7. Queueing Systems. 8. Petri nets vs. State Automata
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1 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 for... FSA Nondeterminism Data Flow Computation Communication Protocols Hans Vangheluwe Modelling and Simulation: Petri Nets 1/69
2 7. Queueing Systems 8. Petri nets vs. State Automata 9. Analysis of Petri nets Boundedness Liveness and Deadlock State Reachability State Coverability Persistence Language Recognition 10. The Coverability Tree 11. Extensions: colour, time,... Hans Vangheluwe Modelling and Simulation: Petri Nets 2/69
3 X x0 Finite State Automaton E f F E is a finite alphabet X is a finite state set f is a state transition function, f : X E X x 0 is an initial state, x 0 X F is the set of final states Dynamics (x is next state): x f x e Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 3/69
4 FSA graphical notation: State Transition Diagram 1 Init 1 End_ End_0 0 Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 4/69
5 FSA Operational Semantics Rule 1 (priority 3) Locate Initial Current State 1 1 <ANY> ::= 3 <COPIED> 2 Current State Rule 2 (priority 1) State Transition 2 <ANY> 1 Current State 3 5 Rule 3 (priority 2) 4 <ANY> / <ANY> condition: matched(4).input == input[0] <ANY> ::= action: remove(input[0]) 3 <COPIED> 1 Current State 2 4 <COPIED> COPIED> 5 <COPIED> Local State Transition 1 Current State 1 Current State 2 condition: matched(4).input == input[0] 2 4 <ANY> / <ANY> ::= 4 <COPIED> COPIED> 3 3 <ANY> action: remove(input[0]) <COPIED> Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 5/69
6 Simulation steps Hans Vangheluwe Modelling and Simulation: Petri Nets 6/69
7 1 1 Init 1 End_1 Init 1 End_1 0 input 0 0 Rule 1 Rule End_0 End_0 Current State 0 Current State Init 1 End_1 Init 1 End_1 input 1 Rule input 0 Rule End_0 End_0 Current State 0 Current State 0 end of input Final Action "Accept Input" Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 7/69
8 X X State Automaton E Γ f x0 E is a countable event set X is a countable state space Γ x x X is the set of feasible or enabled events Γ x E f is a state transition function, f : X E X, only defined for e Γ x 0 is an initial state, x 0 X x E Γ f omits x 0 and describes a class of State Automata. Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 8/69
9 State Automata for Queueing Systems Arrival Queue Physical View Cashier Departure Departure Arrival [IAT distribution] Queue Cashier [ST distribution] Abstract View Hans Vangheluwe Modelling and Simulation: Petri Nets 9/69
10 State Automata for Queueing Systems: customer centered a a a a a a d d d d d d E a d X Γ x a d x 0 Γ 0 a f x a x 1 x 0 f x d x 1 x 0 Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 10/69
11 State Automata for Queueing Systems: server centered (with breakdown) s I B r c D b Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 11/69
12 c B B D State Automata for Queueing Systems: server centered (with breakdown) E s b r Events: s denotes service starts, c denotes service completes, b denotes breakdown, r denotes repair. X I D State: I denotes idle, B denotes busy, D denotes broken down. Γ I s Γ c b Γ r f I s B f B c I f B b D f D r I Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 12/69
13 Limitiations/extensions of State Automata Adding time? Hierarchical modelling? Concurrency by means of States are represented explicitly Specifying control logic, synchronisation? Hans Vangheluwe Modelling and Simulation: Petri Nets 13/69
14 Petri nets Formalism similar to FSA Graphical notation C.A. Petri 1960s Additions to FSA: Explicitly (graphically) represent when event is enabled describe control logic Elegant notation of concurrency Express non-determinism Hans Vangheluwe Modelling and Simulation: Petri Nets 14/69
15 P T t2 Petri net notation and definition (no dynamics) P A w P p 1 p2 is a finite set of places T t 1 is a finite set of transitions A T T P is a set of arcs w : A is a weight function Note: no need for countable P and T. Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 15/69
16 Derived Entities I t j p i : pi t j A set of input places to transition t j conditions ( for transition) O t j p i : t j pi A set of output places from transition t j affected ( by transition) Transitions events similarly: input- and output-transitions for p i graphical representation: Petri net graph (multigraph) Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 16/69
17 O2 Example Petri net P H2 H 2 O T t A H2 t O2 t t H2O w H2 t 2 w O2 t 1 w t H2O 2 Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 17/69
18 Pure Petri net No self-loops: pi P t T j : pi t j A t j pi A Can convert impure to pure Petri net Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 18/69
19 Impure to Pure Petri net Hans Vangheluwe Modelling and Simulation: Petri Nets 19/69
