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Mapping Petri Nets into Event Graphs

Mapping Petri Nets into Event Graphs. 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

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Mapping Petri Nets into Event Graphs

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  1. Mapping Petri Netsinto Event Graphs

  2. 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

  3. PN->EG MAPPING Transitions become timed edges Places become conditional edges

  4. PN->EG Mapping: eg. G/G/s queue S PN: Q A T T s a

  5. PN->EG 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--}

  6. PN->EG Mapping: eg. G/G/s queue S PN: Q A T T s a (A) EG: (S&Q) (S&Q) T T a s {A++, {A--} {Q--, {S++} Q++} S--}

  7. PN->EG Mapping: eg. G/G/s queue Variable A is unnecessary! (A) (S&Q) (S&Q) T T a s {A++, {A--} {Q--, {S++} Q++} S--}

  8. PN->EG Mapping: eg. G/G/s queue Empty Vertex is unnecessary! (S&Q) (S&Q) T T a s { Q++} {Q--, {S++} S--}

  9. PN->EG Mapping: eg. G/G/s queue S&Q conditions redundant too... (S&Q) (S&Q) T T a s { Q++} {Q--, {S++} S--}

  10. PN->EG 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--}

  11. 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.

  12. Analytical Methods and Conditions • PNs Reachabilty, decidability, liveness, and deadlock • EGs State definition, event reduction, priorities, equivalence, boundary conditions, resource deadlock, MIP representations

  13. 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

  14. R1 R2 A Q1 Q2 W ts2 ta ts1 0 B 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

  15. ts1 ts2 0 Each transition becomes a timed edge... R1 R2 A Q1 Q2 W ts2 ta ts1 0 B ta

  16. (Q2&R2) (Q1&R1&B) ~ ~ ~ (Q1&R1&B) (Q2&R2) 0 ts2 ta ts1 ~ ~ ~ (W) ~ 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) (Q1&R1&B)

  17. A Q1 Q2 W ts2 ta ts1 0 B Mapping Petri Net to Event Graph Finally, increment and decrement tokens as state changes R1 R2 ta (Q2&R2) (A) (Q1&R1&B) ~ ~ ~ (Q1&R1&B) (Q2&R2) 0 ts2 ta (W) ts1 ~ ~ ~ {A++, Q1++} {R2-- Q2--} {A--} {R1++, Q2++} {R2++} {Q1--, R1--. B--} {W++ B++} {W--} ~ (Q1&R1&B)

  18. (Q2&R2) (A) (Q1&R1&B) ~ ~ ~ (Q1&R1&B) (Q2&R2) 0 ts2 ta (W) ts1 ~ ~ ~ {A++, Q1++} {R2-- Q2--} {A--} {R1++, Q2++} {R2++} {Q1--, R1--. B--} {W++ B++} {W--} ~ (Q1&R1&B) Final Event Graph: How can this be simplified? (ref: Seila, et. al. Text)

  19. A Simplified Event Relationship Graph (Q2) (Q1&B) ~ ~ (R1&B) (R2) ta ts2 ts1 ~ ~ {R1++, Q2++} {R2-- Q2-- B++} {Q1--, R1--. B--} {Q1++} {R2++} ~ (Q1&R1)

  20. Implications • PN analysis of Event Graphs State reachability Liveness Deadlock • EG analysis of Petri Nets State space Model reduction Equivalence Boundary Conditions Resource Deadlock

  21. Petri Net Simulator

  22. Fail-Repair PN: EG: D a C B A t 0 t f r (D&B) t t f 0 (C) r (D&B) (A&!B)

  23. PN->EG Mapping Transformed EG Failure-Repair Model (A&&!B) (B&&D) / / / / (B&&D) (C) B-- A++ / / / / 0 A-- D++ C-- C++ B++ D-- / / (A&&!B) D=3

