1. Lecturer: Sebastian Coope Ashton Building, Room G.18 E-mail: coopes@liverpool.ac.uk COMP 201 web-page: http://www.csc.liv.ac.uk/~coopes/comp201 Lecture 9, 10 – Modelling Based on Petri Nets

2. High-Level Petri Nets The classical Petri net was invented by Carl Adam Petri in 1962. A lot of research has been conducted (>10,000 publications). Until 1985 it was mainly used by theoreticians. Since the 80s their practical use has increased because of the introduction of high-level Petri nets and the availability of many tools. High-level Petri nets are Petri nets extended with colour (for the modelling of attributes) time (for performance analysis) hierarchy (for the structuring of models, DFD's) 2COMP201 - Software Engineering

3. Why do we need Petri Nets? Petri Nets can be used to rigorously define a system (reducing ambiguity, making the operations of a system clear, allowing us to prove properties of a system etc.) They are often used for distributed systems (with several subsystems acting independently) and for systems with resource sharing. Since there may be more than one transition in the Petri Net active at the same time (and we do not know which will fire first), they are non-deterministic. 3COMP201 - Software Engineering

4. The Classical Petri Net Model A Petri net is a network composed of places ( ) and transitions ( ). t2 p1 p2 p3 p4 t3 t1 Connections are directed and between a place and a transition, or a transition and a place (e.g. Between p1 and t1 or t1 and p2 above). Tokens ( ) are the dynamic objects. 4COMP201 - Software Engineering

5. The Classical Petri Net Model Another (equivalent) notation is to use a solid bar for the transitions: t2 p1 p2 p3 p4 t3 t1 We may use either notation since they are equivalent, sometimes one makes the diagram easier to read than the other.. The state of a Petri net is determined by the distribution of tokens over the places (we could represent the above state as (1,2,1,1) for (p1,p2,p3,p4)) 5COMP201 - Software Engineering

6. Transition t1 has three input places (p1, p2 and p3) and two output places (p3 and p4). Place p3 is both an input and an output place of t1. p1 p2 p3 p4 t1 6COMP201 - Software Engineering Transitions with Multiple Inputs and Outputs

7. Enabling Condition Transitions are the active components and places and tokens are passive components. A transition is enabled if each of the input places contains tokens. t1t2 Transition t1 is not enabled, transition t2 is enabled. 7COMP201 - Software Engineering

8. Firing An enabled transition may fire. Firing corresponds to consuming tokens from the input places and producing tokens for the output places. t2 Firing is atomic (only one transition fires at a time, even if more than one is enabled) 8COMP201 - Software Engineering

9. An Example Petri Net 9COMP201 - Software Engineering

10. Example: Life-Cycle of a Person bachelor child married puberty marriage divorce deathdead 10COMP201 - Software Engineering

11. Creating/Consuming Tokens 11COMP201 - Software Engineering A transition without any input can fire at any time and produces tokens in the connected places: After firing 3 times.. T1 P1

12. Creating/Consuming Tokens 12COMP201 - Software Engineering A transition without any output must be enabled to fire and deletes (or consumes) the incoming token(s): After firing 3 times.. T1 P1

13. Non-Determinism in Petri Nets Two transitions fight for the same token: conflict. Even if there are two tokens, there is still a conflict. The next transition to fire (t1 or t2) is arbitrary (non-deterministic). t1 t2 13COMP201 - Software Engineering

14. Modelling States of a process can be modelled by tokens in places and state transitions leading from one state to another are modelled by transitions. Tokens can represent resources (humans, goods, machines), information, conditions or states of objects. Places represent buffers, channels, geographical locations, conditions or states. Transitions represent events, transformations or transportations. 14COMP201 - Software Engineering

15. Modelling a Traffic Light 15COMP201 - Software Engineering

16. Modelling Two Traffic Lights 16COMP201 - Software Engineering Imagine that we are designing a traffic light system for a crossroads junction (i.e. with two sets of (simplified) lights). An informal specification of a traffic light junction: o A single traffic light turns from Red to Green to Amber and then back to Red (well ignore red and amber for now). o There are two sets of lights. When one of the traffic lights is Amber or Green, the other must be Red. As a first step, we may decide to model the system as a Petri net. This allows us to make sure the specification is rigorously defined and reduces potential ambiguities later. We can also prove properties about the model if we wish.

