Computing Minimal Siphons in Petri Net Models of Resource Allocation Systems: An Evolutionary Approach

Presented at PNSE 2014

Transcript

1. Computing minimal siphons in Petri net models
   of Resource Allocation Systems: an evolutionary
   approach
   Fernando Tricas1 Jos´
   e Manuel Colom1 Juan Juli´
   an Merelo2
   Depto de Inform´
   atica e Ingenier´
   ıa de Sistemas
   Universidad de Zaragoza
   {ftricas,jm}@unizar.es
   Depto. ATC/CITIC
   Universidad de Granada
   jmerelo@geneura.ugr.es
   19 de junio de 2014
   PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
   e Manuel Colom, Juan Juli´
   an Merelo 1
   

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2. Outline
   Petri Nets, Siphons and Liveness
   The genetic algorithm
   The proposal
   Some experiments
   Conclusions
   PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
   e Manuel Colom, Juan Juli´
   an Merelo 2
   

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3. Petri Nets, Siphons and Liveness
   PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
   e Manuel Colom, Juan Juli´
   an Merelo 3
   

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4. The Genetic Algorithm
   1. (Start) Generate random population
   2. (Fitness) Evaluate the fitness of each chromosome in the
   population
   3. (New population) Create a new population
   3.1 (Selection)
   3.2 (Crossover)
   3.3 (Mutation)
   3.4 (Accepting)
   4. (Replace)
   5. (Test) If the end condition is satisfied, stop
   6. (Loop) Go to step 2
   PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
   e Manuel Colom, Juan Juli´
   an Merelo 4
   

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5. The Proposal
   min p∈P
   vp
   ∀p ∈ P, ∀t ∈ •p, vp ≤ q∈ •t
   vq, vp ∈ {0, 1}
   p∈P\P0
   vp < |P \ P0|
   ∀Y ∈ PS, p∈Y
   vp < Y
   PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
   e Manuel Colom, Juan Juli´
   an Merelo 5
   

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6. The Proposal
   min p∈P
   vp
   ∀p ∈ P, ∀t ∈ •p, vp ≤ q∈ •t
   vq, vp ∈ {0, 1}⇐=
   p∈P\P0
   vp < |P \ P0|
   ∀Y ∈ PS, p∈Y
   vp < Y
   Siphon condition:
   ∀p ∈ P, ∀t ∈ •p, vp ≤
   q∈ •t
   vq, with vq, vp ∈ {0, 1}
   PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
   e Manuel Colom, Juan Juli´
   an Merelo 6
   

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7. The Proposal
   min p∈P
   vp
   ∀p ∈ P, ∀t ∈ •p, vp ≤ q∈ •t
   vq, vp ∈ {0, 1}
   p∈P\P0
   vp < |P \ P0|⇐=
   ∀Y ∈ PS, p∈Y
   vp < Y
   Not all the places:
   p∈P\P0
   vp < |P \ P0|
   PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
   e Manuel Colom, Juan Juli´
   an Merelo 7
   

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8. The Proposal
   min p∈P
   vp
   ∀p ∈ P, ∀t ∈ •p, vp ≤ q∈ •t
   vq, vp ∈ {0, 1}
   p∈P\P0
   vp < |P \ P0|
   ∀Y ∈ PS, p∈Y
   vp < Y ⇐=
   P–Semiflows forbidden:
   ∀Y ∈ PS,
   p∈Y
   vp < Y
   PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
   e Manuel Colom, Juan Juli´
   an Merelo 8
   

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9. The Proposal
   min p∈P
   vp
   ∀p ∈ P, ∀t ∈ •p, vp ≤ q∈ •t
   vq, vp ∈ {0, 1}
   p∈P\P0
   vp < |P \ P0|
   ∀Y ∈ PS, p∈Y
   vp < Y
   Plus: Siphons must be non empty
   PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
   e Manuel Colom, Juan Juli´
   an Merelo 9
   

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10. Example
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 10
    

