Overview and Recent Trends of Petri Net Research

1
Overview and Recent Trends of Petri Net Research

• Tadao Murata
• University of Illinois at Chicago
• murata_at_uic.edu
• Romanian Academy of Science
• Bucharest, Romania
• March 24, 2005

2
Plan of Talk

• 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

3
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
• Concurrent
• Parallel
• Asynchronous
• Distributed
• Non-deterministic
• and/or Stochastic

4
"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.

5
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.h
  tml

6
Analysis Methods

• 1) 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 WorkFast Performance Evaluation
Using Fuzzy Logic and Petri Nets
8
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

9
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, 0???1
• Crisp, Non-Fuzzy ? Fuzzy
• Linear ? Nonlinear
• Deterministic ? Non-Deterministic

10
Example of a crisp set

• The 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
11
Fuzzy Timing is a Generalization of Deterministic
Timing

• Without loss of generality, we can use
  trapezoidal fuzzy time functions or possibility
  distributions, using 4 parameters, p(t) (p1,
  p2, p3, p4).
• Note S (probabilities) 1, but S
  (possibilities) ? 1
• Special Cases
• 1. Deterministic Timing if p1 p2 p3 p4 (p)
• 2. Deterministic Time Interval if p1 p2 and
  p3 p4
• 3. Triangular Distributions (Fuzzy Numbers) if
  p2 p3
• This is not restriction. Any possibility
  distributions can be used in this method.

1
1
p
t
p
p
p
p
p
p
p
t
0
0

1
2
3
4
1
2
3
4
(a)
(b)
12
Typical Building Blocks of Possibility
Distributions

• Normal Possibility Distribution by Triangular
• (Trapezoidal) or Exponential Functions

Special Cases Special Cases
?(x)e-1/2((x-c)/?)2
1
20
0
points
100
C60
13
Example 1 Possibility distribution of a typical
exam in my class

• It is a normal probability distribution which can
  be approximated by the triangular possibility
  distribution (?1, ?2, ?3, ?4) (20, 60, 60,
  100) points.

14
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.
15
Example 3 Possibility distribution of time to
download a big file (of 1Mb to 1Gb)

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

16
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.

1
1
1
0
1
2
3
4
0
2
3
4
0
2
3
1

0
5
7
9
11
hours
17
Fuzzy vs. Vague

• A 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.

18
Negation of a Proposition and its Possibility
Distribution

• The possibility distribution of "young"
• The possibility distribution of "not young"

19
Computing with Words

• In 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).

20
Computing Over-all Possibilities Example

• We 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

1
0
days
6
3
9
15
21
(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
22
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.

23
(Continued from the preceding page)

• Normal playout duration per image100ms
• 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)

24
(Continued from the preceding page)

• The max possible skew440-36080 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 t50 and -50 / Area of
  the whole triangular 0.859375

1
50
0
-80
0
80
-50
t
25
(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 lt 50ms limit.
  Thus the requirement is satisfied 100 this time.

26
Probability vs. Possibility

• Probabilities 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.

27
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.

28
Petri Net Model of a Job-Shop

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

e
e
P
P
1a
2a
1a
out-a
a
P
free
b
e
e
P
P
1b
2b
1b
out-b
29
Meanings of Places and Transitions

• Place 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.

30
Fuzzy-Timing Petri Net (FTPN) Model of a simple
resource sharing system
t
t
P
P
1a
2a
1a
out-a
(0,0,0,0)
(4,5,7,9)
P
a
t
d
(
t
)
d
(
)
1a
2a
t
d
(
)
(4,5,7,9)
2a
P
free
t
d
(
)
(4,5,7,9)
2b
d
(
t
)
d
(
t
)
1b
2b
P
b
(0,0,0,0)
(4,5,7,9)
t
t
P
P
1b
2b
1b
out-b
31
Mutual Exclusion Model

• This 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.