17. Example: Traffic Light rg red amber green yr gy 17COMP201 - Software Engineering

18. Two Traffic Lights rg1 red1 amber1 green1 yr1 gy1 rg2 red2 amber 2 green2 yr2 gy2 18COMP201 - Software Engineering

19. Two Safe Traffic Lights rg1 red1 amber1 green1 yr1 gy1 rg2 red2 amber 2 green2 yr2 gy2 safe 19COMP201 - Software Engineering

20. Two Safe and Fair Traffic Lights rg1 red1 yellow1 green1 yr1 gy1 rg2 red2 yellow2 green2 yr2 gy2 safe2 safe1 20COMP201 - Software Engineering

21. Exercise 1) Can you prove that the Petri net from the previous slide will never allow two red lights to be shown simultaneously? 21COMP201 - Software Engineering

22. Exercise COMP201 - Software Engineering22

23. Arcs in Petri Nets The number of arcs between two objects specifies the number of tokens to be produced/consumed (we can alternatively represent this by writing a number next to a single arc). This can be used to model (dis)assembly processes. blackred bbrr br 23COMP201 - Software Engineering

24. Some Definitions Current state (also called current marking) - The configuration of tokens over the places. Reachable state - A state reachable form the current state by firing a sequence of enabled transitions. Deadlock state - A state where no transition is enabled. blackred bbrr br 24 COMP201 - Software Engineering

25. Some Definitions If we write the places in some fixed order (red, black say), then we can use a tuple: (n,m) to denote the number of tokens in each corresponding place (n tokens in red and m tokens in black). The example below is thus in state (3,2). After firing transition rr, it will move to state (1,3) etc.. blackred bbrr br 25COMP201 - Software Engineering

26. 7 reachable states, 1 deadlock state. blackred bbrr br (3,2) (1,3)(3,1) (1,2)(3,0) (1,1) (1,0) rr br bb\br 26COMP201 - Software Engineering

27. Example: Simple Vending Machine Is there a deadlock state? How could a cancel button be simulated? (i.e. To return the persons money) 10p 27COMP201 - Software Engineering 20p 30p40p50p eat Deposit 10p Deposit 20p

28. Exercise: Readers and Writers How many states are reachable? Are there any deadlock states? How to model the situation with 2 writers and 3 readers? How to model a "bounded mailbox" (buffer size =4)? rest mail_box receive_mail type_mail ready rest begin send_mail read_mail 28COMP201 - Software Engineering

29. Exercise COMP201 - Software Engineering29

30. The Four Seasons 30COMP201 - Software Engineering Let us try to model the four seasons of the year together with their properties by a Petri net. We would like to denote the current season {spring, summer, autumn, winter}, the temperature {hot, cold} and the light level {bright, dark}. As a first step, let us model the seasons (with a token to represent that it is currently autumn).

31. The Four Seasons 31COMP201 - Software Engineering 0 Summer Autumn Winter Spring

32. The Four Seasons 32COMP201 - Software Engineering 0 Summer Autumn Winter Spring Hot Cold Dark Bright

33. High-Level Petri Nets In practice, classical Petri nets have some modelling problems: The Petri net becomes too large and too complex. It takes too much time to model a given situation. It is not possible to handle time and data. Therefore, we use high-level Petri nets, i.e. Petri nets extended with: colour time hierarchy 33COMP201 - Software Engineering

34. To explain the three extensions we use the following example of a hairdresser's salon : start waiting finish busy free client waiting hairdresser ready to begin Note how easy it is to model the situation with multiple hairdressers.. 34COMP201 - Software Engineering Example - High-Level Petri Nets finished

35. The Extension with Colour A token often represents an object having all kinds of attributes. Therefore, each token has a value (colour) with refers to specific features of the object modelled by the token. start waiting finish busy free name: Harry age: 28 experience: 2 name: Sally age: 28 hairtype: BL 35COMP201 - Software Engineering finished

36. Each transition has an (in)formal specification which specifies: the number of tokens to be produced, the values of these tokens, and (optionally) a precondition. The complexity is divided over the network and the values of tokens. This results in a compact, manageable and natural process description. 36COMP201 - Software Engineering The Extension with Colour

37. Examples c := a+b a b c + b := -a b neg a if a> 0 then b:= a else c:=a fi a b c select a >=0 | b := a b sqrt a Exercise: calculate |a+b| using these buiding blocks 37COMP201 - Software Engineering