20 T w p1 pn Introducing State: Petri net Markings Conditions met? Use tokens in places Token assignment marking x x : P A marked Petri net P A x 0 x 0 is the initial marking The state x of a marked Petri net x x x p2 x Number of tokens need not be bounded (cfr. State Automata states). Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 20/69
21 pi State Space of Marked Petri net All n-dimensional vectors of nonnegative integer markings X n Transition t T j is enabled if x w pi t j pi I t j Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 21/69
22 Example with marking, enabled Hans Vangheluwe Modelling and Simulation: Petri Nets 22/69
23 T w pi w pi Petri Net Dynamics State Transition Function f of marked Petri net P A x 0 f : n T n is defined for transition t T j if and only if x w pi t j pi I t j If f x t j is defined, set x f x t j where x x pi p i t j w t j pi State transition function f based on structure of Petri net Number of tokens need not be conserved (but can) Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 23/69
24 2 1 Example firing Use PNS tool braunl/pns/ Select Sequential Manual execution Transition: Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 24/69
25 Example order of firing not determined (due to untimed model) selfloop dead net Hans Vangheluwe Modelling and Simulation: Petri Nets 25/69
26 Conflict, choice, decision Hans Vangheluwe Modelling and Simulation: Petri Nets 26/69
27 Semantics sequential vs. parallel Handle nondeterminism: 1. User choice 2. Priorities 3. Probabilities (Monte Carlo) 4. Reachability Graph (enumerate all choices) Hans Vangheluwe Modelling and Simulation: Petri Nets 27/69
28 Application: Critical Section Hans Vangheluwe Modelling and Simulation: Petri Nets 28/69
29 Reachability Graph t1e [1,0,1,0,1] t2e t1 t2 [0,1,0,0,1] [1,0,0,1,0] Hans Vangheluwe Modelling and Simulation: Petri Nets 29/69
30 w 0 Algebraic Description of Dynamics Firing vector u: transition j firing u Incidence matrix A : a ji w t j pi p i t j State Equation x x ua Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 30/69
31 Infinite Capacity Petri net Add Capacity Constraint: K : P New transition rule Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 31/69
32 p Can transform to infinite capacity net 1. Add complimentary place p with initial marking x 0 p K 2. Between each transition t and complimentary places p add arcs w w p t p t t p w p w t t p p or t where Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 32/69
33 Capacity Constrained Petri net Hans Vangheluwe Modelling and Simulation: Petri Nets 33/69
34 Equivalence proof: use Reachability Graph [1, 0] t4 t1 t2 t1 [2, 0] t3 [0, 0] [0, 1] t1 [1, 1] t1 t2 t4 [2, 1] [p1k2, p2k1] Hans Vangheluwe Modelling and Simulation: Petri Nets 34/69
35 Petri net as State Machine Hans Vangheluwe Modelling and Simulation: Petri Nets 35/69
36 Representing a Petri net as a State Machine Construct Reachability Graph Reachability Graph is State Machine States are tuples p1 p2 pn Events correspond to t i firing May be infinite Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 36/69
37 Representing a State Machine as a Petri net 1. no output 2. with output automatic (though inefficient) transformation Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 37/69
38 FSA without output Hans Vangheluwe Modelling and Simulation: Petri Nets 38/69
39 FSA with output Hans Vangheluwe Modelling and Simulation: Petri Nets 39/69
40 Petri net models for Queueing Systems Arrival Queue Physical View Cashier Departure Departure Arrival [IAT distribution] Queue Cashier [ST distribution] Abstract View Capacity Constraints for Resource Conservation Hans Vangheluwe Modelling and Simulation: Petri Nets 40/69
41 Simple Server/Queue Model Hans Vangheluwe Modelling and Simulation: Petri Nets 41/69
42 Model departure explicitly Hans Vangheluwe Modelling and Simulation: Petri Nets 42/69
43 Model Server Breakdown Hans Vangheluwe Modelling and Simulation: Petri Nets 43/69
44 Modular Composition: Communication Protocol Build incrementally: 1. Single transmitter: FSA vs. Petri net 2. Two transmitters competing for channel Pros/Cons of Petri net models (depends on goals!): Petri net is more complex than FSA for single transmitter More insight Incremental modelling Modular modelling Intuitive modelling of concurrency Hans Vangheluwe Modelling and Simulation: Petri Nets 44/69
45 Single Transmitter FSA ack received transmit I M T arr timeout arr arr Idle Message present Transmitting Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 45/69
46 Single Transmitter Petri net Hans Vangheluwe Modelling and Simulation: Petri Nets 46/69
47 Concurrent, Non-interacting Transmitters Hans Vangheluwe Modelling and Simulation: Petri Nets 47/69
48 Concurrent, Interacting Transmitters Hans Vangheluwe Modelling and Simulation: Petri Nets 48/69
49 Analysis of Petri nets Analysis of logical or qualitative behaviour. Resource sharing fair usage of resources: Boundedness Conservation Liveness and Deadlock State Reachability State Coverability Persistence Language Recognition Hans Vangheluwe Modelling and Simulation: Petri Nets 49/69
