Modeling Worldviews Activity Scanning – Petri Nets and Event Graphs Event Scheduling – Event...

Modeling Worldviews

• Activity Scanning – Petri Nets and Event Graphs

• Event Scheduling – Event Graphs

State Machines

• Process Interaction –Block Flow Diagrams and Event Graphs

Petri Nets: Definition and BehaviorBipartite Directed Graph with Place (balls) and

Transition (bars) Nodes

Marking: Multiple Tokens at Places

(Indicate State of Resident Entities)

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.

Petri Nets: Definition and Behavior

How might we keep Net Alive?

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

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).

Non-Zero (Random) Firing Times: Random process indices

PN Example: G/G/1 Queue

sTaT

(Random Firing Times)

Number in Queue

Number of Idle Servers

Petri Nets Example: Tandem queue with limited input

t21tQA

100

Activity Modeling View

Time = Duration of Activity

. . .

Things needed to STARTActivity

What you get when FINISHActivity

Activity

sT

Q

2sT

S S

Q

B

Tandem Queue with Buffer – Production Blocking

What about Communication Blocking?

Ta

T1 T2

2

3

PN Enrichments for Simulation Transition Input and Output Counts

PN Enrichments for Simulation

• Inhibitor Arcs (Fail/Repair Model)

tfrt

Working Machines

Failed Machines

Time Bound Sequence

1 2

T1 T2

Tc

Job must start step 2 within TC of finishing step 1.

Give a Petri Net for this system?

Ta between arrivals

Time Bound Sequence...Error?

Ta

(timer running)

Tc

(timer expired)

T1 T2

(scrap)

Time Bound Sequence...Error?

Ta

(timer running)

Tc

(timer expired)

T1 T2

(scrap)

(kill timer)

Time Bound Sequence...Error?

Ta

(timer running)

Tc

(timer expired)

T1 T2

(scrap)

(kill timer)

What if there are no tokens at Q2 and 1 active transition in Tc? (“Timer expired” place may grab next part from Q2 early!)

Q2

Time Bound Sequence...

Ta

(timer terminator)

Tc

(timer expired)

T1 T2

(scrap)

(kill timer)

(zero-timetransition)

(inhibitorarc)

Q1 Q2

R2R1

Note: need to set transition priorities to make killing the timer have priority if resource R2 becomes idle simultaneously with the timer expiring.

Petri Nets - Summary

• Modeling: Concurrency and contention (Simultaneous

Resource Usage)

• Analysis: Reachability tree, CTMC Analysis, and

much more!

Mapping Petri Netsinto

Event Graphs

Alternative ImplementationsActivity World View

Petri Net Formalism: Simple fundamental elements and behaviorConcurrent resource usage and contention

Event Scheduling ViewEvent Graph Formalism:

Simple fundamental element and behaviorConcurrency and contention - large model

representations

Transitions become timed edges

Places become conditional edges

PN->EG MAPPING

PN->EG Mapping: eg. G/G/s queue

PN:

Q

sTaT

S

A

PN->EG Mapping: eg. G/G/s queue

PN:

EG:

Q

sTaT

S

A

{A--} {A++, Q++}

{Q--, S--}

{S++}

TsaT

PN->EG Mapping: eg. G/G/s queue

PN:

EG:

Q

sTaT

S

A

(S&Q)(A)

(S&Q)

{A--} {A++, Q++}

{Q--, S--}

{S++}

TsaT

PN->EG Mapping: eg. G/G/s queue

(S&Q)(A)

(S&Q)

{A--} {A++, Q++}

{Q--, S--}

{S++}

TsaT

Variable A is unnecessary!

PN->EG Mapping: eg. G/G/s queue

(S&Q)

(S&Q)

{ Q++} {Q--, S--}

{S++}

TsaT

Empty Vertex is unnecessary!

PN->EG Mapping: eg. G/G/s queue

(S&Q)

(S&Q)

{ Q++} {Q--, S--}

{S++}

TsaT

S&Q conditions redundant too...

PN->EG Mapping: eg. G/G/s queue

(Q)

(S)

{ Q++} {Q--, S--}

{S++}

Ts

aT

Result: a conventional G/G/s queue EG model

P = set of Places, T = set of transitions in PN d(t) = delay time (RV) for transition tT {Ip(t),Op(t)}= Set of Input and Output Places for tT {It(p),Ot(p)}= Set of Input and Output Transitions for pP

Step 0. pP: define an integer state variable, X(p). Step 1. tT: create an edge (O(t), D(t)) with delay d(t). Step 2. pP with unique (It(p),Ot(p)) pair: create the edge

(D(It(p)),O(Ot(p))) with the condition that all pIp(Ot(p)) are marked. (For inhibitor arcs these places must not be marked.)

Step 3. Add State changes: for O(t), decrement X(p) pIp(t); for D(t), increment X(p) pOp(t).

