arrives at 3 sec and job b arrives at 5

sec. The machine is available at 0 sec; it takes zero time to get the machine (d1 =

0), takes 2 sec to perform each job (d2 = 2); and takes another 2 sec to clean and return the machine (d3 = 4).

Using the First-Come-First-Serve policy, job a will be completed at 3 + 2 = 5 sec, and job b will be completed at

max{(3+4), 5}+2 = 7+2 = 9 sec.


Fuzzy-Timing CaseSuppose that the request of jobs a and b arrive at 32 sec and 52 sec, respectively, i.e. their possibility distributions are given below.

0

1

5

sec

1 3 7

Job a Job b


Case 1: Job a’s request arrives before Job b’s.Suppose d1 = 0 sec and d2 = d3 = (4,5,7,9).

Then job a is completed at: (1,3,3,5) (4,5,7,9) = (5,8,10,14)and job b is completed at: (5,8,10,14) (4,5,7,9) = (9,13,17,23)

0

1

5

sec

14108

1

0

1

23

sec

17139

1

Job a

Job b


Case 2: Job b’s request arrives before Job a’s.But there are smaller possibilities that job b is completed before job a: that possibility distribution is given by the intersection of the two possibility distributions of job a and job b arrivals:

min{(1,3,3,5), (3,4,4,7)} = 0.5(3,4,4,5).

Therefore, job b could be completed at 0.5(3,4,4,5) (4,5,7,9) = 0.5(7,9,11,14)

0

1

7

sec

9 1411

Job b20.5


(Continued) Case 2: Job b’s request arrives before Job a’s.and job a be could be completed at

0.5(7,9,11,14) (4,5,7,9) = 0.5(11,14,18,23):

Since there are two possible orders a-b and b-a in which jobs are completed, we combine the two possibility distributions to get the overall possibility distributions of completing jobs by taking the union (fuzzy max operation) in the next two slides.

0

1

23

sec

181411

Job a20.5


Union of Job a1 and Job a2

0

1

5

sec

14108

Job a 1

0

1

23

sec

181411

Job a 20.5

0

1

5

sec

14108

0.5

11 2318


Union of Job b1 and Job b2

0

1

23

sec

17139

Job b 1

0

1

7

sec

9 1411

Job b 20.5

0

1

7

sec

9 1411

0.5

2317139


(Continued)

Defuzzification to get “Average”“Average” completion times for Job a and Job b can be computed by one of “Defuzzification” methods, e.g. by the Moment Method:

sec5.12)(

)(23

5

23

5*

d

da

sec0.15)(

)(23

7

23

7*

d

db

For Job a

For Job b


Possibilistic Performance Analysis Examples1) If the deadline to finish both jobs a and b is 24 sec, then the possibility to finish both jobs before the deadline is one (100%).

2) The possibility to finish job a before the 20 sec deadline ≈ (area B) / (area A) ≈ 92%, where (area B) = the shaded area, and (area A) = the total area under the curve.

0

1

5

sec

14108

0.5

11 231820

area B


Possibilistic Performance Analysis Examples (Continued)3) The possibility to finish job b before the 15 sec deadline ≈ (area B’) / (area A’) ≈ 50%, where (area B’) = the shaded area, and (area A’) = the total area under the curve.

0

1

7

sec

9 1411

0.5

231713915

area B’


Computation Steps in FTPN1) Given or compute Fuzzy Time Stamps, i().

2) Compute Fuzzy Enabling Times byet()=latest {i() | i=1,2, …}.

3) Compute Fuzzy Occurrence Times byot()= min {et(), earliest {ei() | i=1,2, …}}.

4) Update Fuzzy Time Stamps:tp() = ot() dtp()

= sup min{ot(1), dtp(2)}. =1+2

5) Repeat the above Steps 1 to 4.


The latest and earliest Operatorslatest {i() | i=1,2, …, n}

≈ extended max* {i() | i=1,2, …, n}≈ latest {hi(i1, i2, i3, i4), i=1,2, …, n}≈ min{hi} (max{i1}, max{i2}, max{i3}, max{i4})

i=1,2, …, n

earliest {ei() | i=1,2, …, n}≈ extended min* {ei() | i=1,2, …, n}≈ earliest {hi(ei1, ei2, ei3, ei4), i=1,2, …, n}≈ max{hi} (min{ei1}, min{ei2}, min{ei3}, min{ei4})

i=1,2, …, n

* D. Dubois and H. Prade, “Possibility Theory: an approach to computerized processing of uncertainty”, Plenum Press, 1988


Illustration of the latest operator The red line is latest{1(), 2()}= latest{0.5(0,1,5,6), (1,3,3,4)} = 0.5(1,3,5,6)

21 3 4 5 6

0

1

0.5

2()1()


Illustration of the latest operator (continued)

The red line is latest{1(t), 2(t), 3(t)}= latest{0.5(0,1,5,6), (1,2,3,4), (6,7,7,8)} = 0.5(6, 7, 7, 8)

21 3 4 5 6 0

1

0.5

2(t)1(t)

7 8

3(t)


