ALl Lab Writeup SM 2015 2016

7/26/2019 ALl Lab Writeup SM 2015 2016

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KJSCE/IT/BE/SEMVIII/SM/2015-16

(Autonomous College Affiliated to University of Mumbai)

Experiment No. 1

Title: Random Number Generator


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Batch: Roll No.: Experiment No.:1

Aim: To study and implement a PseudoRandom Number Generator (PRNG) using Linear

Congruential Method

Resources needed: Turbo C / Java

Theory

Problem Definition:Write a Program for generating random numbers using Linear Congruential method such that

i) Period of the numbers generated is >=100

ii) Density of the numbers generated is maximum (average gap between random

numbers is < 0.1).

Concepts:

Random Numbers: Random numbers are a necessary basic ingredient in simulation of

almost all discrete systems. Most computer languages have a subroutine, object or function

that will generate a random number. A simulation language generates random numbers that

are used to generate event times and other random variables.

Properties of random Numbers:A sequence of random number R1, R2 must have two important statistical properties,uniformity and independence.

Uniformity :

If the interval (0, 1) is divided inton

classes or subintervals of equal length , the expectednumber of observations in each interval is N/n, where N is total number of observations.

Independence:

The probability of observing a value in a particular interval is independent of the previous

drawn value.

Problems faced in generating random numbers:1. The generated number may not be uniformly distributed.2. The number may be discrete valued instead of continuous values.

3. The mean of the numbers may be too high or low

4. The variance of the number may be too high or low.

5. The numbers may not be independente.g. a. Autocorrelation between numbers

b. Numbers successively higher or lower than adjacent numbers.

Criteria for random no. generator:

1. The routine should be fast.2. The routine should be portable.

3. The routine should have a sufficient long cycle. The cycle length or period represents

the length of random number sequence before previous numbers begin to repeat

themselves in an earlier order. A special case of cycling is degenerating. A routine

degenerates when some number appears repeatedly which is unacceptable.

4. The random number should be replicable.

5. Most important, the generated random numbers should closely approximate to theideal statistical properties of uniformity and independence.


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Procedure:

Linear Congruential Method:The Linear Congruential method produces a sequence of integers X1, X2,between 0 and

m-1 according to the following recursive relationship.X i+1= (a X i + c) mod m , i = 0, 1, 2

The initial value X0 is called the seed, a is constant multiplier, c is the increment and m is themodulus. Maximal period can be achieved by a, c, m, X0 satisfying one of the followingconditions

1. For m, a power of 2 (m = 2b) and c0 period p = 2b is achieved provided c is relatively

prime to m and a = 1+4k , k = 0,1,2,

2. For m = 2b and c = 0 , period p = 2b-2 is achieved provided X0 is odd and multiplier a= 3+8k or a = 5+8k , k = 0,1,2,

3. For m a prime number and c = 0, period p = m-1 is achieved provided a has the

property that the smallest integer is such that a k-1 is divisible by m is k = m-1.

Results: (Program printout with output)

Questions:1. List down a few real life applications using random numbers as input.

2. List the methods for generating random numbers.


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Outcomes:

Conclusion:The random number generator using Linear Congruential method fairly satisfies the

properties of random number.

The set of data that provide adequate periodicity and density are as follows:

m = a = x0 = c =


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The Periodicity and Density obtained are as follows:Periodicity =

Density =

Grade: AA / AB / BB / BC / CC / CD /DD

Signature of faculty in-charge with date

References:

Books/ Journals/ Websites:1. "Linear Congruential Generators"by Joe Bolte,Wolfram Demonstrations Project.

2. Severance, Frank (2001). System Modeling and Simulation. John Wiley & Sons, Ltd. p.86.ISBN 0-471-49694-4.

3. The GNU C library's rand() in stdlib.h uses a simple (single state) linear congruential

generator only in case that the state is declared as 8 bytes. If the state is larger (an array), the

generator becomes an additive feedback generator and the period increases. See thesimplified

code that reproduces the random sequence from this library.

4. "A collection of selected pseudorandom number generators with linear structures, K.

Entacher, 1997". Retrieved 16 June 2012.

5. "How Visual Basic Generates Pseudo-Random Numbers for the RND Function".

Microsoft Support. Microsoft. Retrieved 17 June 2011.

6. In spite of documentation onMSDN, RtlUniform uses LCG, and not Lehmer's algorithm,

implementations before Windows Vista are flawed, because the result of multiplication iscut to 32 bits, before modulo is applied

7. GNU Scientific Library: Other random number generators

8. Novice Forth library
http://demonstrations.wolfram.com/LinearCongruentialGenerators/http://demonstrations.wolfram.com/LinearCongruentialGenerators/http://en.wikipedia.org/wiki/Wolfram_Demonstrations_Projecthttp://en.wikipedia.org/wiki/Wolfram_Demonstrations_Projecthttp://en.wikipedia.org/wiki/Wolfram_Demonstrations_Projecthttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Stdlib.hhttp://en.wikipedia.org/wiki/Stdlib.hhttp://www.mscs.dal.ca/~selinger/random/http://www.mscs.dal.ca/~selinger/random/http://www.mscs.dal.ca/~selinger/random/http://www.mscs.dal.ca/~selinger/random/http://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://support.microsoft.com/kb/231847http://support.microsoft.com/kb/231847http://msdn.microsoft.com/en-us/library/bb432429%28VS.85%29.aspxhttp://msdn.microsoft.com/en-us/library/bb432429%28VS.85%29.aspxhttp://msdn.microsoft.com/en-us/library/bb432429%28VS.85%29.aspxhttp://en.wikipedia.org/wiki/Windows_Vistahttp://en.wikipedia.org/wiki/Windows_Vistahttp://www.gnu.org/software/gsl/manual/html_node/Other-random-number-generators.htmlhttp://www.gnu.org/software/gsl/manual/html_node/Other-random-number-generators.htmlhttp://www.forth.org/novice.htmlhttp://www.forth.org/novice.htmlhttp://www.forth.org/novice.htmlhttp://www.gnu.org/software/gsl/manual/html_node/Other-random-number-generators.htmlhttp://en.wikipedia.org/wiki/Windows_Vistahttp://msdn.microsoft.com/en-us/library/bb432429%28VS.85%29.aspxhttp://support.microsoft.com/kb/231847http://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://www.mscs.dal.ca/~selinger/random/http://www.mscs.dal.ca/~selinger/random/http://en.wikipedia.org/wiki/Stdlib.hhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Wolfram_Demonstrations_Projecthttp://demonstrations.wolfram.com/LinearCongruentialGenerators/


