1. Dr. Eng Amr T. Abdel-Hamid
NETW 703
Network Protocols
Petri Nets
Dr. Eng Amr T. Abdel-Hamid
Spring 2008
Based on lectures from:
* Dr Chris Ling, School of Computer Science & Software Engineering Monash University
* Claus Reinke, Computing Lab, UKC

2. Introduction First introduced by Carl Adam Petri in 1962.
A diagrammatic tool to model concurrency and synchronization in distributed systems.
Very similar to FSMs.
Used as a visual communication aid to model the system behaviour.
Based on strong mathematical foundation.
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3. Example: EFTPOS System (STD of an FSM)
Initial
1 digit d1 d2 d3 d4 OK
pressed
approve
Approved
Rejected
Initial state
Final state
Reject
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4. Example: EFTPOS System (A Petri net)
Initial
1 digit d1 d2 d3 d4 OK
pressed
approve
approved
Reject
Rejected!
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5. Petri Net Example Arc with w=1 Arc Place Token Transition
I(tj) = {pi P : (pi, tj) A}
O(tj) = {pi P : (tj, pi) A}
I(pi) ={tj T : (tj, pi) A}
O(pi) ={tj T : (pi, tj) A}
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6. EFTPOS System Scenario 1: Normal Enters all 4 digits and press OK.
Scenario 2: Exceptional
Enters only 3 digits and press OK.
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7. Example: EFTPOS System (Token Games)
Initial
1 digit d1 d2 d3 d4 OK
pressed
approve
approved
Reject
Rejected!
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8. A Petri Net Specification ...
consists of three types of components: places (circles), transitions (rectangles) and arcs (arrows):
Places represent possible states of the system.
Transitions are events or actions which cause the change of state.
Every arc simply connects a place with a transition or a transition with a place.
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9. A Change of State …
is denoted by a movement of token(s) (black dots) from place(s) to place(s); and is caused by the firing of a transition.
The firing represents an occurrence of the event or an action taken.
The firing is subject to the input conditions, denoted by token availability.
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10. A Change of State
A transition is firable or enabled when there are sufficient tokens in its input places.
After firing, tokens will be transferred from the input places (old state) to the output places, denoting the new state.
Note that the EFTPOS example is a Petri net representation of a finite state machine (FSM).
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11. Example: Vending Machine
The machine dispenses two kinds of snack bars – 20c and 15c.
Only two types of coins can be used – 10c coins and 5c coins.
The machine does not return any change.
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12. Example: Vending Machine (STD of an FSM)
0 cent
5 cents
10 cents
15 cents
20 cents
Deposit 5c
Deposit 10c
Take 20c snack bar
Take 15c snack bar
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13. Modeling FSMs vend 15¢ candy 10 15 5 5 5 5 5 10 10 20 105 10 10 20 10
vend 20¢ candy
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14. Modeling FSMs state machines: each transition has exactly
vend 15¢ candy 10 5
state machines:
each transition
has exactly
one input and
one output 5 5 5 10 10
vend 20¢ candy
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15. Example: Vending Machine (A Petri net)
Take 15c bar
Deposit 5c 0c
Deposit 10c
Deposit c
10c
Deposit
20c
15c
Take 20c bar
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16. Example: Vending Machine (3 Scenarios)
Deposit 5c, deposit 5c, deposit 5c, deposit 5c, take 20c snack bar.
Scenario 2:
Deposit 10c, deposit 5c, take 15c snack bar.
Scenario 3:
Deposit 5c, deposit 10c, deposit 5c, take 20c snack bar.
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17. Example: Vending Machine (Token Games)
Take 15c bar
Deposit 5c 0c
Deposit 10c
Deposit c
10c
Deposit
20c
15c
Take 20c bar
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18. Multiple Local States
In the real world, events happen at the same time.
A system may have many local states to form a global state.
There is a need to model concurrency and synchronization.
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19. Example: In a Restaurant (A Petri Net)
Waiter
free
Customer 1
Customer 2
Take
order
Take
order
wait
Order
taken
wait
eating
eating
Tell
kitchen
Serve food
Serve food
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20. Example: In a Restaurant (Two Scenarios)
Waiter takes order from customer 1; serves customer 1; takes order from customer 2; serves customer 2.
Scenario 2:
Waiter takes order from customer 1; takes order from customer 2; serves customer 2; serves customer 1.
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21. Example: In a Restaurant (Scenario 1)
Waiter
free
Customer 1
Customer 2
Take
order
Take
order
wait
Order
taken
wait
eating
eating
Tell
kitchen
Serve food
Serve food
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22. Example: In a Restaurant (Scenario 2)
Waiter
free
Customer 1
Customer 2
Take
order
Order
taken
Tell
kitchen
wait
Serve food
eating
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23. Net Structures A sequence of events/actions: Concurrent executions: e1
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24. Net Structures
Non-deterministic events - conflict, choice or decision: A choice of either e1, e2 … or e3, e4 ... e1 e2 e3 e4
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25. Net Structures
Synchronization e1
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26. Net Structures
Synchronization and Concurrency e1
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27. Modeling communication protocols
ready
to send
ready
to receive
buffer
full
send
msg.
receive
msg.
wait
for ack.
proc.2
proc.1
msg.
received
receive
ack.
send
ack.
buffer
full
ack.
received
ack.
sent
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28. Modeling Protocols Using Petri Nets (Stop and Wait Protocol)
The transmitter sends a frame and stops waiting for an acknowledgement from the receiver (ACK)
Once the receiver correctly receives the expected packet, it sends an acknowledgement to let the transmitter send the next frame.
When the transmitter does not receive an ACK within a specified period of time (timer) then it retransmits the packet.
Timer
Transmitter
Receiver
Frame 0
Frame 1
ACK 0
ACK 1
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29. Stop and Wait Protocol: Normal Operation
Ack 1
pkt 0
Ack 0
pkt 1
Get Pkt 0
Get Pkt 1
Send Pkt 0
Send Pkt 1
Wait Ptk 1
Wait Pkt 0
Wait Ack 0
Wait Ack 1
Transmitter State
Receiver State
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Channel State

