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Petri Nets

Sunday, November 18, 2012

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Definition of Petri Net

C = ( P, T, I, O) Places

P = { p1, p2, p3, …, pn} Transitions

T = { t1, t2, t3, …, tn} Input

I : T Pr (r = number of places) Output

O : T Pq (q = number of places)

marking µ : assignment of tokens to the places of Petri net µ = µ1, µ2, µ3, … µn

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Basics of Petri Nets

Petri net is primarily used for studying the dynamic concurrent behavior of network-based systems where there is a discrete flow.

Petri net consist two types of nodes: places and transitions. And arc exists only from a place to a transition or from a transition to a place.

A place may have zero or more tokens.

Graphically, places, transitions, arcs, and tokens are represented respectively by: circles, bars, arrows, and dots.

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Basics of Petri Nets … Below is an example Petri net with two

places and one transaction. Transition node is ready to fire if and

only if there is at least one token at each of its input places

state transition of form (1, 0) (0, 1)p1 : input place p2: output place

p2

p1

t1

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Basics of Petri Nets …

Sequential ExecutionTransition t2 can fire only after the firing of t1. This impose the precedence of constraints "t2 after t1."

SynchronizationTransition t1 will be enabled only when a token there are at least one token at each of its input places.

MergingHappens when tokens from several places arrive for service at the same transition.

p2

t1

p1 p3

t2

t1

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Basics of Petri Nets …

Concurrency t1 and t2 are concurrent. - with this property, Petri net is able to model systems of distributed control with multiple processes executing concurrently in time.

t1

t2

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Basics of Petri Nets …

Conflictt1 and t2 are both ready to fire but the firing of any leads to the disabling of the other transitions.

t1

t2

t1

t2

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Example: In a Restaurant (A Petri Net)

WaiterfreeCustomer 1 Customer 2

Takeorder

Takeorder

Ordertaken

Tellkitchen

wait wait

Serve food Serve food

eating eating

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Example: In a Restaurant (Two Scenarios)

Scenario 1:Waiter takes order from customer 1;

serves customer 1; takes order from customer 2; serves customer 2.

Scenario 2:Waiter takes order from customer 1;

takes order from customer 2; serves customer 2; serves customer 1.

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Example: In a Restaurant (Scenario 1)

WaiterfreeCustomer 1 Customer 2

Takeorder

Takeorder

Ordertaken

Tellkitchen

wait wait

Serve food Serve food

eating eating

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Example: In a Restaurant (Scenario 2)

WaiterfreeCustomer 1 Customer 2

Takeorder

Takeorder

Ordertaken

Tellkitchen

wait wait

Serve food Serve food

eating eating

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Example: Vending Machine (A Petri net)

5c

Take 15c bar

Deposit 5c

0c

Deposit 10c

Deposit 5c

10c

Deposit 10c

Deposit5c

Deposit 10c20c

Deposit5c

15c

Take 20c bar

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Example: Vending Machine (3 Scenarios)

Scenario 1: Deposit 5c, deposit 5c, deposit 5c, deposit 5c,

take 20c snack bar. Scenario 2:

Deposit 10c, deposit 5c, take 15c snack bar. Scenario 3:

Deposit 5c, deposit 10c, deposit 5c, take 20c snack bar.

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Example: Vending Machine (Token Games)

5c

Take 15c bar

Deposit 5c

0c

Deposit 10c

Deposit 5c

10c

Deposit 10c

Deposit5c

Deposit 10c20c

Deposit5c

15c

Take 20c bar

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Petri Net examples (Dining Philosophers)

Five philosophers alternatively think and eating

Chopsticks: p0, p2, p4, p6, p8

Philosophers eating: p10, p11, p12, p13, p14

Philosophers thinking/meditating: p1, p3, p5, p7, p9

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Time in Petri Net

Original model of Petri Net was timeless. Time was not explicitly considered.

Even though there are arguments against the introduction of time, there are several applications that require notion of time.

General approach: Transition is associated with a time for which

no event/firing of a token can occur until this delay time has elapsed.

This delay time can be deterministic or probabilistic.

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Modeling of Time

Constant times Transition occurs at pre-determined times

(deterministic)

Stochastic times Time is determined by some random variable

(probabilistic) Stochastic Petri Nets(SPN)

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Stochastic Petri Nets

An SPN is defined as a 7-tuple SPN= (P, T, I(.), O(.), H(.), W(.), M0) where PN = (P, T, I(.), O(.), H(.), M0) is the P/T system

underlying the SPN Transitions have an exponentially distributed

delay W(.): T --> R assigns a rate to each transition (inverse of the mean firing time)

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Stochastic Petri Nets …

The stochastic process underlying an SPN

is a CTMC in which the state transition rate diagram is isomorphic

to the reachability graph

the transition labels are computed from the

W(.) functions of the transitions enabled in a

state.

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Stochastic Petri Nets …

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Generalized Stochastic Petri Net (GSPN)

Two types of transitions timed with an exponentially distributed delay immediate, with constant zero delay immediate have priority over timed

Why immediate transitions: to account for instantaneous actions (typically

choices) to implement logical actions (e.g. emptying a

place) to account for large time scale differences (e.g.

bus arbitration vs. I/O accesses)

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Generalized Stochastic Petri Net (GSPN) …

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Stochastic Activity Network (SAN)

The desire to represent system characteristics of parallelism and timeliness, as well as fault tolerance and degradable performance, precipitated the development of general level performability models known as stochastic activity networks.

Stochastic activity networks are probabilistic extensions of activity networks the nature of the extension is similar to the extension that constructs stochastic Petri nets from (classical) Petri nets.

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Stochastic Activity Network (SAN) …