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Experiment No. : 2

Title: Case Study 1

(Technical Paper)


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Experiment No. : 3

Title: Single Server system(The Grocery store problem)


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Aim: To simulate Single Channel Queuing System

Resources needed: Microsoft Excel / Open Office

Problem Statement:A small grocery store has only one checkout counter. Customer arrives at this check outcounter at random from 1 to 8 minutes. The service time varies from 1 to 6 minutes. The

problem is to analyze the system by simulating the arrival & service of 20 customers.

In order to evaluate the system performance the evaluation is done by varying the arrival time

& service time probability distribution.

a) Let the arrival distribution be uniformly distributed between 1 to 10 minutes.

b) Let service time distribution be changedServiceTime(minutes)

1 2 3 4 5 6

Probability 0.05 0.1 0.2 0.3 0.25 0.1

c) Perform simulation for 20 more customers & compare result.

Concepts:The key elements of queuing system are customers & servers. The term customer can refer to

people, machines, and trucks. The server might refer to receptionist, person etc.

A queing system is described by its calling population, the nature of the arrivals, the service

mechanism, the system capacity and the queing discipline.

1. Calling population: The population of potential customer is referred to as calling

population. In systems with large population the calling population is usually assumed to be

infinity. E.g. population of potential customer of a bank. The actual population may be finite.

The main difference between finite & infinite population is based on the definition of the

arrival rate.

2. System Capacity: The system capacity has no limit meaning that any no. of units comes

and waits in the queue. In many queuing system there is a limit to the no. of customers that

may be waiting.

3. Nature of Arrivals: Arrivals for service occur at a time in random fashion and once they

join the waiting line, they are served.

4. Service Mechanism: The units are served in order of their arrival by a single server or a

channel.

5. Queuing Discipline: It refers to the logical ordering of customers that will be chosen for

service when a server becomes busy.

6. State Of System: It is the no. of units in the system & status of server (busy / idle).


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7. Events: It is a set of circumstances that causes an instantaneous change in the state ofsystem

Possible events in a single server system:

The Entry of a unit in the system.

Departure of unit from system on completion of service.

8. Simulation Clock: It is used to track simulation time.

Conceptual Model:The grocery store with one checkout counter is simulated by using a Semi automaticapproach by using simulation table to record successive system snapshots as time advances.

The simulation requires mainly a service time distribution and an interarrival time

distribution of customers.

Characteristics of Grocery Store checkout counter System:I) Calling Population: Infinite.II) System capacity: Infinite.

III) Nature of Arrival: Random arrival uniformly distributed.

IV) Service Mechanism: At a time only one customer is served; service time is random,

probability distribution is given.

V) Queuing Discipline: FIFO.

System State:I) Waiting time in queue.II) Status of Able and Baker (Busy / Idle)

III) Time customer spends in the system.

Entities:

The entities in single channel queue are queue & server.

Events:a) Arrival Event

b) Departure Event

Activities:a) Interarrival time.

b) Service time.

Delay:

Waiting time in the queue.

Use of Random Nos.:

- For generating interarrival time- For generating service time- RAND () function of Excel can be used to generate random nos. in simulation.

Real life Examples:

1. Customers queuing in the Telephone Bill Payment System

Customers form single channel queue.Customer is chosen in FIFO manner.


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Performance measures:

1. Average Waiting Time = (Total time customers wait in queue) / (Total no. ofCustomers)

2. Probability. Of Customers waiting =(No. of Customers who waits) / (Total no. ofCustomers)

3.Probability. Of Idle Server = (Total Idle Time Of Server) / (Total runtime of simulation)

4.Average Time between Arrival = (Total Time between arrivals) / (No. of arrivals)

5.Average Waiting Time of Those Who Wait = (Total Time Customer waits in system) /

(Total no. of Customers)

6.Average Time Customers Spends In System = (Total Time Customer spends in system) /

(Total no. of Customers).

Activity Diagram:

1) Flowchart Arrival event:

Arrival Event

NO YES

Server

Busy?

Unit Enters

service

Unit Enters

queue

2) Flowchart Departure event:

Departure Event

NO YES

Another unit

waiting?

Begin server Idle

Time

Remove waiting

unit from queue


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Results: (Program printout as per the format)

Program:(Problem solved using simulation table and the given inputs with MS-Excel.)

Output:(In printed form i.e. Simulation table along with the performance measures and histogram

showing number of customers against waiting time)

Questions:1) What is simulation? List the different types of simulation.

2) List the queue disciplines used in simulation?


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Outcomes:

Conclusion:



References:

Books/ Journals/ Websites:

1. Jerry Banks, John Carson, Barry Nelson, and David M. Nicol; Discrete Event SystemSimulation, Fifth Edition, Pearson Education.

2. Jerry Banks, John Carson, Barry Nelson, and David M. Nicol; Discrete Event SystemSimulation, Third Edition, Pearson Education.

3. Real Queuing Examples:http://www2.uwindsor.ca/hlynka/qreal.html This site containsexcerpts from news articlesthat deal with aspects of waiting lines.