  24. PN->EG Mapping Reduced EG Failure-Repair Model t f t r Fail F i x {N++} {N- -}

  25. EG -> PN Mapping • Conditions: Need to have non-negative, integral state changes Test only for non-negativity of integers Needs to “edge bi-partite” pure-timed and pure- conditional edges alternate Single state change at each vertex - upstream of timed decrement, downstream of timed increment • Expansion Algorithm of Yucesan and Schruben Add conditions upstream when breaking up a self-scheduling edge (Yucesan and Schruben)

  26. EG -> PN Mapping 1. Split all self-scheduling edges (add conditions upstream of new vertex) 2. Add void edges (either conditional or timed but not both) to make graph “edge-bipartite” (expansion rules) 3. Timed edges become transitions and conditional edges become places

  27. EG -> PN Mapping EG->PN not possible ( C L K < 1 0 0 ) ( Q > 0 ) t a ( S > 0 ) t s ENTER START LEAVE { S = 1 } { Q = Q + 1 } { S = 0 , Q = Q - 1 }

  28. EG -> PN Mapping (Q) (S) t s START QUEUE FINISH t a {Q=Q - 1, {S=S+1} {Q=Q+1} S=S - 1}

  29. EG -> PN Mapping (Q) (S) t s START QUEUE FINISH t a {Q=Q - 1, {S=S+1} {Q=Q+1} (Q) S=S - 1} t a (S) t s ? START QUEUE FINISH “Edge Bipartite”

  30. EG -> PN Mapping (Q) (S) t s START QUEUE FINISH t a {Q=Q - 1, {S=S+1} {Q=Q+1} (Q) S=S - 1} t a (S) t s ? START QUEUE FINISH Timed Edges become Transitions t t s a

  31. EG -> PN Mapping (Q) (S) t s START QUEUE FINISH t a {Q=Q - 1, {S=S+1} {Q=Q+1} (Q) S=S - 1} t a (S) t s ? START QUEUE FINISH ??? Conditional Edges become PN places t t s a

  32. EG -> PN Mapping (Q) (S) t s START QUEUE FINISH t a {Q=Q - 1, {S=S+1} {Q=Q+1} (Q) S=S - 1} t a (S) t s ? START QUEUE FINISH S State changes on conditional edges determine labels ? Q t t s a

  33. COMMUNICATIONS BLOCKING: Edges Mapped from Petri Net (i) ~ (i) (ii) Run Enter Start1 ts1 End1 Start2 ~ ~ ta ts2 ~ (ii) ~ (i) End2 Conditions to Start “Activities”: (i )= (R1&B&Q1) (ii) = (R2&Q2)

  34. COMMUNICATIONS BLOCKING: State Changes {R1--, B--, Q1--} {R1=1, R2=1, B=4} {R1++, Q2++} {R2--, Q2--} {Q1--} Run Enter Start1 End1 Start2 End2 R1,R2 = Number of idle resources Q1,Q2 = Number of waiting jobs B = Number of empty buffer spaces {R2++, B++}

  35. COMMUNICATIONS BLOCKING: Final EGM Model {R1--, B--, Q1--} {R1=1, R2=1, B=4} {R1++, Q2++} {R2--, Q2--} (R1&B&Q1) {Q1--} ~ Run Enter Start1 End1 Start2 ts1 (R2&Q2) ~ ~ ta (R1&B&Q1) ts2 ~ ~ (R2&Q2) (R1&B&Q1) End2 R1,R2 = Number of idle resources Q1,Q2 = Number of waiting jobs B = Number of empty buffer spaces {R2++, B++}

  36. DELAY SEIZE RELEASE Queue Idle Resources Using Activity Scanning to Develop Process Models “Color” tokens that represent Transient Entities - - track these tokens’ paths. Parse Activities into “SEIZE”, “DELAY” and “RELEASE” Blocks.

  37. Process -> EG Mapping Process World View: • Automated SLAM to Event Graphs (Woodward and Mackulak, ASU) Resource deadlock detection • SIMAN to Event Graphs (Barton and Gonzalez, PSU) Premature run termination

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