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11. Example
    Set of equations:
    Related to Siphons Equation:
    vp 0 0 ≤ vr 0 0
    vp 0 1 ≤ vp 0 0 + vr 0 1
    vp 1 0 ≤ vp 1 1 + vr 0 0
    vp 1 1 ≤ vr 0 1
    vr 0 0 ≤ vp 0 0 + vr 0 1
    vr 0 0 ≤ vp 1 0
    vr 0 1 ≤ vp 1 1 + vr 0 0
    vr 0 1 ≤ vp 0 1
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 11
    

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12. Example
    Set of equations:
    Related to not all the places Equa-
    tion:
    vp 0 0 + vp 0 1 + vp 1 0 +
    vp 1 1 + vr 0 0 + vr 0 1 < 6
    Related to no P–Semiflows Equation:
    vp 0 0 + vp 1 0 + vr 0 0 < 3
    vp 0 1 + vp 1 1 + vr 0 1 < 3
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 12
    

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13. Example
    Set of equations:
    vp 0 0 ≤ vr 0 0
    vp 0 1 ≤ vp 0 0
    + vr 0 1
    vp 1 0 ≤ vp 1 1
    + vr 0 0
    vp 1 1 ≤ vr 0 1
    vr 0 0 ≤ vp 0 0
    + vr 0 1
    vr 0 0 ≤ vp 1 0
    vr 0 1 ≤ vp 1 1
    + vr 0 0
    vr 0 1 ≤ vp 0 1
    –
    vp 0 0
    + vp 0 1
    + vp 1 0
    +
    vp 1 1
    + vr 0 0
    + vr 0 1 < 6
    –
    vp 0 0
    + vp 1 0
    + vr 0 0 < 3
    vp 0 1
    + vp 1 1
    + vr 0 1 < 3
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 13
    

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14. Example
    So, for example:
    vp 0 1 = vp 1 0 = vr 0 0 = vr 0 1 = 1
    {p 0 1, p 1 0, r 0 0, r 0 1}
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 14
    

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15. Example
    But:
    vp 0 0 = 1
    {p 0 1, p 1 0, r 0 0, r 0 1, p 0 0}
    vp 0 1 = vp 1 0 = vr 0 0 = vr 0 1 = 1
    {p 0 1, p 1 0, r 0 0, r 0 1}
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 15
    

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16. The Proposal
    We need to combine all together
    For this we need to define an adequate fitness function
    returns:
    the number of places of the siphon when the restrictions are
    met (minimizing).
    (big) negative number for empty or ‘full’ siphons
    (number of unmet restrictions) - (number of restrictions) when
    there are unmet restrictions
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 16
    

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17. The experiments
    Initial population
    Other parameters
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 17
    

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18. The experiments
    Initial population
    Size:
    We start with a population of 8 individuals and run the experiment
    30 times. If it fails more than once, we double the size until the
    experiment does 30 runs with at most one failed.
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 18
    

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19. The experiments
    Initial population
    Size:
    We start with a population of 8 individuals and run the experiment
    30 times. If it fails more than once, we double the size until the
    experiment does 30 runs with at most one failed.
    =⇒ Less than 3.3 % of probability of not finding a solution
    And:
    Elitism and rank-based selection.
    Mutation: bitflip operation with 33 % of probability
    Two-point crossover operator with probability of 66 %.
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 19
    

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20. The experiments
    Taking advantage of structural knowledge?
    Initial population is random
    Initial population includes the P–Semiflows
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 20
    