4. ClearQ :http://clearq.com/ This company produces take-a-number systems for

servicefacilities (e.g., delis), but also providesperformance information about the waitingline.

5. Qmatic:http://us.q-matic.com/index.htmlThiscompany produces informational displaysandother products to keep customers informedabout waiting times.

6. Queuing Presentation by Richard Larson, givenat the Institute for Operations Researchand the Management.

7. ciences:http://caes.mit.edu/people/larson/MontrealINFORMS1/sld001.htm.8. Queuing Tutor :http://www.dcs.ed.ac.uk/home/jeh/Simjava/queueing/mm1_q/mm1_q.html

This site has twoanimated displays of waiting lines. The user canchange arrival and service

rates to see howperformance is affected.

9. Myron Hlynkas Queuing Page:http:www2.uwindsor.ca/hlynka/queue.html This web site

contains information about waiting linesas well as links to other interesting sites.10. Queuing ToolPak:http://www.bus.ualberta.ca/aingolfsson/qtp/11.The Queuing ToolPak is an Excel add-in that allowsyou to easily compute performance

measures fora number of different waiting line models
http://www2.uwindsor.ca/http://clearq.com/http://us.q-matic.com/index.htmlhttp://caes.mit.edu/people/larson/MontrealINFORMS1/sld001.htmhttp://www.dcs.ed.ac.uk/home/jeh/Simjava/queueihttp://www.bus.ualberta.ca/aingolfsson/qtp/http://www.bus.ualberta.ca/aingolfsson/qtp/http://www.dcs.ed.ac.uk/home/jeh/Simjava/queueihttp://caes.mit.edu/people/larson/MontrealINFORMS1/sld001.htmhttp://us.q-matic.com/index.htmlhttp://clearq.com/http://www2.uwindsor.ca/


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Experiment No. : 4

Title: Multi-Serve system (The Able-Baker car hop

problem)


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Aim: To simulate Multi-Serve system (The Able-Baker carhop problem) using Spreadsheet

Resources needed: Microsoft Excel / Open Office

Theory

Problem Statement:Consider a drive in restaurant where carhops take order and bring food to the cars.

Cars arrive in manner as shown:

Time betweenArrival(minutes)

1 2 3 4

Probability 0.25 0.4 0.2 0.15

There are 2 carhops Able & Baker. Able is better to do the job and works a bit faster than

Baker.

Their service distribution is as follows:

Service time Distribution of Able:

ServiceTime(minutes)

2 3 4 5

Probability 0.3 0.28 0.25 0.17

Service time Distribution of Baker:Servicetime(minutes)

3 4 5 6

Probability 0.35 0.25 0.2 0.2

Able gets the customer if both carhops are idle. The problem is to find how well the current

arrangement is working. Simulate the problem for 1 hour.

Conceptual Model:

- Discrete event model of system used for multichannel queuing. e.g. Able & Baker

problem.- A simulation table is used to record the system snapshots as time proceeds.

- The simulation requires mainly an activity table representing a service timedistribution of Able & Baker & interarrival time of customers.

- Activity duration is specified by a modeller.

Characteristics of System:1. Calling Population: Infinite.2. System capacity: Infinite.

3. Nature of Arrival: Random, probability distribution is given.

4. Service Mechanism: At a time maximum two customers can be served one by Able and


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the other by Baker. If Able & Baker both are busy, the customer has to wait. If bothservers are free, priority goes to Able. Service time for both is random; probability

distribution is given.

5. Queuing Discipline: FIFO.

System State:

1. Status of Able (indicating Able being idle or busy at a given instant).

2. Status of Baker (indicating Baker being idle or busy at a given instant).

3. Time customer spends in the queue

4. Time customer spends in the system

Entities:

Neither the customers nor the server needs to be explicitly represented except in terms

of state variable unless customer averages are desired.

Events:

- Arrival Event- Service Completion by Able- Service completion by Baker.

Delay:A customer waits in queue until Able or Baker becomes free.

Use of Random Numbers:

- To generate random numbers in simulation packages, RANDBETWEEN() of Excelcan be used.- In Able & Baker problem random numbers are used for arranging interarrival times

and service times required for customers.

Real life example:

- Scenarios in shopping centre where two cash counter are available.o Here n number of customers pay the bill.o Customer is chosen in FIFO manner.


1. Average Waiting Time = (Total time customers wait in queue) / (Total no. of

Customers)

2. Probability Of Customers waiting = (No. of Customers who waits) / (Total no. of

Customers)

3. Probability Of Idle Server = (Total Idle Time Of Server) / (Total runtime of simulation)

4. Average Service Time = (Total Service Time) / (Total no. of Customers)


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5. Average Time between Arrival = (Total Time between arrivals) / (No. of arrivals)

6. Average Waiting Time Of Those Who Wait = (Total Time Customer waits in system) /

(Total no. of Customers)

7. Average Time Customers Spends In System = (Total Time Customer spends in system) /(Total no. of Customers)

Procedure / Approach /Algorithm / Activity Diagram:Semiautomatic approach by using simulation table.

1) Arrival Event:Arrival Event

Able

Busy?

Server

Busy?

Enter service

from Able

Unit Entersservice

Unit Entersqueue

Results: (Program printout as per the format)

Program:(Problem solved using simulation table and the given inputs with MS-Excel)

Output:(In printed form i.e. Simulation tables along with the performance measures and histogram

showing number of customers against waiting time)

Questions:1. What are the steps carried out for System Simulation using Simulation Table


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2. What is the difference between primary event and secondary event? Give oneexample for each event.


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Outcomes:

Conclusion:( Mention the subjective inference on the basis of the performance measures youhave analyzed )


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References:





4. ClearQ :http://clearq.com/ This company produces take-a-number systems forservicefacilities (e.g., delis), but also providesperformance information about the waiting

line.