PNEG MAPPING

Advanced homework: Confirm this or find a counter example.

Analytical Methods and Conditions

• PNsReachabilty, decidability, liveness, and

deadlock

• EGsState definition, event reduction, priorities,

equivalence, boundary conditions, resource deadlock, MIP representations

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

...

COMMUNICATIONS BLOCKING: Petri Net

A Q1

R1

ta ts1

Q2

R2

Bts20

W

Q1 = number of jobs waiting in queue 1Q2 = number waiting in queue 2R1 = idle resources for step 1R2 = idle resources for step 2B = available buffer slots

Each transition becomes a timed edge...

ta

A Q1

R1

ta ts1

Q2

R2

Bts20

W

ts1 0 ts2

Next, each place becomes a conditional edge...(ALL input places marked is condition)

A Q1

R1

ta ts1

Q2

R2

Bts20

W

(A)

(Q1&R1&B)

ta ts1 0 ts2

~

~

~

~

~~

~

(Q1&R1&B)

(Q1&R1&B)

(Q2&R2)

(Q2&R2)

(W)

ta

Mapping Petri Net to Event GraphFinally, increment and decrement tokens as state changes

{A++,Q1++}

{Q1--,R1--.B--}

{A--} {R1++,Q2++}

{R2--Q2--}

{R2++}

A Q1ta ts1

Q2

Bts20

W

ta ts1 0 ts2

~

~

~

~

~~

~(Q1&R1&B)

(Q1&R1&B)

(Q2&R2)

(Q2&R2)

(W)

R1 R2

{W++B++}

{W--}

(Q1&R1&B)

(A)

{A++,Q1++}

{Q1--,R1--.B--}

{A--} {R1++,Q2++}

{R2--Q2--}

{R2++}

ta ts1 0 ts2

~

~

~

~

~~

~

(Q1&R1&B)

(Q1&R1&B)

(Q2&R2)

(Q2&R2)

(W)

{W++B++}

{W--}

(Q1&R1&B)

(A)

Final Event Graph: How can this be simplified? (ref: Seila, et. al. Text)

{Q1++} {Q1--,R1--.B--}

{R1++,Q2++}

{R2--Q2--B++}

{R2++}

ta ts1ts2~

~

~

~

~

(R1&B)

(Q1&B)

(R2)

(Q2)

(Q1&R1)

A Simplified Event Relationship Graph

ImplicationsPN analysis of Event Graphs

State reachability LivenessDeadlock

EG analysis of Petri NetsState spaceModel reductionEquivalenceBoundary ConditionsResource Deadlock

Petri Net Simulator

EG -> PN Mapping

Remove-on-Fire Rule

Remove-on-Enable Rule

Ref: Peter Haas’ book on Stochastic Petri Nets – resets all timers each scan, prob. depositRemove on Fire rule – vs Remove on enable (Ref: Fishwick)Simulation – Activity Scanning algorithm. (Cancel if transition is disabled!)

Inhibitor arc

Examples of Petri Net Models – Activity Scanning Models

Multiple server queues?

Note: is an “event graph” – one out-transition per placeState dep deposit to d11 or d12...

tFM B “failing” – need “new” machine get broken machine

M,R R,BtR “fixing” – need broken machine

and repairman get good machine and repairman

“Modeling Activities with ERGs”

“Implied ERG”

tFM B

R,M B,RtR

{M--} {B++}M = “new” machinesB = broken machines waitingR = idle repairmen

{R--,B--}

{R++,M++}

~

~

(B)

(R)

“Implied ERG”tF

M B

R,M B,RtR

{M--} {B++}M = “new” machinesB = broken machinesR = idle repairmen

{R--,B--}

{R++,M++}

~

~

(B)

(R)

“Reduced ERG – M is not tested”

tF

B

R B,RtR

{B++}

{R--,B--}

{R++}

~~

(B)

(R)

Exercise: can you further reduce this? hint: assume more machines than repairmen. Define new variable(s).

tF

B

R B,RtR

{B++}

{R--,B--}

{R++}

~

~

(B)

(R)

tF

tR

{B++}

{B--}

~(B<=R)

Fix~(B>=R)

Fail

R = total number of repairmen (const.)B = number of broken machines (incl in rep.)

Colored Petri Nets (transitions are enabled by color or tokens)

d1,i,j = part i waitingOr processing on mach j

GSMP->EGM Mapping (Computer Network)

Prop.

Prop.Reset

NewPacket

Clear

EndTrans

Obs.End Tran.

Start Tran

Need Pg 397….

e1 depositsin d2 or d5with equal probabilty

Transition has two times – e3 or e5 with 50/50 probs.

NEED PAGE 137 for this example

1

Inhibitor arcs are note strictly necessary (but very convenient!)

Add d2 had token iff d1 emptyAnd no tokens if d1 has any.

Measuring DelaysNeed Example 1.4 of Chapter 2

Need figure 9.2 of section 2.6 without colors