30. Stop and Wait Protocol: Normal Operation
Ack 1
pkt 0
Ack 0
pkt 1
Get Pkt 0
Get Pkt 1
Send Pkt 0
Send Pkt 1
Wait Ptk 1
Wait Pkt 0
Wait Ack 0
Wait Ack 1
Transmitter State
Receiver State
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Channel State

31. Stop and Wait Protocol: Deadlock
Ack 1
pkt 0
Ack 0
pkt 1
Get Pkt 0
Get Pkt 1
Send Pkt 0
Send Pkt 1
loss
Wait Ptk 1
Wait Pkt 0
Wait Ack 0
Wait Ack 1
Transmitter State
Receiver State
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Channel State

32. Stop and Wait Protocol: Deadlock Avoidance Using Timers
Ack 1
pkt 0
Ack 0
pkt 1
Get Pkt 0
Get Pkt 1
Send Pkt 0
Send Pkt 1
loss
timer
Wait Ptk 1
Wait Pkt 0
Wait Ack 0
Wait Ack 1
timer
Transmitter State
Receiver State
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Channel State

33. Stop and Wait Protocol: Loss of Acknowledgements
pkt 0
Ack 0
pkt 1
Get Pkt 0
Get Pkt 1
Send Pkt 0
Send Pkt 1
loss
Reject 0
timer
Wait Ptk 1
Wait Pkt 0
Wait Ack 0
Wait Ack 1
timer
Reject 1
Transmitter State
Receiver State
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Channel State

34. Behavioural Properties
Reachability
“Can we reach one particular state from another?”
Boundedness
“Will a storage place overflow?”
Liveness
“Will the system die in a particular state?”
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35. Recalling the Vending Machine (Token Game)
Take 15c bar
Deposit 5c 0c
Deposit 10c
Deposit c
10c
Deposit
20c
15c
Take 20c bar
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36. A marking is a state ... M0 = (1,0,0,0,0) M1 = (0,1,0,0,0)p4
M0 = (1,0,0,0,0) t4
M1 = (0,1,0,0,0) p2 t1
M2 = (0,0,1,0,0)
M3 = (0,0,0,1,0) p1
M4 = (0,0,0,0,1) t3 t7 t5
Initial marking:M0 t6 p5 t2 p3 t9
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37. Reachability p1 t8 t1 t2 p2 t3 p3 t4 t5 t6 p5 t7 p4 t9
Initial marking:M0
M0 = (1,0,0,0,0)
M1 = (0,1,0,0,0)
M2 = (0,0,1,0,0)
M3 = (0,0,0,1,0)
M4 = (0,0,0,0,1) M0 M1 M2 M3 M4 t3 t1 t5 t8 t2 t6
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38. Reachability A firing or occurrence sequence:M0 M1 M2 M3 M4 t3 t1 t5 t8 t2 t6
A firing or occurrence sequence:
“M2 is reachable from M1 and M4 is reachable from M0.”
In fact, in the vending machine example, all markings are reachable from every marking.
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39. Boundedness
A Petri net is said to be k-bounded or simply bounded if the number of tokens in each place does not exceed a finite number k for any marking reachable from M0.
The Petri net for vending machine is 1-bounded.
A 1-bounded Petri net is safe.
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40. Liveness
A Petri net with initial marking M0 is live if, no matter what marking has been reached from M0, it is possible to ultimately fire any transition by progressing through some further firing sequence.
A live Petri net guarantees deadlock-free operation, no matter what firing sequence is chosen.
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41. Liveness
The vending machine is live and the producer-consumer system is also live.
A transition is dead if it can never be fired in any firing sequence.
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42. An Example A bounded but non-live Petri net t1 p3 t3 t4 p1 p2 p4 t2
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43. Analysis methods (1) Coverability tree
tree representation of all possible markings
root = M0
nodes = markings reachable from M0
arcs = transition firings
if net is unbounded, then tree is kept finite by introducing the symbol 
Properties
a PN is bounded iff  doesn’t appear in any node
a PN is safe iff only 0’s and 1’s appear in nodes
a transition is dead iff it doesn’t appear in any arc
if M is reachable form M0, then exists a node M’ that covers M
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44. Coverability tree example
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45. Coverability tree example
“dead end” p2 t1 t0 t2 p3
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46. Coverability tree example
“dead end”
M3=(10) p2 t1 t0 t2 p3
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47. Coverability tree example
“dead end”
M3=(10) t1 p2 t1 t0
M4=(01) t2 p3
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48. Coverability tree example
“dead end”
M3=(10) t1 t3 p2 t1 t0
M4=(01)
M3=(10)
“old” t2 p3
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49. Coverability tree example
“dead end”
M3=(10) t1 t3 p2 t1 t0
M4=(01)
M6=(10)
“old” t2 t2 p3
M5=(01)
“old”
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50. Coverability tree example
100 t1 t3 t1 t3
M1=(001)
“dead end”
M3=(10)
001
10 t1 t3 t1 t3
M4=(01)
M6=(10)
“old”
01 t2 t2
M5=(01)
“old”
coverability graph
coverability tree
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51. For a petrinet to be Rechable (Cr = m) r is the rank of C
x is n x 1 column vector of nonnegative integers and is called the firing count vector
For a petrinet to be Rechable (Cr = m)
r is the rank of C
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52.
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