Petri Net :Abstract formal model of information flow Major use: Modeling of systems of events in which it is possible for some events to occur concurrently, but there are constraints on the occurrences, precedence, or frequency of these occurrences. Petri Net 30

Petri Net as a Graph :Models static properties of a system Graph contains 2 types of nodes Circles (Places) Bars (Transitions) Petri net has dynamic properties that result from its execution Markers (Tokens) Tokens are moved by the firing of transitions of the net. Petri Net 30

Petri Net as a Graph (cont.) (Figure 1) A simple graph representation of a Petri net. Petri Net 30

Petri Net as a Graph (cont.) (Figure 2) A marked Petri net. Petri Net 30

Petri Net as a Graph (cont.) (Figure 3) The marking resulting from firing transition t2 in Figure 2. Note that the token in p1 was removed and tokens were added to p2 and p3 Petri Net 30

Petri Net as a Graph (cont.) (Figure 4) Markings resulting from the firing of different transitions in the net of Figure 3. (a) Result of firing transition t1 Petri Net 30

Petri Net as a Graph (cont.) (Figure 4) Markings resulting from the firing of different transitions in the net of Figure 3. (b) Result of firing transition t3 Petri Net 30

Petri Net as a Graph (cont.) (Figure 4) Markings resulting from the firing of different transitions in the net of Figure 3. (c) Result of firing transition t5 Petri Net 30

Petri Net as a Graph (cont.) (Figure 5) A simple model of three conditions and an event Petri Net 30

(Figure 6) Modeling of a simple computer system Petri Net 30

Petri Net as a Graph (cont.) (Figure 7) Modeling of a nonprimitive event Petri Net 30

Petri Net as a Graph (cont.) (Figure 8) Modeling of “simultaneous” which may occur in either order Petri Net 30

Petri Net as a Graph (cont.) (Figure 9) Illustration of conflicting transitions. Transitions tj and tk conflict since the firing of one will disable the other Petri Net 30

Petri Net as a Graph (cont.) (Figure 10) An uninterpreted Petri net. Petri Net 30

(Figure 11) Hierarchical modeling in Petri nets by replacing places or transitions by subnets (or vice versa). Petri Net 30

(Figure 12) A portion of a Petri net modeling a control unit for a computer with multiple registers and multiple functional units Petri Net 30

(Figure 13) Representation of an asynchronous pipelined control unit. The block diagram on the left is modeled by the Petri net on the right Petri Net 30

Petri Net as a Graph (cont.) 30

(Figure 15) A Petri net model of a P/V solution to the mutual exclusion problem Petri Net 30

(Figure 16) L: S0 Do while P0 if P2 then S1 else S2 endif parbegin Example of a Petri net used to represent the flow of control in programs containing certain kind of constructs L: S0 Do while P0 if P2 then S1 else S2 endif parbegin S3,S4,S5, parend enddo goto L Petri Net 30

(Figure 17) A Petri net model for protocol 3 Petri Net 30

Other properties for analysis Boundeness Safe net (bound = 1) K-bounded net Conservation ==> conservative net Live transition Dead transition Petri Net 30

State of a Petri net State - defined by its marking, m State space - set of all markings: (m0, m1, m2, ...) Change in state - caused by firing a transition, defined by partial Fn, d (example) m1 = d (m0 , tj) Note: marking -- For a marking m , m(Pi) = mi A marked Petri net: m = (P, T, I, O, m) Petri Net 30

m0 = (1, 0, 1, 0, 2) d(m0, t3) = (1, 0, 0, 1, 2) = m1 d(m1, t4) = (1, 1, 1, 0, 2) = m2 etc. Petri Net 30

(Figure 19) A Petri net with a nonfirable transition. Transition t3 is dead in this marking Petri Net 30

Petri Net as a Graph (cont.) 30

Petri Net as a Graph (cont.) (1, 0, 1, 0) (1, 0, 0, 1) (1, w, 1, 0) (1, w, 0, 0) (1, w, 0, 1) t3 t2 t1 (Figure 21) The reachability tree of the Petri net of Figure 19 Petri Net 30

Unsolvable Problems Subset problem - given 2 marked Petri nets, is the reachability of one net a subset of the reachability of the other net undecidable (Hack) ...... Complexity reachability problem is exponential time-hard and exponential space-hard. Petri Net 30