Petri Nets – Lecture 2 - CAPSL Research Laboratories
66
Petri Nets – Lecture 2 Chen Chen Oct. 2 nd , 2011 2011/10/5 \course\867-11F\Topic-A.ppt 1
Transcript of Petri Nets – Lecture 2 - CAPSL Research Laboratories
Petri Nets – Lecture 2
Chen ChenOct. 2nd, 2011
2011/10/5 \course\867-11F\Topic-A.ppt 1
Outline
• Sub-class of Petri Nets• Timed Petri Nets
2011/10/5 \course\867-11F\Topic-A.ppt 2
Liveness and Safeness Problem
•
2011/10/5 \course\867-11F\Topic-A.ppt 3
p1 p2
Deadlock
t
Not safe
t1 t2p
Sub-class of Petri Nets
2011/10/5 \course\867-11F\Topic-A.ppt 4
MarkedGraph
StateMachine
Free-choice Petri Nets
Simple Petri Nets
Petri Nets
Summary of Complexity Results for Petri Nets
Petri Net classes Reachability Liveness DeadlockArbitrary decidable
EXPSPACE-harddecidable EXPSPACE-hard
decidable EXPSPACE-hard
1-safe PSPACE-complete PSPACE-complete PSPACE-complete
Acyclic NP-complete linear time linear time
1-safe acyclic NP-complete constant time constant time
Conflict-free NP-complete polynomial time polynomial time
1-safe conflict-free polynomial time polynomial time polynomial time
Free-choice decidable EXPSPACE-hard
NP-complete NP-complete
1-safe free-choice PSPACE-complete polynomial time polynomial time
2011/10/5 \course\867-11F\Topic-A.ppt 5
— Allan Cheng, Javier Esparza, and Jens Palsberg. 1995. Complexity results for 1-safe nets. Theor. Comput. Sci. 147, 1-2 (August 1995), 117-136.
Relation Among Complexity Classes
2011/10/5 \course\867-11F\Topic-A.ppt 6
NL
PNP
PSPACE
EXPTIME
EXPSPACE
EXPSPACE Space 2poly(n)
EXPTIME Time 2poly(n)
PSPACE Space poly(n)
NP Time poly(n)Non-deterministicTuring machine
P Time poly(n)
NL Space O(log n)Non-deterministicTuring machine
From Wikipedia
NP-Complete
E.g. Livenessof normal PN
Example of Marked Graph
2011/10/5 \course\867-11F\Topic-A.ppt 7
• 3 users try to repeatedly access the same CS• Only one user can access CS each time
Critical Section
3 tokens = 3 users 1 token = lock
t2
p2
p3
p1
t1Marked Graph
Each place has one input arc.Each place has one output arc.
Relation Between Marked Graph and Dataflow
2011/10/5 \course\867-11F\Topic-A.ppt 8
Critical Section
3 tokens = 3 users
1 token = lock
t2
p2
p4
p1
t1
t3
p3
Entr.
CS
Exit
Marked GraphDataflow (extended)
Multiple tokens per arc
??
Features of Marked Graph
• Each place has one input arc and one output arc
• Polynomial time to decide liveness and safeness problemsThere exists a live and safe marking iff the marked
graph is strongly connected [Murata80]• Can be converted to dataflow graph and vice
versa in particular conditions
2011/10/5 \course\867-11F\Topic-A.ppt 9
[Murata80] Murata, T.; Koh, J.; , "Reduction and expansion of live and safe marked graphs," Circuits and Systems, IEEE Transactions on , vol.27, no.1, pp. 68- 71, Jan 1980
Outline
• Sub-class of Petri Nets• Timed Petri Nets
2011/10/5 \course\867-11F\Topic-A.ppt 10
Example: Production Line
2011/10/5 \course\867-11F\Topic-A.ppt 11
Robot Arm(R1)
Robot Arm(R2)
Machine(M1)
Machine(M2)
Conveyor(C1)
Conveyor(C2)
3 slots (raw materials)
2 slots (empty)
1 sec load from C11 sec unload to C2
1 sec load from C21 sec unload to final
Output of final products
10 sec process 16 sec process
Example: Production Line
2011/10/5 \course\867-11F\Topic-A.ppt 12
Robot Arm(R1)
Robot Arm(R2)
Machine(M1)
Machine(M2)
Conveyor(C1)
Conveyor(C2)
3 slots (raw materials)
2 slots (empty)
4 sec load from C14 sec unload to C2
1 sec load from C21 sec unload to final
Output of final products
10 sec process 16 sec process
Question: What’s the bottleneck of the production line?
