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1 Overview and Recent Trends of Petri Net Research Tadao Murata University of Illinois at Chicago murata@uic.edu Romanian Academy of Science Bucharest, Romania March 24, 2005

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University of Illinois at ChicagoC 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

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University of Illinois at ChicagoC S Dept. 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. Concurrent 2. Parallel 3. Asynchronous 4. Distributed 5. Non-deterministic 6. and/or Stochastic

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University of Illinois at ChicagoC S Dept. "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.

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University of Illinois at ChicagoC S Dept. 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

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University of Illinois at ChicagoC S Dept. 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.

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7 Our Recent Work: Fast Performance Evaluation Using Fuzzy Logic and Petri Nets

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University of Illinois at ChicagoC S Dept. 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

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University of Illinois at ChicagoC S Dept. 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

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University of Illinois at ChicagoC S Dept. 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 =1 =0.1 =0.5

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University of Illinois at ChicagoC S Dept. 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, ( ) = ( 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)

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University of Illinois at ChicagoC S Dept. Typical Building Blocks of Possibility Distributions Normal Possibility Distribution by Triangular (Trapezoidal) or Exponential Functions 0 1 20 points C=60 100 (x)=e -1/2((x-c)/ )2 Special Cases

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University of Illinois at ChicagoC S Dept. 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. 0 1 20points60100

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University of Illinois at ChicagoC S Dept. 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 20minutes4012030 0 1 20minutes4012030 123456 ( 1, 2, 3, 4 ) = (20, 30, 40, 120) minutes Approximation using 6 trapezoidals

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University of Illinois at ChicagoC S Dept. 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 1sec10100 5

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University of Illinois at ChicagoC S Dept. 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 1234 0 1 57911hours 0 1 2340 1 23 =

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University of Illinois at ChicagoC S Dept. 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. 0 1 2minutes1013

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University of Illinois at ChicagoC S Dept. Negation of a Proposition and its Possibility Distribution The possibility distribution of "young": The possibility distribution of "not young": 0 1 yr. old40200 1 yr. old4020

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University of Illinois at ChicagoC S Dept. 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).

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University of Illinois at ChicagoC S Dept. 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: 0 1 days63915

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University of Illinois at ChicagoC S Dept. (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 1Step 2Step 3 Deterministic: 3 days + 3 days + 3 days = 9 days

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University of Illinois at ChicagoC S Dept. 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.

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University of Illinois at ChicagoC S Dept. (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 = 4 X (90,100,100,110)=(360, 400, 400, 440) 0 1 90ms1001100 1 360ms400440

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University of Illinois at ChicagoC S Dept. (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 -80 080 50 -50

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University of Illinois at ChicagoC S Dept. (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.

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University of Illinois at ChicagoC S Dept. 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.

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University of Illinois at ChicagoC S Dept. 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.

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University of Illinois at ChicagoC S Dept. Petri Net Model of a Job-Shop A job shop has a machine (P free ) which can process two types of job a or b. a b P free e 1a e 1b e 2a e 2b P 1a P 1b P out-b P out-a

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University of Illinois at ChicagoC S Dept. 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 P free gets a token when the machine is available. Transition e 1a or e 1b represents job a or b gets the machine; Transition e 2a or e 2b represents job a or b performs the job and release the machine, respectively; and Place P out-a (or P out-b ) gets a token when Job a (or b) completed its job, respectively.

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University of Illinois at ChicagoC S Dept. Fuzzy-Timing Petri Net (FTPN) Model of a simple resource sharing system P free t 1a t 1b t 2a t 2b P 1a P 1b P out-b P out-a P a P b d 1a ( )d 2a ( ) d 1b ( ) d 2a ( ) d 2b ( ) d ( ) (0,0,0,0) (4,5,7,9)

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University of Illinois at ChicagoC S Dept. Mutual Exclusion Model This Petri net model also represents a mutual exclusion in which a common resource P free is shared by two processes a and b, where: P a (or P b ): process a (or b) is waiting; P 1a (or P 1b ): process a (or b) is using the resource P out-a (or P out-b ): Process a (or b) finishes using the resource P r ; e 1a (or e 1b ): process a (or b) gets the resource; e 2a (or e 2b ): process a (or b) releases the resource.

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University of Illinois at ChicagoC S Dept. 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 (d 1 = 0), takes 2 sec to perform each job (d 2 = 2); and takes another 2 sec to clean and return the machine (d 3 = 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.

