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Petri nets Classical Petri nets: The basic model 1 Process modeling • Emphasis on dynamic behavior rather than structuring the state space • Transition system is too low level • We start with the classical Petri net • Then we extend it with: – Color – Time – Hierarchy 2 Classical Petri net • Simple process model – Just three elements: places, transitions and arcs. – Graphical and mathematical description. – Formal semantics and allows for analysis. • History: – Carl Adam Petri (1962, PhD thesis) – In sixties and seventies focus mainly on theory. – Since eighties also focus on tools and applications (cf. CPN work by Kurt Jensen). – “Hidden”inmanydiagrammingtechniquesand systems. 3 p4 place Elements t34 t43 transition (name) p3 place t23 t32 p2 (name) transition token arc (directed connection) t12 t21 token p1 t01 t10 p0 4 free Rules wait enter before make_picture after leave gone occupied • Connections are directed. • No connections between two places or two transitions. • Places may hold zero or more tokens. • First, we consider the case of at most one arc between two nodes. 5 Enabled • A transition is enabled if each of its input places contains at least one token. free wait enter before make_picture after leave gone occupied enabled Not Not enabled enabled 6 Firing • An enabled transition can fire (i.e., it occurs). • When it fires it consumes a token from each input place and produces a token for each output place. free fired wait enter before make_picture after leave gone occupied 7 Play“TokenGame” • In the new state, make_picture is enabled. It will fire, etc. free wait enter before make_picture after leave gone occupied 8 Remarks • Firing is atomic. • Multiple transitions may be enabled, but only one fires at a time, i.e., we assume interleaving semantics (cf. diamond rule). • The number of tokens may vary if there are transitions for which the number of input places is not equal to the number of output places. • The network is static. • The state is represented by the distribution of tokens over places (also referred to as marking). 9 p4 p4 Non-determinism t34 t43 t34 t43 p3 p3 t23 t32 t23 t32 transition t23 p2 fires p2 t12 t21 t12 t21 p1 p1 Two transitions are enabled but only t01 t10 t01 t10 one can fire p0 p0 10 Example: Single traffic light rg green red go orange or 11 Two traffic lights rg rg rg green green green red go red go OR red go orange orange orange or or or 12 Problem 13 Solution rg1 rg2 g1 g2 r1 go1 x go2 r2 How to make o1 o2 them alternate? or1 or2 14 Playingthe“TokenGame”ontheInternet • Applet to build your own Petri nets and execute them: http://www.tm.tue.nl/it/staff/wvdaalst/Downloads/ pn_applet/pn_applet.html • FLASH animations: http://www.tm.tue.nl/it/staff/wvdaalst/courses/pm /flash/ 15 Exercise: Train system (1) • Consider a circular railroad system with 4 (one- way) tracks (1,2,3,4) and 2 trains (A,B). No two trains should be at the same track at the same time and we do not care about the identities of the two trains. 16 Exercise: Train system (2) • Consider a railroad system with 4 tracks (1,2,3,4) and 2 trains (A,B). No two trains should be at the same track at the same time and we want to distinguish the two trains. 17 Exercise: Train system (3) • Consider a railroad system with 4 tracks (1,2,3,4) and 2 trains (A,B). No two trains should be at the same track at the same time. Moreover the next track should also be free to allow for a safe distance. (We do not care about train identities.) 18 Exercise: Train system (4) • Consider a railroad system with 4 tracks (1,2,3,4) and 2 trains. Tracks are free, busy or claimed. Trains need to claim the next track before entering. 19 WARNING It is not sufficient to understand the (process) models. You have to be able to design them yourself ! 20 Multiple arcs connecting two nodes • The number of arcs between an input place and a transition determines the number of tokens required to be enabled. • The number of arcs determines the number of tokens to be consumed/produced. free wait enter before make_picture after leave gone 21 Example: Ball game red black rb rr bb 22 Exercise: Manufacturing a chair • Model the manufacturing of a chair from its components: 2 front legs, 2 back legs, 3 cross bars, 1 seat frame, and 1 seat cushion as a Petri net. • Select some sensible assembly order. • Reverse logistics? 23 Exercise: Burning alcohol. • Model C2H5OH + 3 * O2 => 2 * CO2 + 3 * H2O • Assume that there are two steps: first each molecule is disassembled into its atoms and then these atoms are assembled into other molecules. 