9. Myron Hlynkas Queuing Page:http:www2.uwindsor.ca/hlynka/queue.html This web sitecontains information about waiting linesas well as links to other interesting sites.

10. Queuing ToolPak:http://www.bus.ualberta.ca/aingolfsson/qtp/11.The Queuing ToolPak is an Excel add-in that allowsyou to easily compute performance



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Experiment No. : 5

Title: Single Server system (One-operator Barbershop

problem) using a general purpose language


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Aim: To simulate a Single Server system (One-operator Barbershop problem) using a general

purpose language

Resources needed: C / C++ / java

Theory

Problem Statement:Implement a single-server queuing system for which the interarrival times A1,A2.areindependent and identically distributed (IID) random variables. A customer who arrives and

finds the server idle enters service immediately, and service times S1,S2.of the successive

customers are IID random variables that are independent of the interarrival time. A customer

who arrives and finds the server busy joins the end of the single queue. Upon completing

service for a customer, the server chooses a customer from the queue (if any) in a first-in,

first-out (FIFO) manner.

The simulation will begin in the empty-and-idlestate. The simulation should stop when thenth customer enters service.

Concepts:

Software that is used to develop simulation models can be divided into three categories.

First,there are general purpose languages,such as C,C++. Second,there are simulation

programming languages,examples being GPSS,SIMSCRIPT. Third, there are number of

simulation packages such as ARENA.There are several reasons for choosing a general purpose language, rather than more

powerful simulation software, for introducing computer simulation at this point:

By learning to simulate in a general-purpose language, one must pay attention to

every detail.There will be a greater understanding of how simulations actually operate

and thus less chance of conceptual errors if a switch is later made to high-level

simulation package.

Despite the fact that there is now very good and powerful simulation software

available, it is sometimes necessary to write at least parts of complex simulations in a

general-purpose language if the specific, detailed logic of complex systems is to be

represented faithfully. General-purpose languages are widely available, and entire simulations are sometimes

still written in this way.

Characteristics of Queuing System:

The key elements of queuing system are customers & servers. The term customer can refer to

people, machines, and trucks. The server might refer to receptionist, person etc.

1. Calling population: The population of potential customer is referred to as calling

population. In systems with large population the calling population is usually assumed to be

infinity. E.g. population of potential customer of a bank.


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The actual population may be finite. The main difference between finite & infinite populationis based on the definition of the arrival rate.

2. System Capacity: The system capacity has no limit meaning that any no. of units comes

and waits in the queue. In many queuing system there is a limit to the no. of customers that

may be waiting.

3. Nature of Arrivals: Arrivals for service occur at a time in random fashion and once they

join the waiting line, they are served.

4. Service Mechanism: The units are served in order of their arrival by a single server or a

channel.

5. Queuing Discipline: It refers to the logical ordering of customers that will be chosen for

service when a server becomes busy.

6. State Of System: It is the no. of units in the system & status of server (busy / idle).

7. Events: It is a set of circumstances that causes an instantaneous change in the state of

system

Possible events in a single server system:

The Entry of a unit in the system.

Departure of unit from system on completion of service.

8. Simulation Clock: It is used to track simulation time.

Conceptual Model:The Oneoperator barbershop is simulated by using a program written in a general-purposelanguage to record successive system snapshots as time advances. The simulation requires

mainly a service time distribution and an interarrival time distribution of customers.

Characteristics of Grocery Store checkout counter System:1) Calling Population: Infinite.2) System capacity: Infinite.

3) Nature of Arrival: Random arrival exponentially distributed with mean 1 minute

4) Service Mechanism: At a time only one customer is served; service time is random,

exponentially distributed with mean 0.5 minute

5) Queuing Discipline: FIFO.

System State:1) Waiting time in queue.2) Status of operator (Busy / Idle)

3) Time customer spends in the system.

Entities:The entities in single channel queue are queue & server.


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Events:1) Arrival Event2) Departure Event

Activities:

1. Interarrival time.2. Service time.

Delay:Waiting time in the queue.

Stopping event:

1000th customer enters service

Use of Random Nos.:

- For generating interarrival time- For generating service time

Real life Examples:

1. Customers queuing in the Telephone Bill Payment System

Customers form single channel queue.

Customer is chosen in FIFO manner.


1. Average delay in queue =(Total time customers wait in queue) / (Total no. of

Customers)

2. Probability. Of Customers waiting =(No. of Customers who waits) / (Total no. of

Customers)

3. Server utilization = (Total busy Time Of Server) / (Total runtime of simulation)

4. Average Time between Arrival = (Total Time between arrivals) / (No. of arrivals)

5. Average Waiting Time of Those Who Wait = (Total Time Customer waits in

system) / (Total no. of Customers)

6. Average Time Customers Spends In System = (Total Time Customer spends in

system) / (Total no. of Customers)

Time simulation ends


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Activity Diagram:

Flowchart Arrival event:

Arrival event

Schedule the next

arrival event

Is the

server

busy?

Add 1 to the

number in queue

Set delay=0

For this customer and

gather statistics

Is the

queue

full?

Add 1 to the number of

customers delayed

Make the srver busy

Store time of arrival

of this customer

Schedule a departure

event for this customer

Return


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Flowchart Departure event:

Departure

event

Is queue

empty?

Make the server

idle

Substract 1 from the

number in queue

Eliminate departure

event from the

consideration

Compute delay of

customer entering service

and gather statistics

Add 1 to the number of

customers delayed

Schedule a departureevent for this customer

Move each customer in

queue (if any)up one place

Return


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Results: (Program printout with output as per the format)

Questions:

1. Name a few special purpose simulation languages and simulation packages?

2. Name any two features that are relevant when selecting simulating simulation

software.


