Livelock example. p 1. Assume this as initial marking. t 1 R. t 2. t 4. t 3. t 6. t 5. t 7. t 8. (taken from Fundamentals of SE by C.

Size: px

Start display at page:

Download "Livelock example. p 1. Assume this as initial marking. t 1 R. t 2. t 4. t 3. t 6. t 5. t 7. t 8. (taken from Fundamentals of SE by C."

Briana Booker
5 years ago
Views:

1 p 1 Livelock example (taken from Fundamentals of SE by C. Ghezzi) p 1 Assume this as initial marking. t 1 R t 3 t 4 t 5 t 6 t 7 t 8 Slide No.106

2 Livelock example analysis The tokens in R cannot be divided between transitions t 5 and t 6 any more - either one of the transitions will have both tokens at a given moment and the PN will evolve. Some possible firing sequences: <t 1, t 3, t 5, t 7, t 1, t 3,...> <, t 4, t 6, t 8,, t 4,...> <t 1,, t 3, t 4, t 5, t 7,, t 4,...> Slide No.107

3 Starvation example (taken from Fundamentals of SE by C. Ghezzi) t 1 p i t 3 t 4 Assume this as initial marking. Slide No.108

4 Starvation example analysis Starvation of the right side of the previous starvation example. The initial firing sequence could be one of the following: <t 1,, t 3, t 4 > or <, t 1, t 3, t 4 > or etc. At this point the part of the system will go into starvation and the repeated firing sequence will be: <t 1, t 3 > (once the token from p i is used up it can never be replenished) Slide No.109

5 PN extensions Assigning values to tokens Associating predicates and functions with transitions Prioritising transitions Timed PNs Slide No.110

6 Extending PNs (example) (1) Consider the following system described using an FSM: B_done wait digit 1 digit 0 state A A_done state B This situation cannot be adequately modelled using basic PN notation (see next slide). Slide No.111

7 Extending PNs (example) (2) p 1 t 3 p 2 t 1 p 3 t 4 System states: <p 1 >: wait <p 2 >: state A <p 3 >: state B IT IS POSSIBLE TO ASSUME THAT Firing sequence leading to state A: <t 1 > Firing sequence leading to state B: < >...BUT THIS IS NOT EVIDENT FROM THE ABOVE PN. Slide No.112

8 Assigning values to PN tokens Giving definite values to PN tokens could result in the following diagram: p t 3 t 1 t 4 p 2 p 3 It is now clear that the system accepts the digits 0 and 1. Which transitions t 1 or these fire is, however, still unclear. Slide No.113

9 Predicate association (2) However, if we extend the concept of PN transitions to include predicate association and associate predicate p 1 =1 with t 1 and predicate p 1 =0 with the situation becomes clear. p t 3 t 1 t 4 p p 1 =1 p 1 =0 2 p3 Slide No.114

10 Function association However, sometimes the tokens themselves get changed while passing through a transition in this way the tokens do not only serve as control markers but also as data carriers. In the example if we associate the function p 2 =p 1 and p 3 =p 1 then the tokens are left unchanged as follows. Firing sequence was: <t 1, > or <, t 1 > t 3 t 1 p 1 p p 1 =1 p 1 =0 2 p 3 t Slide No.115

11 Textbook example (taken from Fundamentals of SE by C. Ghezzi) Transition association table: trans. predicates functions t 1 p 2 >p 1 p 4 :=p 2 +p 1 p 1 p 2 p p 3 =p 2 p 4 :=p 3 -p 2 p 5 :=p 2 +p 3 t 1 Firing sequence and outcome: Starting <t 1 > p 4 :=7 or p 4 :=10 if p 4 :=7 then < > is disabled. if p 4 :=10 and < > then p 4 :=0 and p 5 :=8 Starting < > p 4 :=0 and p 5 :=8 after <t 1 > p 4 :=10 <t 1 >: non-determinate <, t 1 >: p 4 :=10 and p 5 :=8 p 4 p 5 Slide No.116

12 Prioritising PN transitions Re-definition of firing rule: If at any one moment more than one transition is enabled, then only the one with the highest priority can fire. Priorities are generally assigned in ascending integer order (1-least, 2-higher, etc.) Priorities can be defined either statically or dynamically as functions of transitions Slide No.117

13 PN prioritisation example p 1 p 2 p 3 p t 1 t 3 Assuming the following static priorities: t 1 : priority = 2; : priority = 1; t 3 : priority = 3 ONLY the following firing sequence is possible: <t 3, t 1, > Slide No.118

14 Introducing timing in PNs The idea is to associate a minimum and maximum response time (t min, t max ) with each transition. Any transition t i can fire if t i min< t < t i max Any timed PN regular PN if for each of its transitions t min = 0 and t max = Slide No.119

15 Timed PN example p 1 p 2 p 3 p Arrival of tokens 0 t 1 t min = 3 t max = 6 priority = 2 t min = 0 t max = 8 priority = 1 t 3 t min = 1 t max = 4 priority = 3 p 2 &p 4 < > <t 3 > <t 3 > <t 3 > < t 1 > < t 1 > < > < > none p 4 none <t 3 > <t 3 > <t 3 > none none none none none p 2 < > < > < > < t 1 > < t 1 > < t 1 > < > < > none none none none none none none none none none time Slide No.120

7. Queueing Systems. 8. Petri nets vs. State Automata

7. Queueing Systems. 8. Petri nets vs. State Automata Petri Nets 1. Finite State Automata 2. Petri net notation and definition (no dynamics) 3. Introducing State: Petri net marking 4. Petri net dynamics 5. Capacity Constrained Petri nets 6. Petri net models