pn and spn - theory, modeling and applications

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Petri nets (PN) and Stochastic Petri

nets (SPN): Theory, Modeling and

Applications

Kamlesh Joshi

Advisor: Dr. Sahra Sedigh

CpE 417 – Network Performance Analysis and Modeling

Semester Project



• Petri nets

Contents

• Introduction

•

Modeling with PN and SPN• Formal definition

• Example system and applications

• Conclusion



• Introduction

• Petri nets are formal graph models that are well

suited for representing the flow of the information

and controls in the system

• PN have been used for the study of the qualitativeproperties of systems exhibiting concurrency and

synchronization characteristics

• Two types of nodes: Circles (Places) and Bars

(Transitions)

• These nodes are connected by “directed arcs”



• Introduction (Contd.)

Fig 1. Simple representation

of PN

• In addition to the static properties, a PN has dynamic properties that result from itsexecution

• The execution of a PN is controlled by “Markers” (Tokens – a black dot, resides in the

places)

• A PN with tokens is called a marked PN

• The distribution of tokens in a marked Petri net defines the state of the system, called

“Markings”

Fig 2. A marked PNFig 3. The marking resulting

from firing transition t2 in Fig

2.



• Modeling with PN

• Petri nets are also a modeling tool, devised for use in the modeling of the class of

discrete –event systems with concurrent or parallel events

• Places are used to represent conditions and Transitions to represent events

Conditions of interest:

1. The processor is idle

2. A job is on the input list

3. A job is being processed

4. A job is on the output list

Events of interest

1. A new job enters the system2. Job processing is started

3. Job processing is completed

4. A job leaves the system Fig 4. PN model for example

of a Computer system



• Modeling with PN (Contd.)

•Properties of PN useful in modeling

1. Concurrency or Parallelism

2. Asynchronous nature – No inherent measure of time or the flow of time in PN

3. Nondeterminism – increases complexity in analysis

4. The firing of a transition is considered to be instantaneous – to reduce the

complexity of the model5. Uninterpreted models

6. Ability to model a system hierarchically

Fig 5. Hierarchical modeling in PN

replacing places or transitions by

subnets

• Extended PN

1. PN that allows multiple arcs are called as Generalized PN2. Zero testing/Inhibitor arcs

Fig 6. PN model of a chemical

reaction



• Formal definition and structure of PN

C = (P, T, I, O) M = (P, T, I, O, μ)1. P: set of Places;

2. T: set of Transitions;

3. The input function I, for each transition t j, the set of input places for the transition

I(t j). The output function O defines, for each transition t j, the set of output places for

the transition O(t j)

4. The vector μ = (μ1, μ2, μ3, μ4,..., μn)

Fig 7. A marked PN

• The Reachability set of a Petri

net1. The reachability set of a marked Petri net is

the set of all states into which the PN can enterby any possible execution

2. From a marking μ, if we can fire some

enabled transition resulting in marking μ’, then

we say that μ’ is reachable from μ

3. Define the reachability set R(M) for a

marked PN as the set of all markings which canbe reached from μ

Example:

P = {p1, p2, p3, p4, p5}, T = {t1,t2,t3,t4}

I(t1) = {p1}, I(t2) = {p2, p3, p5}, . . .

O(t1) = {p2, p3, p5}, O(t2) = {p5}, . . .

μ = (1, 0, 1, 0, 2)



• Stochastic Petri nets (SPN)

Contents• Introduction

• Prerequisites

1. Stochastic processes

2. Continuous - time Markov chain (CTMC)

• Formal definition

• Applications

1. Example system 12. Example system 2



• Introduction•

The use of PN modeling for quantitative analysisof systems requires temporal specifications andcertain characteristics as follows:

1. The transition firing is atomic; i.e. tokens are

removed from input places and put on outputplaces in a single indivisible operation

2. The specification of the firing delay is of probabilistic in nature; hence either pdf or CDF

of the delay associated with a transition needs tobe specified



• Prerequisites

• Stochastic Processes and Markov chain

1. A stochastic process {X(t),t ϵT} is a family of random variable

defined over the same probability space, taking values in the

state space S, and indexed by parameter t, which assumes

values in the set T

2. A Markov process is a stochastic process that satisfies the

Markovian property

3. Markov processes with discrete state space are called Markov

chains

4. Earlier proposals to introduce SPN models were aimed at an

aspect to obtain an equivalence between SPN and CTMC models



• Formal Definition

•An SPN is a six tuple

where

(P, T, I, O, M0) is the earlier definition of PN model

And is an array of firing rates associated withtransitions

Again the firing is assumed to be instantaneous

• The parameter of the pdf associated with transition ti isthe firing rate associated with ti i.e. λi



• Applications• Example System 1

The CTMC is said to ergodic if an equilibrium or steadystate pdf exists and is independent of initial state

4. Denoting by,

is the probability that the CTMC is in state i at time tand is the pmf at time t

Fig 8. State transition

diagram for light bulbexample

Fig 9. SPN Model for example system 1



• Applications (Contd.)• Example system 2

Modeling of a two station single buffer polling system• The place P1represents the condition that

station 1 is idle, and P2 represents the

condition that station 2 is idle.

• The place P5 represents the condition that

the server is serving at station 1, and the

place P6 represents the condition that the

server is serving at station 2.

• The place P3, represents the condition that

station 1 has generated a message, and P4

represents the condition that station 2

has generated a message.• Finally, the place P8, represents the

condition that the server is polling station 2,

and the place P7, represents the condition

that the server is polling station 1

Fig 10. SPN Model for example system 2



• Applications (Contd. )• Corresponding CTMC Model

Where,The time until station i generates a

message is assumed to be

exponentially distributed with mean

1/ λi .

The service time at station i isassumed to be exponentially

distributed with mean

1/ μi.

The polling time of station i is

assumed to be exponentially

distributed with mean

1 /ϒi.

i= 1 , 2.

When station 1 (station 2) generates a

message, the timed transition t1 (t2)

fires.

Fig 11. Corresponding CTMC Model

for example system 2



• Conclusion

• A comprehensive study of concepts of modeling PN and SPN is explained along withthe applications

• The steady state solution of the SPN model is

obtained by same equations for ergodic CTMCmodel

• The complexity while modeling CTMC

considerably reduces with the modelingtechniques using SPN as an alternative

pn and spn - theory, modeling and applications

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