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Outcomes:

Conclusion:



References:





4. ClearQ :http://clearq.com/ This company produces take-a-number systems forservicefacilities (e.g., delis), but also providesperformance information about the waiting

line.






9. Myron Hlynkas Queuing Page:http:www2.uwindsor.ca/hlynka/queue.html This web sitecontains information about waiting linesas well as links to other interesting sites.

10. Queuing ToolPak:http://www.bus.ualberta.ca/aingolfsson/qtp/11.The Queuing ToolPak is an Excel add-in that allowsyou to easily compute performance



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Experiment No.: 6

Title: Implementation of Uniformity test


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Batch: Roll No.: Experiment No.: 6

Aim: To implement Kolmogorov -Smirnov test or Chi-square test to perform uniformity test

of generated random numbers.


Theory

Problem Statement:Write function in C / C++ / java or macros in MS-excel to implement Kolmogorov-Smirnovtest or Chi-square test.

Concepts:

Random Numbers generated using a known process or algorithm is called Pseudo random

Number.The random numbers generates must possess the property of :

1. Uniformity

2. Independence

Uniformity :

If the interval (0, 1) is divided into nclasses or subintervals of equal length , the expected

number of observations in each interval is N/n, where N is total number of observations.

Independence:


drawn value.

Each random number R must be an independent sample drawn from a continuous uniformdistribution between 0 & 1

i.e.p.d.f. is given byf(x) = 1 0 x 1

0 otherwise

The expected value of each Ri is given by

E(R) = 1

1 0

0x dx = [x2/2 ] = 1/2And variance is given by

1V(R) = 0x2 dx = [x3/3 ] = 1/31/4 = 1/12

Tests for Random numbers

1) Uniformity Test

A basic test that is to be performed to validate a new generator is the test of uniformity. Two

different testing methods are available, they are

a. Kolmogorov- Smirnov Test

b. Chi-square Test


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Both of these measure the degree of agreement between distance of sample of generatedrandom numbers and the theoretical uniform distributions.

1) Kolmogorov-Smirnov Test: This test compares the continuous cdf F(x) of the uniform

distribution to the empirical cdf SN(x) of sample of N distribution

By definition,F(x) = x 0 x 1

If the sample from random no. generated is R1, R2, ,RN then the empirical cdf SN(x) is

defined as

SN(x) =

No. of R1, R2, ,RN which are x

N

As N becomes larger SN(x) should become better approximation to F(x) provided the null

hypothesis is true. The Kolmogorov-Smirnov distance test is best on largest absolute

deviation between F(x) & SN(x) over range of random variable.

2) Chi-square Test: The chi-square test uses sample statistic

2

O i E i

(0)2 =

=1E i

Where, Oi = Observed frequency in ith classEi = Expected frequency in ith class

n = is the no. of classes

Procedure:(Write the algorithm for the test to be implemented and follow the steps given below)Steps:

Implement either Kolmogorov-Smirnov Test or Chi-square Test using C / C++ / java

or macros in MS-excel

Generate 5 sample sets (Each set consisting of 100 random numbers) of Pseudorandom numbers using Linear Congruential Method.

Execute the test using all the five sample sets of random numbers as input and using

=0.05.

Draw conclusions on the acceptance or rejection of the null hypothesis of

independence.



31/50



Questions:

3. List down the pros and cons of the Kolmogorov - Smirnov test and Chi- Square test.

Outcomes:

Conclusion:




32/50



References:


1. "Linear Congruential Generators"by Joe Bolte,Wolfram Demonstrations Project.2. Severance, Frank (2001). System Modeling and Simulation. John Wiley & Sons, Ltd.

p. 86.ISBN 0-471-49694-4.

3. The GNU C library's rand() instdlib.h uses a simple (single state) linear congruential

generator only in case that the state is declared as 8 bytes. If the state is larger (an

array), the generator becomes an additive feedback generator and the period increases.

See thesimplified code that reproduces the random sequence from this library.





6. In spite of documentation on MSDN, RtlUniform uses LCG, and not Lehmer's

algorithm, implementations beforeWindows Vista are flawed, because the result of

multiplication is cut to 32 bits, before modulo is applied


8. Novice Forth library9. Matsumoto, Makoto, and Takuji Nishimura (1998) ACM Transactions on Modeling

and Computer Simulation

10. S.K. Park and K.W. Miller (1988)."Random Number Generators: Good Ones Are

Hard To Find". Communications of the ACM31 (10): 11921201.

doi:10.1145/63039.63042.

11.D. E. Knuth. The Art of Computer Programming, Volume 2: SeminumericalAlgorithms, Third Edition. Addison-Wesley, 1997.ISBN 0-201-89684-2. Section

3.2.1: The Linear Congruential Method, pp. 1026.

12. P. L'Ecuyer (1999)."Tables of Linear Congruential Generators of Different Sizes and

Good Lattice Structure".Mathematics of Computation68 (225): 249260.

doi:10.1090/S0025-5718-99-00996-5.

13. Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007),"Section 7.1.1.

Some History",Numerical Recipes: The Art of Scientific Computing (3rd ed.), New

York: Cambridge University Press,ISBN 978-0-521-88068-8

14. Gentle, James E., (2003).Random Number Generation and Monte Carlo Methods, 2nd

edition, Springer,ISBN 0-387-00178-6.