24 Exercise: Manufacturing a car • Model the production process shown in the Bill- Of-Materials. car subassembly2 engine 2 chair subassembly1 4 chassis wheel 25 Formal definition A classical Petri net is a four-tuple (P,T,I,O) where: • P is a finite set of places, • T is a finite set of transitions, • I : P x T -> N is the input function, and • O : T x P -> N is the output function. Any diagram can be mapped onto such a four tuple and vice versa. 26 Formal definition (2) The state (marking) of a Petri net (P,T,I,O) is defined as follows: • s: P-> N, i.e., a function mapping the set of placesonto.}…,2,1,0{ 27 Exercise: Map onto (P,T,I,O) and S red black rb rr bb 28 Exercise: Draw diagram Petri net (P,T,I,O): • P = {a,b,c,d} • T = {e,f} • I(a,e)=1, I(b,e)=2, I(c,e)=0, I(d,e)=0, I(a,f)=0, I(b,f)=0, I(c,f)=1, I(d,f)=0. • O(e,a)=0, O(e,b)=0, O(e,c)=1, O(e,d)=0, O(f,a)=0, O(f,b)=2, O(f,c)=0, O(f,d)=3. State s: • s(a)=1, s(b)=2, s(c)=0, s(d) = 0. 29 Enabling formalized Transition t is enabled in state s1 if and only if: 30 Firing formalized If transition t is enabled in state s1, it can fire and the resulting state is s2 : 31 Mapping Petri nets onto transition systems A Petri net (P,T,I,O) defines the following transition system (S,TR): 32 Reachability graph • The reachability graph of a Petri net is the part of the transition system reachable from the initial state in graph-like notation. • The reachability graph can be calculated as follows: 1. Let X be the set containing just the initial state and let Y be the empty set. 2. Take an element x of X and add this to Y. Calculate all states reachable for x by firing some enabled transition. Each successor state that is not in Y is added to X. 3. If X is empty stop, otherwise goto 2. 33 Example red black rb (3,2) (3,1) (3,0) (1,3) (1,2) (1,1) rr bb (1,0) Nodes in the reachability graph can be represented by a vector”)2,3(“oras 3“red + 2 black”.Thelatterisusefulfor “sparsestates”(i.e.,fewplacesaremarked). 34 Exercise: Give the reachability graph using both notations rg1 rg2 g1 g2 r1 go1 x go2 r2 o1 o2 or1 or2 35 Different types of states • Initial state: Initial distribution of tokens. • Reachable state: Reachable from initial state. • Final state (alsoreferredtoas“deadstates”): No transition is enabled. • Home state (also referred to as home marking): It is always possible to return (i.e., it is reachable from any reachable state). How to recognize these states in the reachability graph? 36 Exercise: Producers and consumers • Model a process with one producer and one consumer, both are either busy or free and alternate between these two states. After every production cycle the producer puts a product in a buffer. The consumer consumes one product from this buffer per cycle. • Give the reachability graph and indicate the final states. • How to model 4 producers and 3 consumers connected through a single buffer? • How to limit the size of the buffer to 4? 37 Exercise: Two switches • Consider a room with two switches and one light. The light is on or off. The switches are in state up or down. At any time any of the switches can be used to turn the light on or off. • Model this as a Petri net. • Give the reachability graph. 38 Modeling • Place: passive element • Transition: active element • Arc: causal relation • Token: elements subject to change The state (space) of a process/system is modeled by places and tokens and state transitions are modeled by transitions (cf. transition systems). 39 Role of a token Tokens can play the following roles: • a physical object, for example a product, a part, a drug, a person; • an information object, for example a message, a signal, a report; • a collection of objects, for example a truck with products, a warehouse with parts, or an address file; • an indicator of a state, for example the indicator of the state in which a process is, or the state of an object; • an indicator of a condition: the presence of a token indicates whether a certain condition is fulfilled. 40 Role of a place • a type of communication medium, like a telephone line, a middleman, or a communication network; • a buffer: for example, a depot, a queue or a post bin; • a geographical location, like a place in a warehouse, office or hospital; • a possible state or state condition: for example, the floor where an elevator is, or the condition that a specialist is available. 