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University of Illinois at ChicagoC S Dept. 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. 0 1 5 sec 137 Job aJob b

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University of Illinois at ChicagoC S Dept. Case 1: Job a’s request arrives before Job b’s. Suppose d 1 = 0 sec and d 2 = d 3 = (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

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University of Illinois at ChicagoC S Dept. 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 91411 Job b 2 0.5

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University of Illinois at ChicagoC S Dept. (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 a 2 0.5

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University of Illinois at ChicagoC S Dept. Union of Job a 1 and Job a 2 0 1 5 sec 14108 Job a 1 0 1 23 sec 181411 Job a 2 0.5 0 1 5 sec 14108 0.5 112318

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University of Illinois at ChicagoC S Dept. Union of Job b 1 and Job b 2 0 1 23 sec 17139 Job b 1 0 1 7 sec 91411 Job b 2 0.5 0 1 7 sec 91411 0.5 2317139

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University of Illinois at ChicagoC S Dept. (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 5 * d d a sec0.15 )( )( 23 7 7 * d d b For Job a b

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University of Illinois at ChicagoC S Dept. 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. 0 1 5 sec 14108 0.5 112318 20 area B

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University of Illinois at ChicagoC S Dept. 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 91411 0.5 2317139 15 area B’

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University of Illinois at ChicagoC S Dept. Computation Steps in FTPN 1) Given or compute Fuzzy Time Stamps, i ( ). 2) Compute Fuzzy Enabling Times by e t ( )=latest { i ( ) | i=1,2, …}. 3) Compute Fuzzy Occurrence Times by o t ( )= min {e t ( ), earliest {e i ( ) | i=1,2, …}}. 4) Update Fuzzy Time Stamps: tp ( )= o t ( ) d tp ( ) = sup min{o t ( 1 ), d tp ( 2 )}. = 1 + 2 5) Repeat the above Steps 1 to 4.

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University of Illinois at ChicagoC S Dept. The latest and earliest Operators latest { i ( ) | i=1,2, …, n} ≈ extended max* { i ( ) | i=1,2, …, n} ≈ latest {h i ( i1, i2, i3, i4 ), i=1,2, …, n} ≈ min{h i } (max{ i1 }, max{ i2 }, max{ i3 }, max{ i4 }) i=1,2, …, n earliest {e i ( ) | i=1,2, …, n} ≈ extended min* {e i ( ) | i=1,2, …, n} ≈ earliest {h i (e i1, e i2, e i3, e i4 ), i=1,2, …, n} ≈ max{h i } (min{e i1 }, min{e i2 }, min{e i3 }, min{e i4 }) i=1,2, …, n * D. Dubois and H. Prade, “Possibility Theory: an approach to computerized processing of uncertainty”, Plenum Press, 1988

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University of Illinois at ChicagoC S Dept. 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) 213 4 5 6 0 1 0.5 2()2() 1()1()

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University of Illinois at ChicagoC S Dept. 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) 213 4 5 6 0 1 0.5 2 (t) 1 (t) 7 8 3 (t)

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University of Illinois at ChicagoC S Dept. Illustration of the earliest operator o( )=earliest{e 1 ( ), e 2 ( ), e 3 ( )} = earliest{0.5(0,1,6,7), (1,3,3,5), (6,7,7,8)} = (0,1,3,5) 213 4 5 6 0 1 0.5 e2()e2() e1()e1() 7 8 e3()e3()

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University of Illinois at ChicagoC S Dept. Finding occurrence times by the min (intersection) operator o 1 ( )=min{e 1 ( ), o( )} = min{0.5(0,1,5,6), (0,1,3,5)} = 0.5(0,1,4,5) 213 4 5 6 0 1 0.5 7 8 O1()O1()

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University of Illinois at ChicagoC S Dept. Finding occurrence times by the min operator (continued) o 2 ( )=min{e 2 ( ), o( )} =min{(1,3,3,5), (0,1,3,5)} = (1,3,3,5) 21360 4 5 7 8 1 0.5 O2()O2() o 3 (t)=min{e 3 (t), o(t)} = min{(6,7,7,8), (0,1,3,5)}=

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University of Illinois at ChicagoC S Dept. 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.

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University of Illinois at ChicagoC S Dept. 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”.

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University of Illinois at ChicagoC S Dept. 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.

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University of Illinois at ChicagoC S Dept. 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.

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University of Illinois at ChicagoC S Dept. 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.

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University of Illinois at ChicagoC S Dept. Degree of Satisfaction Example: Degree of satisfaction for completing a job by the deadline of 9 days 9 15 1 days 0 µ

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University of Illinois at ChicagoC S Dept. Open Question: Find a Method to Maximize “Total Degree of Satisfaction” Given n degrees of satisfaction for n parameters of a system, µ 1, µ 2, …, µ n ; Find a method to maximize a “total satisfaction degree,” in some sense, e.g., Max{f(µ 1 ) + f(µ 2 ) + … + f(µ n )} 1 µ1µ1 µ2µ2 1 µmµm 1

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