15. Joan Boyar (1989)."Inferring sequences produced by pseudo-random numbergenerators".Journal of the ACM36 (1): 129141.doi:10.1145/58562.59305. (in this

paper, efficient algorithms are given for inferring sequences produced by certain

pseudo-random number generators).
http://demonstrations.wolfram.com/LinearCongruentialGenerators/http://demonstrations.wolfram.com/LinearCongruentialGenerators/http://demonstrations.wolfram.com/LinearCongruentialGenerators/http://en.wikipedia.org/wiki/Wolfram_Demonstrations_Projecthttp://en.wikipedia.org/wiki/Wolfram_Demonstrations_Projecthttp://en.wikipedia.org/wiki/Wolfram_Demonstrations_Projecthttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Stdlib.hhttp://en.wikipedia.org/wiki/Stdlib.hhttp://en.wikipedia.org/wiki/Stdlib.hhttp://www.mscs.dal.ca/~selinger/random/http://www.mscs.dal.ca/~selinger/random/http://www.mscs.dal.ca/~selinger/random/http://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://support.microsoft.com/kb/231847http://support.microsoft.com/kb/231847http://msdn.microsoft.com/en-us/library/bb432429%28VS.85%29.aspxhttp://msdn.microsoft.com/en-us/library/bb432429%28VS.85%29.aspxhttp://en.wikipedia.org/wiki/Windows_Vistahttp://en.wikipedia.org/wiki/Windows_Vistahttp://en.wikipedia.org/wiki/Windows_Vistahttp://www.gnu.org/software/gsl/manual/html_node/Other-random-number-generators.htmlhttp://www.gnu.org/software/gsl/manual/html_node/Other-random-number-generators.htmlhttp://www.gnu.org/software/gsl/manual/html_node/Other-random-number-generators.htmlhttp://www.forth.org/novice.htmlhttp://www.forth.org/novice.htmlhttp://portal.acm.org/citation.cfm?id=63042http://portal.acm.org/citation.cfm?id=63042http://portal.acm.org/citation.cfm?id=63042http://portal.acm.org/citation.cfm?id=63042http://portal.acm.org/citation.cfm?id=63042http://portal.acm.org/citation.cfm?id=63042http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Donald_Knuthhttp://en.wikipedia.org/wiki/Donald_Knuthhttp://en.wikipedia.org/wiki/Donald_Knuthhttp://en.wikipedia.org/wiki/Special:BookSources/0201896842http://en.wikipedia.org/wiki/Special:BookSources/0201896842http://en.wikipedia.org/wiki/Special:BookSources/0201896842http://citeseer.ist.psu.edu/132363.htmlhttp://citeseer.ist.psu.edu/132363.htmlhttp://citeseer.ist.psu.edu/132363.htmlhttp://citeseer.ist.psu.edu/132363.htmlhttp://citeseer.ist.psu.edu/132363.htmlhttp://citeseer.ist.psu.edu/132363.htmlhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://apps.nrbook.com/empanel/index.html#pg=343http://apps.nrbook.com/empanel/index.html#pg=343http://apps.nrbook.com/empanel/index.html#pg=343http://apps.nrbook.com/empanel/index.html#pg=343http://apps.nrbook.com/empanel/index.html#pg=343http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0387001786http://en.wikipedia.org/wiki/Special:BookSources/0387001786http://en.wikipedia.org/wiki/Special:BookSources/0387001786http://portal.acm.org/citation.cfm?id=59305&dl=ACM&coll=portalhttp://portal.acm.org/citation.cfm?id=59305&dl=ACM&coll=portalhttp://portal.acm.org/citation.cfm?id=59305&dl=ACM&coll=portalhttp://portal.acm.org/citation.cfm?id=59305&dl=ACM&coll=portalhttp://portal.acm.org/citation.cfm?id=59305&dl=ACM&coll=portalhttp://portal.acm.org/citation.cfm?id=59305&dl=ACM&coll=portalhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://portal.acm.org/citation.cfm?id=59305&dl=ACM&coll=portalhttp://portal.acm.org/citation.cfm?id=59305&dl=ACM&coll=portalhttp://en.wikipedia.org/wiki/Special:BookSources/0387001786http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://apps.nrbook.com/empanel/index.html#pg=343http://apps.nrbook.com/empanel/index.html#pg=343http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://citeseer.ist.psu.edu/132363.htmlhttp://citeseer.ist.psu.edu/132363.htmlhttp://en.wikipedia.org/wiki/Special:BookSources/0201896842http://en.wikipedia.org/wiki/Donald_Knuthhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://portal.acm.org/citation.cfm?id=63042http://portal.acm.org/citation.cfm?id=63042http://www.forth.org/novice.htmlhttp://www.gnu.org/software/gsl/manual/html_node/Other-random-number-generators.htmlhttp://en.wikipedia.org/wiki/Windows_Vistahttp://msdn.microsoft.com/en-us/library/bb432429%28VS.85%29.aspxhttp://support.microsoft.com/kb/231847http://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://www.mscs.dal.ca/~selinger/random/http://en.wikipedia.org/wiki/Stdlib.hhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Wolfram_Demonstrations_Projecthttp://demonstrations.wolfram.com/LinearCongruentialGenerators/


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Experiment No.: 7

Title: Implementation of Frequency test


34/50




Aim: To implement Autocorrelation test to perform frequency test for the generated

random numbers.


Theory

Problem Statement:Write function in C / C++ / java or macros in MS-excel to implement Autocorrelation test .

Concepts:

Random Numbers generated using a known process or algorithm is called Pseudo random

Number. The random numbers generates must possess the property of :

3. Uniformity

4. Independence

Uniformity :

If the interval (0, 1) is divided into nclasses or subintervals of equal length , the expected

number of observations in each interval is N/n, where N is total number of observations.

Independence:


drawn value.Each random number R must be an independent sample drawn from a continuous uniform

distribution between 0 & 1

i.e.p.d.f. is given by

f(x) = 1 0 x 1

1 otherwise

The expected value of each Ri is given by

E(R) = 1

1 0

0x dx = [x2/2 ] = 1/2

And variance is given by1

V(R) = 0x2 dx = [x3/3 ] = 1/31/4 = 1/12

Tests for Independence:

These tests are done to check the independence of sequence of random numbers.

