Lecturer Sebastian Coope Ashton Building Room G18 Email coopesliverpoolacuk COMP 201 webpage httpwwwcsclivacukcoopescomp201 Lecture 9 10 Modelling Based on Petri Nets ID: 1001415
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1. Software EngineeringCOMP 201Lecturer: Sebastian CoopeAshton Building, Room G.18E-mail: coopes@liverpool.ac.uk COMP 201 web-page:http://www.csc.liv.ac.uk/~coopes/comp201Lecture 9, 10 – Modelling Based on Petri Nets
2. High-Level Petri NetsThe 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 80’s 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 withcolour (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 ModelA Petri net is a network composed of places ( ) and transitions ( ).t2p1p2p3p4t3t1Connections 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 ModelAnother (equivalent) notation is to use a solid bar for the transitions:t2p1p2p3p4t3t1We 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.p1p2p3p4t16COMP201 - Software EngineeringTransitions with Multiple Inputs and Outputs
7. Enabling ConditionTransitions are the active components and places and tokens are passive components.A transition is enabled if each of the input places contains tokens.t1t2Transition t1 is not enabled, transition t2 is enabled.7COMP201 - Software Engineering
8. FiringAn enabled transition may fire.Firing corresponds to consuming tokens from the input places and producing tokens for the output places.t2t2Firing is atomic (only one transition fires at a time, even if more than one is enabled)8COMP201 - Software Engineering
9. An Example Petri Net9COMP201 - Software Engineering
10. Example: Life-Cycle of a Personbachelorchildmarriedpubertymarriagedivorcedeathdead10COMP201 - Software Engineering
11. Creating/Consuming Tokens11COMP201 - Software EngineeringA transition without any input can fire at any time and produces tokens in the connected places:After firing 3 times..T1T1T1T1P1P1P1P1
12. Creating/Consuming Tokens12COMP201 - Software EngineeringA transition without any output must be enabled to fire and deletes (or consumes) the incoming token(s):After firing 3 times..T1T1T1T1P1P1P1P1
13. Non-Determinism in Petri NetsTwo 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).t1t213COMP201 - Software Engineering
14. ModellingStates 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 Light15COMP201 - Software Engineering
16. Modelling Two Traffic Lights16COMP201 - 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: A single traffic light turns from “Red” to “Green” to “Amber” and then back to “Red” (we’ll ignore “red and amber” for now). 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 Lightrgredambergreenyrgy17COMP201 - Software Engineering
18. Two Traffic Lightsrg1red1amber1green1yr1gy1rg2red2amber 2green2yr2gy218COMP201 - Software Engineering
19. Two Safe Traffic Lightsrg1red1amber1green1yr1gy1rg2red2amber 2green2yr2gy2safe19COMP201 - Software Engineering
20. Two Safe and Fair Traffic Lightsrg1red1yellow1green1yr1gy1rg2red2yellow2green2yr2gy2safe2safe120COMP201 - Software Engineering
21. Exercise1) 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. ExerciseCOMP201 - Software Engineering22
23. Arcs in Petri NetsThe 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.blackredbbrrbr23COMP201 - Software Engineering
24. Some DefinitionsCurrent 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. blackredbbrrbr24COMP201 - Software Engineering
25. Some DefinitionsIf 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.. blackredbbrrbr25COMP201 - Software Engineering
26. 7 reachable states, 1 deadlock state.blackredbbrrbr(3,2)(1,3)(3,1)(1,2)(3,0)(1,1)(1,0)rrrrrrbrbrbb\brbb\brbb\br26COMP201 - Software Engineering
27. Exercise: Readers and WritersHow 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)?restmail_boxreceive_mailtype_mailreadyrestbeginsend_mailread_mail27COMP201 - Software Engineering
28. ExerciseCOMP201 - Software Engineering28
29. The Four Seasons29COMP201 - Software EngineeringLet 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).
30. The Four Seasons30COMP201 - Software Engineering0SummerAutumnWinterSpring
31. The Four Seasons31COMP201 - Software Engineering0SummerAutumnWinterSpringHotColdDarkBright
32. High-Level Petri NetsIn 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:colourtimehierarchy32COMP201 - Software Engineering
33. To explain the three extensions we use the following example of a hairdresser's salon:startwaitingfinishbusyfreeclient waitinghairdresser ready to beginNote how easy it is to model the situation with multiple hairdressers..33COMP201 - Software EngineeringExample - High-Level Petri Netsfinished
34. The Extension with ColourA 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.startwaitingfinishbusyfreename: Harryage: 28experience: 2name: Sallyage: 28hairtype: BL34COMP201 - Software Engineeringfinished
35. 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.35COMP201 - Software EngineeringThe Extension with Colour
36. Examplesc := a+babc+b := -abnegaif a> 0then b:= aelse c:=afiabcselecta >=0 | b := Ö absqrtaExercise:calculate Ö |a+b| using these buiding blocks36COMP201 - Software Engineering
37. The Extension with TimeTo analyse performance, we must model durations, delays, etc.A timed Petri net associates a pair tmin and tmax with each transition (there are other possible definitions for timed Petri net, but we shall only consider this one).startwaitingfinishbusyfreeTmin = 0Tmax = 337COMP201 - Software EngineeringTmin = 5Tmax = 10finished
38. The Extension with TimeThe values tmin and tmax, 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.startwaitingfinishbusyfreeTmin = 0Tmax = 338COMP201 - Software EngineeringTmin = 5Tmax = 10finished
39. The Extension with TimeQuestion: 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?startwaitingfinishbusyfreefinishedTmin = 0Tmax = 339COMP201 - Software EngineeringTmin = 5Tmax = 10
40. ExerciseCOMP201 - Software Engineering40
41. The Extension with HierarchyA 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..41COMP201 - Software Engineering
42. The Extension with Hierarchywaitingreadyh1h2h3startfinishbusyfree42COMP201 - Software EngineeringHere we expand subnet h3..
43. Exercise: Remove Hierarchywaitingreadyh1h2h3startfinishbusyfreebeginendpendingbeginendpending43COMP201 - Software Engineering
44. Another ExampleRecall 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?44[1] - C. Ghezzi, M. Jazayeri, D. Mandrioli, “Fundamentals of Software Engineering”, Prentice Hall, Second Edition, page 196 - 198
45. Message TriplicationCOMP201 - Software Engineering45P1P2P3Original MessageTvoting1Tvoting2Tvoting3Message CopiesTmin = c1Tmax = k1Tmin = c2Tmax = k2Tvoting1: P1 = P2Tvoting2: P1 = P3Tvoting3: P2 = P3Tmin = c3Tmax = k3
46. Message Triplication (2)COMP201 - Software Engineering46P1P2P3Original MessageTvotingMessage CopiesTmin = c1Tmax = k1Tmin = c2Tmax = k2Tvoting: (P1 = P2) or (P2 = P3) or (P1 = P3) else “ERROR” Tmin = c3Tmax = k3
47. A Final Note on Petri NetsWe 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.47
48. A Final Note on Petri NetsImagine 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..48
49. Lecture Key PointsPetri 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 Engineering49