41 Role of a transition • an event: for example, starting an operation, the death of a patient, a change seasons or the switching of a traffic light from red to green; • a transformation of an object, like adapting a product, updating a database, or updating a document; • a transport of an object: for example, transporting goods, or sending a file. 42 Typical network structures • Causality • Parallelism (AND-split - AND-join) • Choice (XOR-split – XOR-join) • Iteration (XOR-join - XOR-split) • Capacity constraints – Feedback loop – Mutual exclusion – Alternating 43 Causality 44 Parallelism 45 Parallelism: AND-split 46 Parallelism: AND-join 47 Choice: XOR-split 48 Choice: XOR-join 49 Iteration: 1 or more times XOR-join before XOR-split 50 Iteration: 0 or more times XOR-join before XOR-split 51 Capacity constraints: feedback loop AND-join before AND-split 52 Capacity constraints: mutual exclusion AND-join before AND-split 53 Capacity constraints: alternating AND-join before AND-split 54 We have seen most patterns, e.g.: rg1 rg2 Example of mutual exclusion g1 g2 r1 go1 x go2 r2 How to make o1 o2 them alternate? or1 or2 55 Exercise: Manufacturing a car (2) • Model the production process shown in the Bill-Of-Materials with car resources. subassembly2 • Each assembly step engine requires a dedicated machine and an 2 operator. chair • There are two operators subassembly1 and one machine of 4 each type. chassis wheel • Hint: model both the start and completion of an assembly step. 56 Modeling problem (1): Zero testing • Transition t should fire if place p is empty. t ? p 57 Solution • Only works if place is N-bounded N input and t Initially there output arcs are N tokens p’ p 58 Modeling problem (2): Priority • Transition t1 has priority over t2 t1 ? t2 Hint: similar to Zero testing! 59 A bit of theory • Extensions have been proposed to tackle these problems, e.g., inhibitor arcs. • These extensions extend the modeling power (Turing completeness*). • Without such an extension not Turing complete. • Still certain questions are difficult/expensive to answer or even undecidable (e.g., equivalence of two nets). * Turing completeness corresponds to the ability to execute any computation. 60 Exercise: Witness statements • As part of the process of handling insurance claims there is the handling of witness statements. • There may be 0-10 witnesses per claim. After an initialization step (one per claim), each of the witnesses is registered, contacted, and informed (i.e., 0-10 per claim in parallel). Only after all witness statements have been processed a report is made (one per claim). • Model this in terms of a Petri net. 61 Exercise: Dining philosophers • 5 philosophers sharing 5 chopsticks: chopsticks are located in-between philosophers • A philosopher is either in state eating or thinking and needs two chopsticks to eat. • Model as a Petri net. 62 High level Petri nets Extending classical Petri nets with color, time and hierarchy (informal introduction) Prof.dr.ir. Wil van der Aalst Eindhoven University of Technology, Faculty of Technology Management, Department of Information and Technology, P.O.Box 513, NL-5600 MB, Eindhoven, The Netherlands. 63 Limitations of classical Petri nets • Inability to test for zero tokens in a place. • Models tend to become large. • Models cannot reflect temporal aspects • No support for structuring large models, cf. top- down and bottom-up design 64 Inability to test for zero tokens in a place t ? p “Tricks”onlyworkifpisbounded 65 Models tend to become (too) large r1 r2 r3 r4 r5 incr1 incr2 incr3 incr4 incr5 wheel bell steering bike frame wheel decr1 decr2 decr3 decr4 decr5 Size linear in the number of l1 l2 l3 l4 l5 products. 66 Models tend to become (too) large (2) 1 b 2 b 3 b 4 b rn r f tase rn r f tase tase rn r f rn r f tase 1 c 2 c 3 c 4 c lm c a t r c i _ak 1 f 2 f 3 f 4 f lm c a t c i _ak r a t c i _ak lm c r lm c c i _ak a t r Size linear in the number of tracks. 67 Models cannot reflect temporal aspects rg1 rg2 g1 g2 r1 go1 x go2 r2 Duration of each o1 o2 phase is highly relevant. or1 or2 68 No support for structuring large models rg1 rg2 g1 g2 r1 go1 x go2 r2 o1 o2 or1 or2 69 High-level Petri nets • To tackle the problems identified. • Petri nets extended with: – Color (i.e., data) – Time – Hierarchy • For the time being be do not choose a concrete language but focus on the main concepts. • Later we focus on a concrete language: CPN. • These concepts are supported by many variants of CPN including ExSpect, CPN AMI, etc. 70 Running example: Making punch cards free free desk employees wait done start stop busy waiting served patients patient/ patients employees 71 Extension with color (1) • Tokens have a color (i.e., a data value) {Brand="BMW", RegistrationNo="GD-XW-11", Year=1993, Colour="blue", Owner= "Inge"} {Brand="Lada", RegistrationNo="PH-14-PX", Year=1986, Color="grey", Owner="Inge"} 72 Extension with color (2) • Places are typed (also referred to as color set). record Brand:string * RegistrationNo:string * Year:int * Color:string * Owner:string {Brand="BMW", RegistrationNo="GD-XW-11", Year=1993, Colour="blue", Owner= "Inge"} {Brand="Lada", RegistrationNo="PH-14-PX", Year=1986, Color="grey", Owner="Inge"} 73 Extension with color (3) • The relation between production and consumption needs to be specified, i.e., the value of a produced token needs to be related to the values of consumed tokens. 