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Modeling Worldviews

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Modeling Worldviews. Activity Scanning Petri Nets. and Event Graphs ... Bipartite Directed Graph with Place (balls) and Transition (bars) Nodes ... – PowerPoint PPT presentation

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Title: Modeling Worldviews


1
Modeling Worldviews
  • Activity Scanning Petri Nets and Event
    Graphs
  • Event Scheduling Event Graphs
  • State Machines
  • Process Interaction Block Flow Diagrams and
    Event Graphs

2
Petri Nets Definition and Behavior
Bipartite Directed Graph with Place (balls) and
Transition (bars) Nodes Marking
Multiple Tokens at Places
(Indicate State of Resident Entities)
3
Petri Nets Definition and Behavior
Firing Rule A transition fires when ALL of
its input places are marked. A token is removed
from each input place and added to each output
place.
4
Petri Nets Definition and Behavior
How might we keep Net Alive?
5
PN Enrichments for Simulation
  • Non-Zero (Random) Firing Times
  • Random process indices
  • Colored Tokens
  • Different classes of transient entities - change
    colors?
  • Input and Output Counts
  • Analytical and aggregation production
  • Inhibitor arcs
  • Prevents firing...resource priorities

6
Non-Zero (Random) Firing Times Random process
indices
  • When a transition is enabled, a timer starts.
    After it times out, the transition fires.
  • Remove on Fire rule remove and deposit tokens
    when the transition fires.
  • Remove on Enable rule remove tokens when timer
    starts, deposit tokens when it times out (used
    here to save space).

7
PN Example G/G/1 Queue
Number of Idle Servers
Number in Queue
T
T
s
a
(Random Firing Times)
8
Petri Nets
  • Example Tandem queue with limited input

A
Q
t
t
2
1
100
9
Activity Modeling View
What you get when FINISH Activity
Things needed to START Activity
Activity
. . .
Time Duration of Activity
10
Tandem Queue with Buffer Production Blocking
B
T
T
s
s
2
Q
Q
S
S
What about Communication Blocking?
11
PN Enrichments for Simulation
  • Transition Input and Output Counts

2
T
T
T
1
a
2
3
12
PN Enrichments for Simulation
  • Inhibitor Arcs (Fail/Repair Model)

Working Machines
t
f
Failed Machines
t
r
13
Time Bound Sequence
Tc
1
2
Ta between arrivals
T1
T2
Job must start step 2 within TC of finishing step
1. Give a Petri Net for this system?
14
Time Bound Sequence...Error?
T1
Ta
T2
(timer running)
15
Time Bound Sequence...Error?
T1
Ta
T2
(timer running)
16
Time Bound Sequence...Error?
Q2
T1
Ta
T2
(timer running)
What if there are no tokens at Q2 and 1 active
transition in Tc? (Timer expired place may
grab next part from Q2 early!)
17
Time Bound Sequence...
R2
R1
Q1
Q2
T1
Ta
T2
(timer terminator)
(zero-time transition)
Note need to set transition priorities to make
killing the timer have priority if resource R2
becomes idle simultaneously with the timer
expiring.
(inhibitor arc)
18
Petri Nets - Summary
  • Modeling
  • Concurrency and contention (Simultaneous Resource
    Usage)
  • Analysis
  • Reachability tree, CTMC Analysis, and much more!

19
Mapping Petri Netsinto Event Graphs
20
Alternative Implementations
  • Activity World View
  • Petri Net Formalism
  • Simple fundamental elements and behavior
  • Concurrent resource usage and contention
  • Event Scheduling View
  • Event Graph Formalism
  • Simple fundamental element and behavior
  • Concurrency and contention - large model
    representations

21
PN-gtEG MAPPING
Transitions become timed edges Places become
conditional edges
22
PN-gtEG Mapping eg. G/G/s queue
S
PN
Q
A
T
T
s
a
23
PN-gtEG Mapping eg. G/G/s queue
S
PN
Q
A
T
T
s
a
EG
T
T
a
s
A,
A--
Q--,
S
Q
S--
24
PN-gtEG Mapping eg. G/G/s queue
S
PN
Q
A
T
T
s
a
EG
(A)
(SQ)
(SQ)
T
T
a
s
A,
A--
Q--,
S
Q
S--
25
PN-gtEG Mapping eg. G/G/s queue
Variable A is unnecessary!
(A)
(SQ)
(SQ)
T
T
a
s
A,
A--
Q--,
S
Q
S--
26
PN-gtEG Mapping eg. G/G/s queue
Empty Vertex is unnecessary!
(SQ)
(SQ)
T
T
a
s

Q
Q--,
S
S--
27
PN-gtEG Mapping eg. G/G/s queue
SQ conditions redundant too...
(SQ)
(SQ)
T
T
a
s

Q
Q--,
S
S--
28
PN-gtEG Mapping eg. G/G/s queue
Result a conventional G/G/s queue EG model
(Q)
T
a
(S)
T
s