Illustration of the earliest operator

o()=earliest{e1(), e2(), e3()}

= earliest{0.5(0,1,6,7), (1,3,3,5), (6,7,7,8)}

= (0,1,3,5)

21 3 4 5 6 0

1

0.5

e2()e1()

7 8

e3()


Finding occurrence times by the min (intersection) operator

o1()=min{e1(), o()}

= min{0.5(0,1,5,6), (0,1,3,5)}

= 0.5(0,1,4,5)

21 3 4 5 6 0

1

0.5

7 8

O1()


Finding occurrence times by the min operator (continued)

o2()=min{e2(), o()}

=min{(1,3,3,5), (0,1,3,5)} = (1,3,3,5)

21 3 60 4 5 7 8

1

0.5

O2()

o3(t)=min{e3(t), o(t)} = min{(6,7,7,8), (0,1,3,5)}=


Concluding Remarks (1) The computations involved in the above FTPN method

are mostly additions and comparisons of real numbers and do not require solving any equations. Therefore, they can be done very fast. Thus this method is suitable for applications to time-critical systems.

The FTPN method is considered to be complementary to existing probabilistic or stochastic approaches.

The FTPN method is more general but approximate and subjective in many cases.


Concluding Remarks (2) FTPN and other fuzzy approaches are suitable for :

Complex systems for which complicated mathematical systems must be solved;

Large-scale systems which have intractable computational complexity/cost; and

Applications that involve human descriptive or intuitive thinking. Fuzzy logic has no memory and lacks learning

capabilities. Thus it is good to combine fuzzy logic with neural networks and to work with so-called “neurofuzzy systems”.


Some of Our Application Examples (1)

T. Murata, "Temporal uncertainty and fuzzy-timing high-level Petri nets," in Application and Theory of Petri Nets 1996, Lecture Notes in Computer Science, pp. 11-28, Vol. 1091, Springer, New York, June 1996.

T. Murata, T. Suzuki and S. Shatz, “Fuzzy-timing high-level Petri nets (FTHNs) for

time-critical systems,” in J. Cardoso and H. Camargo (editors) “Fuzziness in Petri Nets” Vol. 22 in the series "Studies in Fuzziness and Soft Computing" by Springer Verlag, New York, pp. 88-114, 1999.

T. Murata and Chun-Pin Chen, “Fuzzy-timing Petri-net modeling and analysis of

video-on-demand system response times,” Procs. of the 5th World Conference on Integrated Design & Process Technology, pp. 298-306, June 4-8, 2000.

K. Watanuki and T. Murata, “Evaluation method for assembly / disassembly by Petri

nets’” Procs. of the International Conf. on Engineering Design (ICED’99), pp.519-522, Vol.1, Munich, August 24-26, 1999.


Some of Our Application Examples (2)K. Watanuki and T. Murata, "Fuzzy-timing Petri net model of temperature control

for car air conditioning system," Procs. of 1999 IEEE International Conference on Systems, Man, and Cybernetics, Vol. IV, Tokyo, Japan, pp.618-622, October 12-15, 1999.

Y. Zhou and T. Murata, “Fuzzy-timing Petri net model for distributed multimedia synchronization,” Procs. of the 1998 IEEE International Conference on Systems, Man, and Cybernetics, Lolla, California, pp. 244 - 249, October 11-14, 1998.

Y. Zhou and T. Murata, “Petri net model with fuzzy-timing and fuzzy-metric

temporal logic,” the special issue on fuzzy Petri nets: concepts and intelligent system modeling, International Journal of Intelligent Systems, vol. 14, no. 8, pp. 719-746, August 1999.


Some of Our Application Examples (3)Y. Zhou, T. Murata, and T. DeFanti, "Modeling and performance analysis using

extended fuzzy-timing Petri nets for networked virtual environments," IEEE Transactions on Systems, Man, and Cybernetics, Part B, Vol. 30, No.5, pp.737-756, October 2000.

Y. Zhou and T. Murata, "Modeling and analysis of distributed multimedia synchronization by extended fuzzy-timing Petri nets," Journal of Integrated Design and Process Science, Journal of Integrated Design and Process Science, Vol. 4, No. 4, pp. 23-38, December 2001.

T. Murata, J. Yim, H. Yin and O. Wolfson, "Fuzzy-Timing Petri-Net Model for Updating Moving Objects Database," Proceedings of the 2003 VIP Scientific Forum of International Conference on IPSI (Internet, Processing, Systems, and Interdisciplinaries), Sveti Stefan, Montenegro, Yugoslavia, pp. 1-7, October 4-11, 2003.


Degree of SatisfactionExample: Degree of satisfaction for completing a

job by the deadline of 9 days

9 15

1

days0

µ


Open Question: Find a Method to Maximize “Total Degree of Satisfaction”

Given n degrees of satisfaction for n parameters of a system, µ1, µ2, …, µn;

Find a method to maximize a “total satisfaction

degree,” in some sense, e.g.,

Max{f(µ1) + f(µ2) + … + f(µn)}

1

µ1 µ2

1

µm

1

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Overview and Recent Trends of Petri Net Research