1)Runs Test

This test analyses an orderly grouping of numbers in a sequence to test the hypothesis of

independence


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The Runs test used are:i) Runs Up and Down

ii) Runs above and below the mean

iii) Runs test for testing length of runs

2) Auto correlation Test: The test for auto correlation is concerned with dependencebetween numbers in a sequence. The test computes auto correlation between every m

numbers starting with the ith number. Thus autocorrelation limit between following numbers

would be of interest.

Ri, Ri+m, Ri+2m, Ri+(M+1)mwhereM is the largest integer such that i+(M+1)m N where N is total number of values in

the sequence.Since the nonzero autocorrelation implies a lack of independence, the following test is

appropriate:

H 0 :

H1 :

im 0,

im 0,

if numbers are independen t

if numbers are dependent

For large values of M, the distribution of the estimator of im, denoted

normal,if the valuesRi, Ri+m, Ri+2m, Ri+(M+1)m are uncorrelated.

The test statistics is

im is approximately

Z0

im

im

with a mean of 0 and variance of 1,under the assumption of independence , for large M.If -Z/2 Z0 Z/2 , H0 is rejected for the significance level .

3)Gap Test: The gap test is used to determine the significance of the interval between

reoccurrence of the same digit. A gap of length x occurs between reoccurrence of same digit.

4)Poker Test: The poker test for independence is based on frequency with which certain

digits are repeated in a series of numbers in each case a pair of like digits appear in the

numbers that were generated. In 3 digit sample of numbers there are three possibilities which

are as follows:

i) The individual numbers can all be different

ii) The individual numbers can all be same

iii) There can be one pair of like digits.

Procedure:(Write the algorithm for the Autocorrelation test and follow the steps given below)

Steps:

Implement Autocorrelation test using C / C++ / java or macros in MS-excel.

Generate 5 sample sets (Each set consisting of 100 random numbers) of random

numbers using random number generator developed in the first experiment.

Execute the test using all the five sample sets of random number as input and using

=0.05.

Draw conclusions on the acceptance or rejection of the null hypothesis of

independence.


36/50




Questions:

1. Define the hypothesis for

a) Testing of Uniformity

b) Testing of Independence

Outcomes:

Conclusion:


37/50





References:


1. Linear Congruential Generators"by Joe Bolte,Wolfram Demonstrations Project.

2. Severance, Frank (2001). System Modeling and Simulation. John Wiley & Sons, Ltd.

p. 86.ISBN 0-471-49694-4.

3. The GNU C library's rand() instdlib.h uses a simple (single state) linear congruential

generator only in case that the state is declared as 8 bytes. If the state is larger (an

array), the generator becomes an additive feedback generator and the period increases.

See thesimplified code that reproduces the random sequence from this library.





6. In spite of documentation onMSDN, RtlUniform uses LCG, and not Lehmer's

algorithm, implementations beforeWindows Vista are flawed, because the result ofmultiplication is cut to 32 bits, before modulo is applied


8. Novice Forth library

9. Matsumoto, Makoto, and Takuji Nishimura (1998) ACM Transactions on Modeling

and Computer Simulation

10. S.K. Park and K.W. Miller (1988)."Random Number Generators: Good Ones Are

Hard To Find". Communications of the ACM31 (10): 11921201.

doi:10.1145/63039.63042.

11.D. E. Knuth. The Art of Computer Programming, Volume 2: Seminumerical

Algorithms, Third Edition. Addison-Wesley, 1997.ISBN 0-201-89684-2. Section

3.2.1: The Linear Congruential Method, pp. 1026.12. P. L'Ecuyer (1999)."Tables of Linear Congruential Generators of Different Sizes and

Good Lattice Structure".Mathematics of Computation68 (225): 249260.

doi:10.1090/S0025-5718-99-00996-5.

13. Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007),"Section 7.1.1.

Some History",Numerical Recipes: The Art of Scientific Computing (3rd ed.), New

York: Cambridge University Press,ISBN 978-0-521-88068-8

14. Gentle, James E., (2003).Random Number Generation and Monte Carlo Methods,

2nd edition, Springer,ISBN 0-387-00178-6.