Solution of Production Line Problem
• Each component works in eager mode• Figure out bottleneck with timed Petri Net
modelingThe problem can be efficiently solved if the
timed Petri Net is a timed Marked Graph
2011/10/5 \course\867-11F\Topic-A.ppt 13
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 14
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Definition of Timed Petri Nets (Self-reading)
•
2011/10/5 \course\867-11F\Topic-A.ppt 15
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 16
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
InitialMarkingTime = 0
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 17
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 0Firing of t1 is initiated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 18
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 1Firing of t1 is terminated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 19
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 1Firing of t2 is initiated (will be done at 11)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 20
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 11Firing of t2 is terminated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 21
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 11Firing of t3 is initiated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 22
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 12Firing of t3 is terminated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 23
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 12Firing of t1 is initiatedFiring of t4 is initiated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 24
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 13Firing of t1 is terminatedFiring of t4 is terminated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 25
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 13Firing of t2 is initiatedFiring of t5 is initiated (will be done at 29)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 26
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 23Firing of t2 is terminatedFiring of t5 is executing (will be done at 29)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 27
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 23Firing of t3 is initiatedFiring of t5 is executing (will be done at 29)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 28
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 24Firing of t3 is terminatedFiring of t5 is executing (will be done at 29)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 29
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 24Firing of t1 is initiatedFiring of t5 is executing (will be done at 29)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 30
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 25Firing of t1 is terminatedFiring of t5 is executing (will be done at 29)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 31
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 25Firing of t2 is initiated (will be done at 35)Firing of t5 is executing (will be done at 29)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 32
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 29Firing of t2 is executing (will be done at 35)Firing of t5 is terminated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 33
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 29Firing of t2 is executing (will be done at 35)Firing of t6 is initiated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 34
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 30Firing of t2 is executing (will be done at 35)Firing of t6 is terminated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 35
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 30Firing of t2 is executing (will be done at 35)Firing of t4 is initiated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 36
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 31Firing of t2 is executing (will be done at 35)Firing of t4 is terminated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 37
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 31Firing of t2 is executing (will be done at 35)Firing of t5 is initiated (will be done at 47)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 38
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 35Firing of t2 is terminatedFiring of t5 is executing (will be done at 47)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 39
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 35Firing of t3 is initiatedFiring of t5 is executing (will be done at 47)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 40
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 36Firing of t3 is terminatedFiring of t5 is executing (will be done at 47)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 41
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 36Firing of t1 is initiatedFiring of t5 is executing (will be done at 47)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 42
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 37Firing of t1 is terminatedFiring of t5 is executing (will be done at 47)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 43
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 37Firing of t2 is initiated (will be done at 47)Firing of t5 is executing (will be done at 47)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 44
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 47Firing of t2 is terminated Firing of t5 is terminated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 45
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 47Firing of t3 is initiatedFiring of t6 is initiated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 46
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 48Firing of t3 is terminatedFiring of t6 is terminated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 47
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 48Firing of t3 is terminatedFiring of t6 is terminated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 48
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 48Firing of t3 is terminatedFiring of t6 is terminated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 49
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 48Firing of t1 is initiatedFiring of t4 is initiated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 50
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 49Firing of t1 is terminatedFiring of t4 is terminated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 51
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 49Firing of t2 is initiated (will be done at 59)Firing of t5 is initiated (will be done at 65)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 52
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 59Firing of t2 is terminatedFiring of t5 is initiated (will be done at 65)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 53
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 59Firing of t3 is initiatedFiring of t5 is initiated (will be done at 65)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 54
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 60Firing of t3 is terminatedFiring of t5 is initiated (will be done at 65)
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 55
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 65Firing of t5 is terminated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 56
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 65Firing of t6 is initiated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 57
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 66Firing of t6 is terminated
Solution of Production Line Problem (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 58
C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
Time = 66Firing of t6 is terminated
Same marking as time = 48Go to time = 48
Process Pattern
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0
Starting of process
48 66 84
Starting of stable cycle
Starting of 2nd cycle
Starting of 3rd cycle
Time (sec)
Minimum Cycle Time = 18
18 seconds per product
Minimum Cycle Time
• If the Timed PN is a strongly connected Timed Marked Graph
• Min Cycle Time = max {Di / Ni}• i indicates the i-th simple loop. Di = total delay time in the i-th loop Ni = # of tokens in the i-th loop
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Figure Out Bottleneck
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C1
C2R1 R2
M1 M2
t1 t2
t3 t4
t5 t6
t1: R1 load from C1 (1s)t2: M1 process (10s)t3: R1 unload to C2 (1s)t4: R2 load from C2 (1s)t5: M2 process (16s)t6: R2 unload to final (1s)
loop1 (12) loop3 (1)
loop2 (12)
loop4 (18)
loop5 (18)
loop6 (10)
Understanding of Minimum Cycle Time
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t1(5s)p1
Time = 0Initial marking
Min Cycle Time = max {Di / Ni}=2.5
Correct?
Understanding of Minimum Cycle Time (cont.)
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t1(5s)
Min Cycle Time = max {Di / Ni}=2.5
p1
Time = 0Initial marking
t1(5s)p1
Time = 0Firing of t1 is initiated(twice)
t1(5s)p1
Time = 5Firing of t1 is terminated(twice)
Same marking appears every 5 seconds!!
Understanding of Minimum Cycle Time (cont.)
• Minimum Cycle TimeAverage time to fire the same transitionSi(m): Time when transition ti initiates its m-th
firing. O(n3) algorithm
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mmSC i
m
)(lim∞→
=
C. V. Ramamoorthy and G. S. Ho. 1980. Performance Evaluation of Asynchronous Concurrent Systems Using Petri Nets. IEEE Trans. Softw. Eng. 6, 5 (September 1980), 440-449.
Understanding of Minimum Cycle Time (cont.)
2011/10/5 \course\867-11F\Topic-A.ppt 65
t1(5s)p1
Time = 0Initial marking
t1(5s)p1
Time = 0Firing of t1 is initiated(twice)
t1(5s)p1
Time = 5Firing of t1 is terminated(twice)
t1 is fired twice per every 5 seconds
5.225)(lim 1 ===
∞→ mmSC
m
Min Cycle Time = max {Di / Ni}=2.5
Deep Thinking
• Relation between production line problem and loop scheduling problem
• An alternative way to think about minimum cycle time Concept of “Computation Rate” The average number of firings of a transition in unit time Maximum computation rate = minimum cycle time -1
Ramchandani (1973) Ph.D thesis
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