3 in sum 2 add 0 3 1 The value of the token produced for place sum is the sum of the values of the consumed tokens. 74 Running example: Tokens are colored {EmpNo=641112, Experience=7} wait done free start stop busy {Name="Klaas", Address="Plein 10", DateOfBirth="13-Dec-1962", Gender="M"} 75 Running example: Places are typed record EmpNo:int * Experience:int wait done free start stop record Name:string * record Name:string * Address:string * Address:string * busy DateOfBirth:str * record Name:string * DateOfBirth:str * Address:string * Gender:string Gender:string DateOfBirth:str * Gender:string * EmpNo:int * Experience:int 76 Running example: Initial state {EmpNo=641112, Experience=7} wait done free start stop busy {Name="Klaas", Address="Plein 10", DateOfBirth="13-Dec-1962", Gender="M"} start is enabled 77 Running example: Transition start fired New value is created by simply merging the two records. wait done free start stop busy {Name="Klaas", Address="Plein 10", DateOfBirth="13-Dec-1962", Gender="M", EmpNo=641112, Experience=7} stop is enabled 78 Running example: Transition stop fired {EmpNo=641112, Experience=7} wait done free start stop busy {Name="Klaas", Address="Plein 10", DateOfBirth="13-Dec-1962", Gender="M"} New values are created by simply spliting the record into two parts. 79 The number of tokens produced is no longer fixed (1) {sample_number=931101011, measurement_outcomes="XYV"} positive test sample negative {sample_number=931101023, measurement_outcomes="VXY"} Note that the network structure is no longer a complete specification! 80 The number of tokens produced is no longer fixed (2) {sample_number=931101011, measurement_outcomes="XYV"} positive test sample negative The number of tokens produced for each output place is between 0 and 3 and the sum should be 3. 81 r1 r2 r3 r4 r5 Example incr1 incr2 incr3 incr4 incr5 wheel bell steering bike frame wheel decr1 decr2 decr3 decr4 decr5 l1 l2 l3 l4 l5 Model as a colored Petri net. 82 in {prod="bell", num=2} Product and quantity are in the value of the increase token [{prod="bike", num=4}, stock {prod="wheel", num=2}, The entire {prod="bell", number=3}, {prod="steering wheel", num=3}, stock is {prod="frame", num=2}] represented by decrease the value of a single token, i.e., a list of records. out 83 in {prod="bell", num=2} Types StockItem increase [{prod="bike", num=4}, stock {prod="wheel", num=2}, {prod="bell", number=3}, {prod="steering wheel", num=3}, color Product = string; Stock {prod="frame", num=2}] color Number = int; decrease color StockItem = record prod:Product * num:Number; color Stock = list StockItem; StockItem out 84 Extension with time (1) • Each token has a timestamp. • The timestamp specifies the earliest time when it can be consumed. 2 5 85 Extension with time (2) • The enabling time of a transition is the maximum of the tokens to be consumed. • If there are multiple tokens in a place, the earliest ones are consumed first. • A transition with the smallest firing time will fire first. • Transitions are eager, i.e., they fire as soon as they can. • Produced token may have a delay. • The timestamp of a produced token is the firing time plus its delay. 86 Running example: Enabling time • Transition start is enabled at time 2 = max{0,min{2,4,4}}. 0 4 wait done 2 free start stop 4 busy 87 Running example: Delays • Tokens for place busy get a delay of 3 • @+3 = firing time plus 3 time units 0 4 @+0 wait done 2 free start stop @+3 @+0 4 busy 88 Running example: Transition start fired • Transition start fired a time 2. 4 wait @+0 done free start stop @+3 @+0 5 4 busy Continuetoplay(timed)tokengame… 89 Exercise: Final state? a d 1 x @+1 3 b 5 c e y @+2 4 90 Exercise: Final state? 3 red black @+2 1 4 rb 2 2 @+1 @+1 rr bb 91 Extension with hierarchy • Timed and colored Petri nets result in more compact models. • However, for complex systems/processes the model does not fit on a single page. • Moreover, putting things at the same level does not reflect the structure of the process/system. • Many hierarchy concepts are possible. In this course we restrict ourselves to transition refinement. 