38. The Extension with Time To analyse performance, we must model durations, delays, etc. A timed Petri net associates a pair t min and t max with each transition (there are other possible definitions for timed Petri net, but we shall only consider this one). start waiting finish busy free Tmin = 0 Tmax = 3 38COMP201 - Software Engineering Tmin = 5 Tmax = 10 finished

39. The Extension with Time The values t min and t max, tell us the minimum and maximum time that a transition will take to fire once enabled. This allows us to model performance properties of the system, although the analysis of such systems may be more difficult. start waiting finish busy free Tmin = 0 Tmax = 3 39COMP201 - Software Engineering Tmin = 5 Tmax = 10 finished

40. The Extension with Time Question: What is the minimum/maximum time for all three people to have their hair cut in this system? (Harder) Question: What about with n clients and m hairdressers? Is there a general formula for the required time? start waiting finish busy free finished Tmin = 0 Tmax = 3 40COMP201 - Software Engineering Tmin = 5 Tmax = 10

41. Exercise COMP201 - Software Engineering41

42. The Extension with Hierarchy A hierarchy is a mechanism to structure complex Petri nets comparable to Data Flow Diagrams. A subnet is a net composed out of places, transitions and other subnets. This allows us to model a system at different levels of abstraction and can reduce the complexity of the model. We shall see an example of this on the next slide.. 42COMP201 - Software Engineering

43. The Extension with Hierarchy waitingready h1 h2 h3 startfinishbusy free 43COMP201 - Software Engineering Here we expand subnet h3..

44. Exercise: Remove Hierarchy waitingready h1 h2 h3 startfinishbusy free beginendpending beginendpending 44COMP201 - Software Engineering

45. Another Example Recall the following example of an informal specification from a critical system [1] : The message must be triplicated. The three copies must be forwarded through three different physical channels. The receiver accepts the message on the basis of a two-out-of-three voting policy. Questions: Can you identify any ambiguities in this specification? How could we model this system with a Petri net? 45 [1] - C. Ghezzi, M. Jazayeri, D. Mandrioli, Fundamentals of Software Engineering, Prentice Hall, Second Edition, page 196 - 198

46. Message Triplication COMP201 - Software Engineering46 P1 P2 P3 Original Message Tvoting1 Tvoting2Tvoting3 Message Copies Tmin = c1 Tmax = k1 Tmin = c2 Tmax = k2 Tvoting1: P1 = P2 Tvoting2: P1 = P3 Tvoting3: P2 = P3 Tvoting1: P1 = P2 Tvoting2: P1 = P3 Tvoting3: P2 = P3 Tmin = c3 Tmax = k3

47. Message Triplication (2) COMP201 - Software Engineering47 P1 P2 P3 Original Message Tvoting Message Copies Tmin = c1 Tmax = k1 Tmin = c2 Tmax = k2 Tvoting: (P1 = P2) or (P2 = P3) or (P1 = P3) else ERROR Tmin = c3 Tmax = k3

48. A Final Note on Petri Nets We can see from the previous example that the ambiguity (or impreciseness) in the informal specification for the message triplication protocol is clearly highlighted by the more formal Petri net model. We can also perform some analysis on the model itself, for example to see if certain bad states ever occur or if deadlock/livelock is possible in the model. Finally we can represent timing constraints (to encode even more constraints on the system) and use hierarchical models to show different levels of abstration. 48

49. A Final Note on Petri Nets Imagine modelling the elevator system of a skyscraper which contains three elevators and twenty floors. What would be some of the advantages of using a Petri net model for this? We can ensure if someone at a floor pushes the lift button (up or down), the elevator will eventually come. We can attempt to model the timing constraints of the system (Timed Petri net). We can also use hierarchies to simplify the system. Finally we could try to optimize the model in some way if its performance is not optimal. Etc.. 49

50. Lecture Key Points Petri nets have Arcs, Places and Transitions. Petri nets are non-deterministic and thus may be used to model discrete distributed systems. They have a well defined semantics and many variations and extensions of Petri nets exist. The state or marking of a net is an assignment of tokens to places. For those interested, the book Fundamentals of Software Engineering (Prentice Hall) by C. Ghezzi, M. Jazayeri and D. Mandrioli has an extensive example of using Petri nets for an elevator system. COMP201 - Software Engineering50