Q
Q--,
S
S--
29
PN?EG MAPPING
P set of Places, T set of transitions in
PN d(t) delay time (RV) for transition
t?T Ip(t),Op(t) Set of Input and Output
Places for t?T It(p),Ot(p) Set of Input and
Output Transitions for p?P Step 0. ? p?P
define an integer state variable, X(p). Step 1.
? t?T create an edge (O(t), D(t)) with delay
d(t). Step 2. ? p?P with unique (It(p),Ot(p))
pair create the edge (D(It(p)),O(Ot(p))) with
the condition that all p?Ip(Ot(p)) are marked.
(For inhibitor arcs these places must not be
marked.) Step 3. Add State changes
for O(t), decrement X(p) ? p?Ip(t)
for D(t), increment X(p) ? p?Op(t).
Advanced homework Confirm this or find a counter
example.
30
Analytical Methods and Conditions
  • PNs
  • Reachabilty, decidability, liveness, and
    deadlock
  • EGs
  • State definition, event reduction, priorities,
    equivalence, boundary conditions, resource
    deadlock, MIP representations

31
COMMUNICATIONS BLOCKING (R1 needs empty buffer to
Start)
...
...
...
R1,R2 Number of idle resources
Q1,Q2 Number of waiting jobs
B Number of empty buffer spaces
ta,ts1,ts2 Arrival and processing times
32
COMMUNICATIONS BLOCKING Petri Net
Q1 number of jobs waiting in queue 1
Q2 number waiting in queue 2
R1 idle resources for step 1
R2 idle resources for step 2
B available buffer slots
33
Each transition becomes a timed edge...
R1
R2
A
Q1
Q2
W
ts2
ta
ts1
0
B
ta
34
Next, each place becomes a conditional
edge... (ALL input places marked is condition)
R1
R2
A
Q1
Q2
W
ts2
ta
ts1
0
B
(A)
(Q1R1B)
35
Mapping Petri Net to Event Graph Finally,
increment and decrement tokens as state changes
R1
R2
ta
(Q2R2)
(A)
(Q1R1B)



(Q1R1B)
(Q2R2)
ta
0
ts2
(W)
ts1



A, Q1
R2-- Q2--
A--
R1, Q2
R2
Q1--, R1--. B--
W B
W--

(Q1R1B)
36
Final Event Graph How can this be simplified?
(ref Seila, et. al. Text)
37
A Simplified Event Relationship Graph
(Q2)
(Q1B)


(R1B)
(R2)
ta
ts2
ts1


R1, Q2
R2-- Q2-- B
Q1--, R1--. B--
Q1
R2

(Q1R1)
38
Implications
  • PN analysis of Event Graphs
  • State reachability
  • Liveness
  • Deadlock
  • EG analysis of Petri Nets
  • State space
  • Model reduction
  • Equivalence
  • Boundary Conditions
  • Resource Deadlock

39
Petri Net Simulator
40
EG -gt PN Mapping
Remove-on-Fire Rule
Remove-on-Enable Rule
41
Examples of Petri Net Models Activity Scanning
Models
Inhibitor arc
Ref Peter Haas book on Stochastic Petri Nets
resets all timers each scan, prob. deposit Remove
on Fire rule vs Remove on enable (Ref
Fishwick) Simulation Activity Scanning
algorithm. (Cancel if transition is disabled!)
42
Multiple server queues?
43
Note is an event graph one out-transition
per place State dep deposit to d11 or d12...
44
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45
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46
Modeling Activities with ERGs
tF
B
M
failing need new machine
get broken machine
tR
R,B
M,R
fixing need broken machine
and repairman get good machine
and repairman
Implied ERG
tF
B
M
M new machines B broken machines waiting R
idle repairmen
M--
B

(R)
tR
B,R
R,M

R, M
R--, B--
(B)
47
Implied ERG
tF
B
M
M--
B
M new machines B broken machines R idle
repairmen

(R)
tR
B,R
R,M
R, M

R--, B--
(B)
Exercise can you further reduce this?
hint assume more machines
than repairmen. Define new
variable(s).
Reduced ERG M is not tested
B
B

(R)
tF
tR
B,R
R

R
R--, B--
(B)
48
B
B

(R)
tF
tR
B,R
R

R
R--, B--
R total number of repairmen (const.) B
number of broken machines (incl in rep.)
(B)
tR
Fail

(BltR)
B
(BgtR)

Fix
tF
B--
49
Colored Petri Nets (transitions are enabled by
color or tokens)
50
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53
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54
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55
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56
d1,i,j part i waiting Or processing on mach j
57
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58
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59
GSMP-gtEGM Mapping (Computer Network)
End
Trans
New
Packet
Obs. End Tran.
Prop.
Clear
Start Tran
Prop.
Reset
60
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61
Need Pg 397.
62
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63
Transition has two times e3 or e5 with 50/50
probs.
e1 deposits in d2 or d5 with equal probabilty
64
NEED PAGE 137 for this example
65
Inhibitor arcs are note strictly necessary (but
very convenient!)
Add d2 had token iff d1 empty And no tokens if d1
has any.
66
Need Example 1.4 of Chapter 2
Measuring Delays
67
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69
Need figure 9.2 of section 2.6 without colors
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