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21. The experiments
    Times for siphon computation with the proposed method
    Initial Population
    P–Semiflows Random
    Size Pop. Time Eval. Time Eval.
    FMSAD 3 64 0.93 (0.22) 912 (214.83) 0.96 (0.18) 938 (169.27)
    4 64 1.97 (0.33) 1,102 (185.03) 1.92 (0.32) 1,082 (185.82)
    5 64 3.97 (1.01) 1,448 (361.33) 3.33 (0.41) 1,222 (152.79)
    6 128 10.16 (1.19) 2,627 (305.12) 12.61 (12.70) 2,657 (394.46)
    7 256 31.07 (4.65) 5,877 (785.53) 31.71 (3.47) 5,954 (644.96)
    8 256 43.18 (4.65) 6,299 (677.50) 50.95 (17.79) 7,009 (1,231.68)
    FMSLD 3 32 0.19 (0.03) 408 (72.73) 0.50 (1.73) 379 (73.60)
    4 32 0.37 (0.09) 458 (112.10) 0.36 (0.07) 447 (84.49)
    5 32 0.65 (0.10) 519 (79.26) 0.65 (0.10) 526 (79.82)
    6 32 1.01 (0.22) 571 (124.39) 1.00 (0.21) 562 (117.14)
    7 32 1.59 (0.34) 654 (137.96) 1.59 (0.85) 661 (343.29)
    8 64 4.14 (1.18) 1,349 (385.48) 4.05 (0.78) 1,310 (248.29)
    Phil 3 64 0.33 (0.05) 790 (106.88) 0.34 (0.03) 812 (64.70)
    4 64 0.64 (0.06) 887 (92.53) 0.62 (0.07) 863 (91.17)
    5 128 2.02 (0.20) 1,852 (195.65) 2.03 (0.23) 1,859 (195.90)
    6 128 3.08 (0.35) 1,988 (217.09) 3.18 (0.38) 2,042 (240.61)
    7 128 4.77 (0.52) 2,260 (237.10) 4.60 (0.41) 2,185 (183.80)
    8 128 6.61 (0.60) 2,454 (221.54) 6.40 (0.67) 2,362 (234.06)
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 21
    

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22. FMSxD
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 22
    

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23. The experiments
    FMSAD Example
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 23
    

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24. The experiments
    FMSLD Example
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 24
    

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25. A philosopher
    forkR_i
    phil1Waiting_i
    philForkR_i philForkL_i
    forkL_i
    philEating_i
    T5_i
    T6_i
    T3_i
    T2_i
    T1_i
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 25
    

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26. The experiments
    Philosophers Example
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    e Manuel Colom, Juan Juli´
    an Merelo 26
    

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27. How many siphons?
    Number of siphons P R P R Percentage
    FMSAD 3 42 22 28 25 28 52.38 % 66.67 % 59.52 % 66.67 %
    4 78 29 32 34 42 37.18 % 41.03 % 43.59 % 53.85 %
    5 150 42 48 44 49 28 % 32 % 29.33 % 32.67 %
    6 250 48 53 55 68 19.20 % 21.20 % 22 % 27.20 %
    7 490 75 70 90 83 15.31 % 14.29 % 18.37 % 16.94 %
    8 906 59 67 110 78 6.51 % 7.40 % 12.14 % 8.61 %
    FMSLD 3 24 14 11 58.33 % 45.83 %
    4 54 28 32 32 36 51.85 % 59.26 % 59.26 % 66.67 %
    5 116 34 31 38 45 29.31 % 26.72 % 32.76 % 38.79 %
    6 242 31 37 43 48 12.81 % 15.29 % 17.77 % 19.83 %
    7 496 35 36 48 49 7.06 % 7.26 % 9.68 % 9.88 %
    8 1006 38 48 64 58 3.78 % 4.77 % 6.36 % 5.77 %
    Phil 3 10 4 6 2 2 40 % 60 % 20 % 20 %
    4 17 5 5 5 8 29.41 % 29.41 % 29.41 % 47.06 %
    5 26 6 9 7 10 23.08 % 34.62 % 26.92 % 38.46 %
    6 37 10 12 10 12 27.03 % 32.43 % 27.03 % 32.43 %
    7 50 12 15 11 11 24 % 30 % 22 % 22 %
    8 65 11 15 10 13 16.92 % 23.08 % 15.38 % 20 %
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 27
    

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28. Conclusions and future work
    We can compute siphons (but not all!)
    It is not clear how good the method is (but the problem size
    can increase...)
    Adapting other methods (partially done)
    Adapting methods that use siphons
    Adding more information to the fitness function (siphonosity?)
    PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
    e Manuel Colom, Juan Juli´
    an Merelo 28
    

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