92 Instead of rg1 rg2 g1 g2 r1 go1 x go2 r2 o1 o2 or1 or2 93 We can use hierarchy tl1 tl2 x rg1 rg2 g1 g2 r1 go1 go2 r2 x x o1 o2 or1 or2 94 tl1 tl2 Reuse x • Reuse saves design efforts. rg • Hierarchy can have any number of g levels • Transition go r refinement can be x used for top-down and bottom-up o design or 95 Exercise: model three (parallel) punch card desks in a hierarchical manner free wait done start stop busy 96 Analysis of Process Models: Reachability graphs, invariants, and simulation Prof.dr.ir. Wil van der Aalst Eindhoven University of Technology, Faculty of Technology Management, Department of Information and Technology, P.O.Box 513, NL-5600 MB, Eindhoven, The Netherlands. 97 Questions raised when considering the handling of customer orders • How many orders arrive on average? • How many orders can be handled? • Do orders get lost? • Do back orders always have priority? • What is the utilization of office workers? • If the desired product is no longer available, does the order get stuck? • Etc. 98 Questions raised when considering the handling of customers in the canteen • What is the average waiting time from 12.30- 13.00? • What is the variance of waiting times? • What is the effect of an additional cashier on the queue length? • Etc. 99 Questions raised when considering the an intersection with multiple traffic lights • How much traffic can be handled per hour? • Give some volume of traffic, what is the probability to get a red light? • Is the intersection safe, i.e., crossing flows have never a green light at the same time? • Can a light go from yellow to green? • Is the intersection fair (i.e., a traffic light cannot turn green twice while cars are waiting on the other side)? 100 Questions raised when considering a printer shared by multiple users • Can two print jobs get mixed? • Do small jobs always get priority? • Can the settings of one job influence the next job? • Do out-of-paper events cause jobs to get lost? • How many jobs can be handled per day? • What is the probability of a paper jam? 101 Questions raised when considering a teller machine • What is the average response time? • Is there a balance, i.e., the amount of money leaving the machine matches the amount taken from bank accounts? • How often should the machine be filled to guarantee 90% availability? • Is fraud possible? • Etc. 102 Analysis • Analysis is typically model- driven to allow e.g. what-if questions. • Models of both operational Operational process processes and/or the Information System information systems can be analyzed. • Types of analysis: Model – validation – verification – performance analysis 103 Three analysis techniques (Chapter 8) • Reachability graph • Place & transition invariants • Simulation • Each can be applied to both classical and high-level Petri nets. Nevertheless, for the first two we restrict ourselves to the classical Petri nets. • Use: – reachability graph (validation, verification) – invariants (validation, verification) – simulation (validation, performance analysis) 104 Reachability graph rg1 rg2 (0,0,1,1,0,0,0) (1,0,0,0,0,1,0) g1 g2 r1 go1 x go2 r2 (1,0,0,1,0,0,1) o1 o2 (0,1,0,1,0,0,0) (1,0,0,0,1,0,0) or1 or2 Five reachable states. Traffic lights are safe! 105 Alternative notation rg1 rg2 o1+r2 r1+o2 g1 g2 r1 go1 x go2 r2 r1+r2+x o1 o2 g1+r2 r1+g2 or1 or2 106 Reachability graph (2) • Graph containing a node for each reachable state. • Constructed by starting in the initial state, calculate all directly reachable states, etc. • Expensive technique. • Only feasible if finitely many states (otherwise use coverability graph). • Difficult to generate diagnostic information. 107 Infinite reachability graph rg1 rg2 g1 g2 r1 go1 x go2 r2 o1 o2 or1 or2 108 Exercise: Construct reachability graph free wait enter before make_picture after leave gone occupied 109 Exercise: Dining philosophers (1) • 5 philosophers sharing 5 chopsticks: chopsticks are located in-between philosophers • A philosopher is either in state eating or thinking and needs two chopsticks to eat. • Model as a Petri net. 110 Exercise: Dining philosophers (2) • Assume that philosophers take the chopsticks one by one such that first the right-hand one is taken and then the left-hand one. • Model as a Petri net. • Is there a deadlock? 111 Exercise: Dining philosophers (3) • Assume that philosopher take the chopsticks one by one in any order and with the ability to return a chopstick. • Model as a Petri net. • Is there a deadlock? 112 Structural analysis techniques • To avoid state-explosion problem and bad diagnostics. • Properties independent of initial state. • We only consider place and transition invariants. • Invariants can be computed using linear algebraic techniques. 