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# Classical PETRI NETS Assignment Help

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Basic Introduction To PETRI NETS - Classical PETRI NETS
**CLASSICAL PETRI NETS**

Petri nets, ever as Carl Adam Petri has proposed them in 1962 year, have formed a classical model of concurrency that non-determinism and control flow. Petri nets are bipartite graphs and recommend a mathematically and an elegant rigorous modelling framework for discrete event dynamical systems. Modelling via Petri nets is done employing Petri net objects in a top-down fashion. In order to solving the conflicts happening in the Petri net execution, real time high-level control systems are independently modelled and integrated along with the Petri net models.

*Basic Definitions*

In order to attain an in-depth understanding of Petri nets and its applications, we should first start a few fundamental definitions that are concerned to Petri nets.

Definition 1: The Classical Petri Net

A Petri net is defined as a four-tuple. (P, T, IN, OUT) here,

P = { p_{1} , p_{2 }, ... , p_{n }}

T = {t_{1} , t_{2} , ... , t_{n} }

Above is a set of transitions.

P ∪ T ≠ Φ, P ∩ T = Φ

Here, P is a set of places, and

T is a set of transitions.

IN: (P × T) → N is an input function explain the arcs emanating from places to transitions.

OUT: (P × T) → N is an output function explains the arcs emanating from transitions to source.

By graphically, the places are shown by circles and transitions by vertical or horizontal bars. Places in a Petri net presentation normally model the condition or resources or state while the transitions are employed to depict the activities in a system. In order to generate a clear view we take up an easy example as shown below.

**Illustration .1**

We take up an illustration of a manufacturing system constituted of only a machine and processing a job. All part undergoes an operation upon the machine and after such the part is unloaded from the system and the machine is ready for a part's fresh piece to be processed. Following figure depicts the Petri net model of the manufacturing system discussed.

**Figure: Petri Net Model of a Simple Manufacturing System Constituting of a Single Machine Processing a Job**

** **

** Table no.1: Interpretation of Places and Transitions as Shown in Petri Net of Figure**

**Places**

*p*1

Place representing a raw part

*p*2

Place indicating the machine M is available

*p*3

Place standing for the processing of the part

**T****ransitions**

*t*1

Transition depicting that the machine M starts processing the raw part

*t*2

Transition portraying that the machine M finishes processing of the part

In the manufacturing illustration shown here we have the given:

P = { p_{1} , p_{2} , p_{3}}

T = {t_{1} , t_{2} }

The directed arcs shows the output and the input functions IN and OUT respectively such that,

IN (p_{1}, t_{1}) = 1

But, IN (p3, t1) = 0

In the same way,

OUT (p3, t2) = 1

But, OUT (p3, t1) = 0

Additionally, above Petri net has maximum arc weight as 1 hence can explain the above Petri net in a simpler notation employing a set A such as the Petri net is now explained as the 3-tuple (P, T, A)

Here A is,

A ⊆ (P × T ) ∪ (P × P)

Consider the Petri net above in Figure and determine the set A accordingly. By definition we contain the set A given by:

A = {

(p1, t1), (p2, t1), (p3, t2),

(t1, p3), (t2, p1), (t2, p2)

};

**Definition 2: IP (t**_{j}) and OP (t_{j})

Moreover, for all transition t_{j} we have IP (t_{j}) which consequent to a set of all the places such as the arcs emanating from it end up on the transition t_{j}. Likewise, we have the OP or t_{j} that corresponds to a set of all the places such as the arcs emanating from it end up at the output places.

By using definition we have,

IP (t_{j}) = { p_{i} ∈ P : IN ( p_{i} , t_{j} ) ≠ 0}

OP (t_{j}) = { p_{i} ∈ P : OUT ( p_{i} , t_{j} ) ≠ 0}

Illustration 3

In above illustration 1 fined the IP (t_{1}) and OP (t_{1})

Solution

IP (t_{1}) = {p_{1}, p_{2}}

OP (t_{1}) = {p_{3}}

**Definition 3: IT (p**_{i}) and OT (p_{i})

For every place p_{i} we define IT (p_{i}) like a set of input transitions such as all the arcs emanating from those transitions end up on the place p_{i}. Likewise, we define OT (p_{i}) like the set of output transitions such as all the arcs emanating from pi end up at those transitions.

