Slide 1
Slide 1 text
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
[email protected]
19 de junio de 2014
PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
e Manuel Colom, Juan Juli´
an Merelo 1
Slide 2
Slide 2 text
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
Slide 3
Slide 3 text
Petri Nets, Siphons and Liveness
PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
e Manuel Colom, Juan Juli´
an Merelo 3
Slide 4
Slide 4 text
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
Slide 5
Slide 5 text
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
Slide 6
Slide 6 text
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
Slide 7
Slide 7 text
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
Slide 8
Slide 8 text
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
Slide 9
Slide 9 text
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
Slide 10
Slide 10 text
Example
PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
e Manuel Colom, Juan Juli´
an Merelo 10
Slide 11
Slide 11 text
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
Slide 12
Slide 12 text
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
Slide 13
Slide 13 text
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
Slide 14
Slide 14 text
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
Slide 15
Slide 15 text
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
Slide 16
Slide 16 text
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
Slide 17
Slide 17 text
The experiments
Initial population
Other parameters
PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
e Manuel Colom, Juan Juli´
an Merelo 17
Slide 18
Slide 18 text
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
Slide 19
Slide 19 text
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
Slide 20
Slide 20 text
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
Slide 21
Slide 21 text
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
Slide 22
Slide 22 text
FMSxD
PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
e Manuel Colom, Juan Juli´
an Merelo 22
Slide 23
Slide 23 text
The experiments
FMSAD Example
PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
e Manuel Colom, Juan Juli´
an Merelo 23
Slide 24
Slide 24 text
The experiments
FMSLD Example
PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
e Manuel Colom, Juan Juli´
an Merelo 24
Slide 25
Slide 25 text
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
Slide 26
Slide 26 text
The experiments
Philosophers Example
PNSE 2014. June. Tunis, Tunisia Fernando Tricas, Jos´
e Manuel Colom, Juan Juli´
an Merelo 26
Slide 27
Slide 27 text
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
Slide 28
Slide 28 text
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