50 Boundedness Example: upper bound on number of customers in queue. Definition: A place p P i in a Petri net with initial state x0 is k or bounded k safe if x pi k for all states in all possible sample paths. Abounded 1 place is called safe. If a place is bounded k for some k, the place is bounded. If all places are bounded, the Petri net is bounded. Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 50/69
51 Bounded vs. Unbounded Hans Vangheluwe Modelling and Simulation: Petri Nets 51/69
52 Conservation Token represents resource, process,... Sum Busy Idle tokens must be constant for all states in all sample paths Hans Vangheluwe Modelling and Simulation: Petri Nets 52/69
53 Idle Conservation, weighted sum 2 Transm trschannel constant Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 53/69
54 pi Conservation A Petri net with initial state x 0 is conservative with respect to γ γ 1 γ2 γn if Σ n i1γi x constant for all states in all possible sample paths. Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 54/69
55 Liveness and Deadlock Cyclic dependency wait indefinitely Deadlock Deadlock avoidance: avoid certain states in sample paths Hans Vangheluwe Modelling and Simulation: Petri Nets 55/69
56 Queue1 1 Deadlock in Queueing system with Rework QueueFree Rework 0 1 Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 56/69
57 Deadlock resolved Hans Vangheluwe Modelling and Simulation: Petri Nets 57/69
58 Liveness Given initial state x 0, a transition in a Petri net is: L0-live (dead): if the transition can never fire. L1-live: if there is some firing sequence from x 0 such that the transition can fire at least once. L2-live: if the transition can fire at least k times for some given positive integer k. L3-live: if there exists some infinite firing sequence in which the transition appears infinitely often. L4-live: if the transition is L1-live for every possible state reached from x 0. Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 58/69
59 Liveness example Hans Vangheluwe Modelling and Simulation: Petri Nets 59/69
60 State Reachability A state x in a Petri net is reachable from a state x 0 if there exists a sequence of transitions starting at x 0 such that the state eventually becomes x. Build/use reachability graph. Deadlock avoidance is a special case of reachability. Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 60/69
61 State Coverability In a Petri net with initial state x 0, a state y is coverable if there exists a sequence of transitions starting at x 0 such that the state eventually becomes x and x pi y pi. Related to L1-liveness: minimum number of tokens required to enable a transition. Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 61/69
62 Persistence More than one transition enabled by the same set of conditions (choice, undeterminism). If one fires, does the other remain enabled? A Petri net is persistent if, for any two enabled transitions, the firing of one cannot disable the other. Non-interruptedness (of multiple processes). Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 62/69
63 Language Recognition Language defined by Petri net set of transition sequences which can fire Hans Vangheluwe Modelling and Simulation: Petri Nets 63/69
64 Coverability Notation Root node Terminal node Duplicate node Hans Vangheluwe Modelling and Simulation: Petri Nets 64/69
65 p1 pn p1 pn pi pi pi pi pi pi n pi n Coverability Notation Node dominance x x x p2 x y y y p2 y x d y (x dominates y)if 1. x y i 1 2. x y for at least some i 1 The symbol ω represents infinity x d y For all i such that x y, replace x by ω ω k ω ω k Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 65/69
66 Coverability Tree Construction 1. Initialize x x0 (initial state) 2. Fore each new node x, evaluate the transition function f (a) if f (b) if f x x t j t j x t i for all t T j : is undefined for all t T j, then x is a terminal node. is defined for some t j create a new node x f x t j ω for some pi, set x T,. i. if x pi pi ω. ii. If there exists a node y in the path from root node x 0 (included) to x such that xd y, set x pi ω for all pi such that x pi y pi iii. Otherwise, set x f x 3. Stop if all new nodes are either terminal or duplicate t j. Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 66/69
67 Coverability Tree Example: Cashier/Queue Hans Vangheluwe Modelling and Simulation: Petri Nets 67/69
68 Coverability Tree Example: Cashier/Queue Hans Vangheluwe Modelling and Simulation: Petri Nets 68/69
69 Applications of the Coverability Tree Boundedness: ω does not appear in coverability tree Bounded Petri net reachability graph Conservation: γ i 0 for ω positions Inverse problem: what are γ and C? Coverability: inspect coverability tree Limitations: deadlock detection Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Petri Nets 69/69
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