113 Place invariant man • Assigns a weight to each place. • The weight of a couple token depends on marriage divorce the weight of the place. woman • The weighted token sum is invariant, i.e., no transition can 1 man + 1 woman + 2 couple change it 114 Other invariants • 1 man + 0 woman + 1 couple man (Also denoted as: man + couple) • 2 man + 3 woman + 5 couple couple • -2 man + 3 woman + couple marriage divorce • man – woman • woman – man woman (Any linear combination of invariants is an invariant.) 115 Example: traffic light rg1 rg2 • r1 + g1 + o1 • r2 + g2 + o2 g1 g2 • r1 + r2 + g1 + g2 + o1 + o2 • x + g1 + o1 + g2 + o2 r1 go1 x go2 r2 • r1 + r2 - x o1 o2 or1 or2 116 Exercise: Give place invariants start_production start_consumption free producer wait consumer product end_production end_consumption 117 Transition invariant • Assigns a weight to man each transition. • If each transition fires the number of couple times indicated, the marriage divorce system is back in the initial state. woman • I.e. transition invariants indicate potential firing sets without any net 2 marriage + 2 divorce effect. 118 Other invariants • 1 marriage + 1 divorce man (Also denoted as: marriage + divorce) • 20 marriage + 20 divorce couple Any linear combination of marriage divorce invariants is an invariant, but transition invariants with woman negative weights have no obvious meaning. Invariants may be not be realizable. 119 Example: traffic light rg1 rg2 • rg1 + go1 + or1 • rg2 + go2 + or2 g1 g2 • rg1 + rg2 + go1 + go2 + or1 + or2 r1 go1 x go2 r2 • 4 rg1 + 3 rg2 + 4 go1 + 3 go2 + 4 or1 + 3 or2 o1 o2 or1 or2 120 Exercise: Give transition invariants start_production start_consumption free producer wait consumer product end_production end_consumption 121 Exercise: four philosophers st1 • Give place e1 t1 invariants. se1 • Give c4 c1 transition t4 e2 invariants st4 se4 se2 st2 e4 t2 c3 c2 se3 t3 e3 st3 122 Two ways of calculating invariants • "Intuitive way": Formulate the property that you think holds and verify it. • "Linear-algebraic way": Solve a system of linear equations. Humans tend to do it the intuitive way and computers do it the linear-algebraic way. 123 man Incidence matrix of a Petri net • Each row corresponds to couple a place. marriage divorce • Each column corresponds to a transition. woman • Recall that a Petri net is described by (P,T,I,O). • N(p,t)=O(t,p)-I(p,t) where p is a place and t a −1 1 transition. N= −1 1 1 −1 124 man woman marriage Example man couple −1 1 marriage divorce N= −1 1 woman 1 −1 couple divorce 125 Place invariant • Let N be the incidence matrix of a net with n places and m transitions • Any solution of the equation X.N = 0 is a transition invariant – X is a row vector (i.e., 1 x n matrix) – O is a row vector (i.e., 1 x m matrix) • Note that (0,0,... 0) is always a place invariant. • Basis can be calculated in polynomial time. 126 Example −1 1 X − 1 1 = 0,0 man 1 −1 couple marriage divorce woman Solutions: • (0,0,0) −1 1 • (1,0,1) man,woman, couple − 1 1 = 0,0 • (0,1,1) 1 −1 • (1,1,2) • (1,-1,0) 127 Transition invariant • Let N be the incidence matrix of a net with n places and m transitions • Any solution of the equation N.X = 0 is a place invariant – X is a column vector (i.e., m x 1 matrix) – 0 is a column vector (i.e., n x 1 matrix) • Note that (0,0,... 0)T is always a place invariant. • Basis can be calculated in polynomial time. 128 Example −1 1 0 −1 1 X= 0 man 1 −1 couple marriage woman divorce 0 −1 1 0 Solutions: marriage • (0,0)T −1 1 = 0 divorce • (1,1)T 1 −1 0 • (32,32)T 129 start_production start_consumption Exercise free producer wait consumer product end_production end_consumption • Give incidence matrix. • Calculate/check place invariants. • Calculate/check transition invariants. 130 Simulation • Most widely used analysis technique. • From a technical point of view just a "walk" in the reachability graph. • By making many "walks" (in case of transient behavior) or a very "long walk" (in case of steady-state) behavior, it is possible to make reliable statements about properties/ performance indicators. • Used for validation and performance analysis. • Cannot be used to prove correctness! 131 Stochastic process • Simulation of a deterministic system is not very interesting. • Simulation of an untimed system is not interesting. • In a timed and non-deterministic system, durations and probabilities are described by some probability distribution. • In other words, we simulate a stochastic process! • CPN allows for the use of distributions using some internal random generator. 