Hence by definition we have:

IT ( p_{i} ) = {t_{j} ∈ T : OUT ( p_{i} , t_{j} ) ≠ 0}

OT ( p_{i} ) = {t_{j} ∈ T : IN ( p_{i} , t_{j} ) ≠ 0}

Thus proved

(b) From definition 2 and definition 3 as given above we can easily visualize that:

OP (t_{2}) = {p_{1}, p_{2}}

IP (t_{1}) = {p_{1}, p_{2}}

Hence we can say that,

OP (t_{2}) = IP (t_{1}) = {p_{1}, p_{2}}

Thus proved

**Definition 4: Marking of a Petri Net**

A marking of a Petri net is a function M: P → N, here N is a set of natural numbers.

A marked Petri net is a Petri net along with an associated marking.

A marking of a Petri net is a vector (n × 1), here n stands for the number of places in the Petri net. Along with each place specific number of tokens are associated that are shown by dots. Mostly M_{0} is utilized to represent the primary state of the Petri net. Usually the term state and marking are employed interchangeably.

Hence initial marking M_{0} is defined as:

**Illustration .6**

In the Petri net represented in the figure of illustration.1 the initial marking is specified by the following vector as:

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**CLASSICAL PETRI NETS**

*Basic Definitions*_{1}, p

_{2 }, ... , p

_{n }}

_{1}, t

_{2}, ... , t

_{n}}

**Illustration .1**

**Figure: Petri Net Model of a Simple Manufacturing System Constituting of a Single Machine Processing a Job**

**Table no.1: Interpretation of Places and Transitions as Shown in Petri Net of Figure****Places**

*p*1

Place representing a raw part

*p*2

Place indicating the machine M is available

*p*3

Place standing for the processing of the part

**T****ransitions**

*t*1

Transition depicting that the machine M starts processing the raw part

*t*2

Transition portraying that the machine M finishes processing of the part

_{1}, p

_{2}, p

_{3}}

_{1}, t

_{2}}

_{1}, t

_{1}) = 1

**Definition 2: IP (t**

_{j}) and OP (t_{j})_{j}we have IP (t

_{j}) which consequent to a set of all the places such as the arcs emanating from it end up on the transition t

_{j}. Likewise, we have the OP or t

_{j}that corresponds to a set of all the places such as the arcs emanating from it end up at the output places.

_{j}) = { p

_{i}∈ P : IN ( p

_{i}, t

_{j}) ≠ 0}

_{j}) = { p

_{i}∈ P : OUT ( p

_{i}, t

_{j}) ≠ 0}

_{1}) and OP (t

_{1})

_{1}) = {p

_{1}, p

_{2}}

_{1}) = {p

_{3}}

**Definition 3: IT (p**

_{i}) and OT (p_{i})_{i}we define IT (p

_{i}) like a set of input transitions such as all the arcs emanating from those transitions end up on the place p

_{i}. Likewise, we define OT (p

_{i}) like the set of output transitions such as all the arcs emanating from pi end up at those transitions.

_{i}) = {t

_{j}∈ T : OUT ( p

_{i}, t

_{j}) ≠ 0}

_{i}) = {t

_{j}∈ T : IN ( p

_{i}, t

_{j}) ≠ 0}

_{2}) = {p

_{1}, p

_{2}}

_{1}) = {p

_{1}, p

_{2}}

_{2}) = IP (t

_{1}) = {p

_{1}, p

_{2}}

**Definition 4: Marking of a Petri Net**

_{0}is utilized to represent the primary state of the Petri net. Usually the term state and marking are employed interchangeably.

_{0}is defined as:

**Illustration .6**