32
Non-Fuzzy Case

• Suppose 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(34), 52 72 9
  sec.

33
Fuzzy-Timing Case

• Suppose that the request of jobs a and b arrive
  at 3?2 sec and 5?2 sec, respectively, i.e. their
  possibility distributions are given below.

Job a
Job b
1
sec
0
5
1
3
7
34
Case 1 Job as request arrives before Job bs.

• 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)

Job a
1
1
sec
0
5
14
10
8
Job b
1
1
sec
0
23
17
13
9
35
Case 2 Job bs request arrives before Job as.

• 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)

36
(Continued) Case 2 Job bs request arrives
before Job as.

• 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.

37
Union of Job a1 and Job a2
38
Union of Job b1 and Job b2
39
(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

40
Possibilistic Performance Analysis Examples

• 1) 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.

41
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.

42
Computation Steps in FTPN

• 1) Given or compute Fuzzy Time Stamps, pi(t).
• 2) Compute Fuzzy Enabling Times by et(t)latest
  pi(t) i1,2, .
• 3) Compute Fuzzy Occurrence Times by ot(t) min
  et(t), earliest ei(t) i1,2, .
• 4) Update Fuzzy Time Stamps ptp(t) ot(t) ?
  dtp(t) sup minot(t1), dtp(t2).
  tt1t2
• 5) Repeat the above Steps 1 to 4.

43
The latest and earliest Operators

• latest pi(t) i1,2, , n extended max
  pi(t) i1,2, , n latest hi(pi1, pi2,
  pi3, pi4), i1,2, , n minhi (maxpi1,
  maxpi2, maxpi3, maxpi4) i1,2, , n
• earliest ei(t) i1,2, , n extended min
  ei(t) i1,2, , n earliest hi(ei1, ei2,
  ei3, ei4), i1,2, , n maxhi (minei1,
  minei2, minei3, minei4) i1,2, , n
• D. Dubois and H. Prade, Possibility Theory an
  approach to computerized processing of
  uncertainty, Plenum Press, 1988

44
Illustration of the latest operator

• The red line is latest?1(t), ?2(t)
• latest0.5(0,1,5,6), (1,3,3,4) 0.5(1,3,5,6)

?2(t)
?1(t)
1
0.5
2
1
3
6
0
4
5
t
45
Illustration of the latest operator (continued)

• The red line is latest?1(t), ?2(t), ?3(t)
• latest0.5(0,1,5,6), (1,2,3,4), (6,7,7,8)
• 0.5(6, 7, 7, 8)

?2(t)
?3(t)
?1(t)
1
0.5
2
1
3
6
0
5
8
4
7
t
46
Illustration of the earliest operator

• o(t)earlieste1(t), e2(t), e3(t)
• earliest0.5(0,1,6,7), (1,3,3,5), (6,7,7,8)
• (0,1,3,5)

e2(t)
e3(t)
e1(t)
1
0.5
2
1
3
6
0
5
8
4
7
t
47
Finding occurrence times by the min
(intersection) operator

• o1(t)mine1(t), o(t)
• min0.5(0,1,5,6), (0,1,3,5)
• 0.5(0,1,4,5)

48
Finding occurrence times by the min operator
(continued)

• o2(t)mine2(t), o(t)
• min(1,3,3,5), (0,1,3,5) (1,3,3,5)

O2(t)
1
0.5
t
8
2
1
3
6
0
4
5
7
o3(t)mine3(t), o(t) min(6,7,7,8),
(0,1,3,5) ?
49
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.

50
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.

51
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
  (ICED99), pp.519-522, Vol.1, Munich, August
  24-26, 1999.

52
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.
•  
•  

53
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.

54
Degree of Satisfaction

• Example Degree of satisfaction for completing a
  job by the deadline of 9 days

µ
1
15
0
9
days
55
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.,
• Maxf(µ1) f(µ2) f(µn)

µm
µ1
µ2
1
1
1