132 Uniform distribution pdf cumulative 133 Negative exponential distribution 134 Normal distribution 135 Distributions in CPN Tools Assume some library with functions: • uniform(x,y) • nexp(x) • erlang(n,x) • Etc. A nice function is also C.ran() which returns a randomly selected element of finite color set C, e.g., color C = int with 1..5; fun select1to5() = C.ran() returns a number between 1 and 5 136 Example color BT = unit; color Dice = int with 1..6; () () Dice.ran() trigger BT () throw_dice outcome Dice 137 Example(2) color INT = int; color TINT = int timed; color Dice = int with 1..6; color Delay = int with 0..99; 0 x throw_dice trigger Dice.ran() TINT outcome Dice (x+1)@+(Delay.ran()) 138 Subruns and confidence intervals • A single run does not provide information about reliability of results. • Therefore, multiple runs or one run cut into parts: subruns. • If the subruns are assumed to be mutually independent, one can calculate a confidence interval, e.g., the flow time is with 95% confidence within the interval 5.5+/-0.5 (i.e. [5,6]). 139 Example of a simulation model • Gas station with one pump and space for 4 cars (3 waiting and 1 being served). • Service time: uniform distribution between 2 and 5 minutes. • Poisson arrival process with mean time between arrivals of 4 minutes. • If there are more than 3 cars waiting, the "sale" is lost. • Questions: flow time, waiting time, utilization, lost sales, etc. 140 Top-level page: main color Car = string arrive Car HS HS environment gas_station drive_on Car depart Car 141 In arrive Car c c Subpage gas_station [len(q)<3] [len(q)>=3] put_in_queue drive_on q color Car = string; q^^[c] q q color Pump = unit; [] color TCar = Car timed; color Queue = list Car; queue Queue var c:Car; q c::q var q:Queue; fun len(q:Queue) = if q=[] then 0 start else 1+len(tl(q)); c@+uniform(2,5) () c pump_free () fill_up TCar Pump c () end c Out Out depart Car drive_on Car 142 Assuming pages for the environment and measurements the last two pages allow for ... • Calculation of flow time (average, variance, maximum, minimum, service level, etc.). • Calculation of waiting times (average, variance, maximum, minimum, service level, etc.). • Calculation of lost sales (average). • Probability of no space left. • Probability of no cars waiting. For each of these metrics, it is possible to formulate a confidence interval given sufficient observations. 143 In arrive Car c c Alternatives [len(q)<3] [len(q)>=3] put_in_queue drive_on q color Car = string; q^^[c] q q color Pump = unit; [] color TCar = Car timed; color Queue = list Car; queue Queue var c:Car; q c::q var q:Queue; fun len(q:Queue) = if q=[] then 0 start else 1+len(tl(q)); c@+uniform(2,5) () c pump_free () Model the following fill_up TCar Pump alternatives: c () • 5 waiting spaces end • 2 pumps c Out Out • 1 faster pump depart Car drive_on Car 144 Simulation of a Production system S X Y Z C A B C 145 Use distributions Data X Y Z X 2 A 2 5 8 Y 3 B 3 6 9 Z 4 C 4 7 1 Resources per Processing times work center A 2 A 2 A 7 B 1 B 1 B 9 C 2 C 1 C 8 Replenishment Kanbans in- Time in- lead times between work between centers subsequent orders 146 Top level page: main color INT = int; color Prod = string; color PT = product Prod * INT; color PTimed = Prod timed; color PTTimed = PT timed; var p:Prod; var t:INT; 4 5 6 var i:INT; resources_X INT resources_Y INT resources_Z INT 2`"A"++ 2`"A"++ 2`"A"++ 1`"B"++ 1`"B"++ 1`"B"++ 1`"C" 1`"C" 1`"C" Prod Prod Prod Prod HS kanban1 kanban2 kanban3 kanban4 HS work_ work_ work_ supplier center_ HS center_ HS center_ HS customer X Y Z product1 Prod product2 Prod product3 Prod product4 Prod 1`("A",2)++ 1`("A",2)++ 1`("A",5)++ 1`("A",8)++ 1`("A",7)++ 1`("B",1)++ 1`("B",3)++ 1`("B",6)++ 1`("B",9)++ 1`("B",9)++ 1`("C",2) 1`("C",4) 1`("C",7) 1`("C",1) 1`("C",8) io_lead_time PT processing_ PT processing_ PT processing_ PT c_ia_time PTTimed time_X time_Y time_Z supplier work_center work_center work_center customer io_lead_time = io_lead_time processing_time = processing_time = processing_time = c_ia_time = c_ia_time kanban_in = kanban1 processing_time_X processing_time_Y processing_time_Z kanban_out = kanban4 product_out = product1 resources = resources_X resources = resources_Y resources = resources_Z product_in = product4 kanban_in = kanban2 kanban_in = kanban3 kanban_in = kanban4 product_in = product1 product_in = product2 product_in = product3 kanban_out = kanban1 kanban_out = kanban2 kanban_out = kanban3 product_out = product2 product_out = product3 product_out = product4 147 Sub page: supplier accept_order In p (p,t) kanban_in Prod p@+t In/Out (p,t) io_lead_time PT oip PTimed p Out p deliver_order product_out Prod 148 Sub page: customer Out place_order p kanban_out Prod (p,t) (p,t)@+t In/Out c_ia_time PTTimed In p product_in Prod consume 149 Sub page: work_center Out kanban_out Prod p end_proc Out p i+1 product_out Prod p i In/Out wip PTimed INT resources p@+t [i>=1] i-1 In p i (p,t) kanban_in Prod start_proc In/Out In p (p,t) processing_time PT product_in Prod 150 color INT = int; color Prod = string; color PT = product Prod * INT; color PTimed = Prod timed; color PTTimed = PT timed; var p:Prod; var t:INT; 4 5 6 Overview var i:INT; 2`"A"++ 1`"B"++ resources_X INT 2`"A"++ 1`"B"++ resources_Y INT 2`"A"++ resources_Z INT 1`"B"++ 1`"C" 1`"C" 1`"C" Prod Prod Prod Prod HS kanban1 kanban2 kanban3 kanban4 HS Results: supplier work_ center_ X HS work_ center_ Y HS work_ center_ Z HS customer • response time product1 Prod product2 Prod product3 Prod product4 Prod • utilization io_lead_time 1`("A",2)++ 1`("B",1)++ 1`("C",2) PT processing_ 1`("A",2)++ 1`("B",3)++ 1`("C",4) processing_ 1`("A",5)++ 1`("B",6)++ 1`("C",7) PT processing_ 1`("A",8)++ 1`("B",9)++ 1`("C",1) PT c_ia_time 1`("A",7)++ 1`("B",9)++ 1`("C",8) PTTimed • % backorders PT time_X time_Y time_Z supplier work_center work_center work_center customer io_lead_time = io_lead_time processing_time = processing_time = processing_time = c_ia_time = c_ia_time kanban_in = kanban1 processing_time_X processing_time_Y processing_time_Z kanban_out = kanban4 • average stock product_out = product1 resources = resources_X resources = resources_Y resources = resources_Z product_in = product4 kanban_in = kanban2 kanban_in = kanban3 kanban_in = kanban4 product_in = product1 product_in = product2 product_in = product3 kanban_out = kanban1 kanban_out = kanban2 kanban_out = kanban3 product_out = product2 product_out = product3 product_out = product4 • etc. Out accept_order In p place_order kanban_out Prod p Out end_proc p (p,t) kanban_in Prod Out p@+t p (p,t) kanban_out Prod In/Out (p,t) i+1 product_out Prod In/Out (p,t)@+t p i In/Out io_lead_time PT oip PTimed p c_ia_time PTTimed Out wip PTimed INT resources p p@+t In [i>=1] i-1 p In p i deliver_order product_out Prod product_in Prod consume (p,t) kanban_in Prod start_proc In/Out In p (p,t) processing_time PT product_in Prod 151 Classical versus high-level Petri nets • Simulation clearly works for all types of nets. • Hierarchy is never a problem. • Time allows for new types of analysis. • Reachability graphs and invariants can also be extended to high-level nets. – More complex (both technique and computation) • Sometimes abstraction from color is possible to derive invariants (consider previous example). 152 Exercise: Five Chinese philosophers • Recall hierarchical CPN model of five Chinese philosophers alternating between states thinking and eating. – Give place invariants – Give transition invariants • Change the model such that philosophers can take one chopstick at a time but avoid deadlocks and a fixed ordering of philosophers. – Give place invariants – Give transition invariants 153 color BlackToken = unit; var b:BackToken Top-level page philosopher left = CS2 HS right = CS1 () CS2 () CS1 BlackToken PH1 BlackToken HS PH5 PH2 philosopher left = CS1 HS right = CS5 philosopher left = CS3 () CS3 () right = CS2 CS5 BlackToken BlackToken CS4 () PH4 PH3 HS HS philosopher BlackToken philosopher left = CS5 left = CS4 right = CS4 right = CS3 154 Page philosopher think () eat BlackToken BlackToken b b b b take_chopsticks put_down_chopsticks b b b b left right BlackToken In/Out In/Out BlackToken 155 Flat model is obtained by replacing substitution transitions by subpages color BlackToken = unit; var b:BackToken Naming: HS philosopher left = CS2 right = CS1 •PH3.think CS1 BlackToken () CS2 () •PH3.eat PH1 BlackToken •PH3.take_chopsticks PH5 HS philosopher left = CS1 PH2 HS •PH3.put_down_chopsticks right = CS5 philosopher left = CS3 () CS3 () right = CS2 CS5 BlackToken BlackToken think () eat CS4 () BlackToken BlackToken PH4 PH3 HS HS b b philosopher BlackToken philosopher b b left = CS5 left = CS4 take_chopsticks put_down_chopsticks right = CS4 right = CS3 b b b b left right BlackToken In/Out In/Out BlackToken Repeat 5 times... 156 think () eat Alternative page BlackToken BlackToken b b b b start_eating start_thinking b b b b hold_left hold_ right BlackToken BlackToken b b b b return_left take_left take_right return_right b b b b left right BlackToken In/Out In/Out BlackToken 157 You should be able to ... • Construct a reachability graph for a classical Petri net. • Give meaningful place and transition invariants for a classical Petri net. • Construct a reachability graph and give meaningful place and transition invariants for a hierarchical CPN after abstracting from data and time and removing hierarchy. • Build a simple simulation model using CPN. • Motivate the use of each of the analysis techniques. 158