15. Joan Boyar (1989)."Inferring sequences produced by pseudo-random number

generators".Journal of the ACM36 (1): 129141.doi:10.1145/58562.59305. (in this

paper, efficient algorithms are given for inferring sequences produced by certainpseudo-random number generators).
http://demonstrations.wolfram.com/LinearCongruentialGenerators/http://demonstrations.wolfram.com/LinearCongruentialGenerators/http://en.wikipedia.org/wiki/Wolfram_Demonstrations_Projecthttp://en.wikipedia.org/wiki/Wolfram_Demonstrations_Projecthttp://en.wikipedia.org/wiki/Wolfram_Demonstrations_Projecthttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Stdlib.hhttp://en.wikipedia.org/wiki/Stdlib.hhttp://en.wikipedia.org/wiki/Stdlib.hhttp://www.mscs.dal.ca/~selinger/random/http://www.mscs.dal.ca/~selinger/random/http://www.mscs.dal.ca/~selinger/random/http://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://support.microsoft.com/kb/231847http://support.microsoft.com/kb/231847http://msdn.microsoft.com/en-us/library/bb432429%28VS.85%29.aspxhttp://msdn.microsoft.com/en-us/library/bb432429%28VS.85%29.aspxhttp://msdn.microsoft.com/en-us/library/bb432429%28VS.85%29.aspxhttp://en.wikipedia.org/wiki/Windows_Vistahttp://en.wikipedia.org/wiki/Windows_Vistahttp://en.wikipedia.org/wiki/Windows_Vistahttp://www.gnu.org/software/gsl/manual/html_node/Other-random-number-generators.htmlhttp://www.gnu.org/software/gsl/manual/html_node/Other-random-number-generators.htmlhttp://www.forth.org/novice.htmlhttp://www.forth.org/novice.htmlhttp://portal.acm.org/citation.cfm?id=63042http://portal.acm.org/citation.cfm?id=63042http://portal.acm.org/citation.cfm?id=63042http://portal.acm.org/citation.cfm?id=63042http://portal.acm.org/citation.cfm?id=63042http://portal.acm.org/citation.cfm?id=63042http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Donald_Knuthhttp://en.wikipedia.org/wiki/Donald_Knuthhttp://en.wikipedia.org/wiki/Donald_Knuthhttp://en.wikipedia.org/wiki/Donald_Knuthhttp://en.wikipedia.org/wiki/Special:BookSources/0201896842http://en.wikipedia.org/wiki/Special:BookSources/0201896842http://en.wikipedia.org/wiki/Special:BookSources/0201896842http://citeseer.ist.psu.edu/132363.htmlhttp://citeseer.ist.psu.edu/132363.htmlhttp://citeseer.ist.psu.edu/132363.htmlhttp://citeseer.ist.psu.edu/132363.htmlhttp://citeseer.ist.psu.edu/132363.htmlhttp://citeseer.ist.psu.edu/132363.htmlhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://apps.nrbook.com/empanel/index.html#pg=343http://apps.nrbook.com/empanel/index.html#pg=343http://apps.nrbook.com/empanel/index.html#pg=343http://apps.nrbook.com/empanel/index.html#pg=343http://apps.nrbook.com/empanel/index.html#pg=343http://apps.nrbook.com/empanel/index.html#pg=343http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0387001786http://en.wikipedia.org/wiki/Special:BookSources/0387001786http://en.wikipedia.org/wiki/Special:BookSources/0387001786http://portal.acm.org/citation.cfm?id=59305&dl=ACM&coll=portalhttp://portal.acm.org/citation.cfm?id=59305&dl=ACM&coll=portalhttp://portal.acm.org/citation.cfm?id=59305&dl=ACM&coll=portalhttp://portal.acm.org/citation.cfm?id=59305&dl=ACM&coll=portalhttp://portal.acm.org/citation.cfm?id=59305&dl=ACM&coll=portalhttp://portal.acm.org/citation.cfm?id=59305&dl=ACM&coll=portalhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://portal.acm.org/citation.cfm?id=59305&dl=ACM&coll=portalhttp://portal.acm.org/citation.cfm?id=59305&dl=ACM&coll=portalhttp://en.wikipedia.org/wiki/Special:BookSources/0387001786http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://apps.nrbook.com/empanel/index.html#pg=343http://apps.nrbook.com/empanel/index.html#pg=343http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://citeseer.ist.psu.edu/132363.htmlhttp://citeseer.ist.psu.edu/132363.htmlhttp://en.wikipedia.org/wiki/Special:BookSources/0201896842http://en.wikipedia.org/wiki/Donald_Knuthhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://portal.acm.org/citation.cfm?id=63042http://portal.acm.org/citation.cfm?id=63042http://www.forth.org/novice.htmlhttp://www.gnu.org/software/gsl/manual/html_node/Other-random-number-generators.htmlhttp://en.wikipedia.org/wiki/Windows_Vistahttp://msdn.microsoft.com/en-us/library/bb432429%28VS.85%29.aspxhttp://support.microsoft.com/kb/231847http://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.53.3686&rep=rep1&type=pdfhttp://www.mscs.dal.ca/~selinger/random/http://en.wikipedia.org/wiki/Stdlib.hhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Wolfram_Demonstrations_Projecthttp://demonstrations.wolfram.com/LinearCongruentialGenerators/


38/50



Experiment No. 8

Title: Simulation of Cafeteria


39/50


40/50



Procedure:(Write the algorithm for the Autocorrelation test and follow the steps given below)

Steps:Starting a model and setting simulation parameters

The following steps are typical when starting any discrete event model.

Open a new model worksheet Give the command Run > Simulation Setup. In the Setup tab enter the simulation

parameters:

o Global time units: seconds

If they arent already open, open the Item, Plotter, and Value libraries Place an Executive block (Item library) on the top left corner of the model worksheet

Open dialog of the Executive block ; control tab; select options;

o Stop Simulation: when count connector value>= ; enter 100.

The Executive block does event scheduling and manages discrete event simulations. It must

be present in every discrete event model.

Start small

In building any simulation model, it is easiest to start with a simple subset of the process andadd detail until you arrive at a completed model that approximates the system thats being

modelled. This allows you to test at various stages while making the model building process

more manageable.

The following table lists the blocks that will be added to the worksheet and their use in the

model.

Except for the Plotter block from the Plotter library and random number block from value

library, the blocks in the table are from the Item library.

Name (Label) Block Function

Create block (customers) Generates items or values, either randomly or on

schedule. If used to generate items, it pushes them into

the simulation and should be followed by a queue-type

block.

Purpose in Cafeteria Model

Generates customers that arrive as per Poisson process

Set (Item > Properties) Attaches user-assigned properties (attribute, priority,

and quantity) to items passing through.


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Set property as preference (sandwich or meal) and further dessert or no dessert

Queue block(Entry Line) Acts as a sorted queue or as a resource pool queue. As a

sorted queue, holds items in FIFO or LIFO order, orsorts items based on their attribute or priority.


Holds the customers and, when the server is available, releases one by one in first-in, first-out

order.

Select I tem I n (Item > Routing) Selects an input and outputs its item.


For merging two inputs (customers coming from sandwich counter and meal counter

towards) and output one (one cashier).

Select I tem Out (Item > Routing) Sends each item it gets to a selected O/P


Apply the routing rule 40% go to sandwich counter, others go to meal counter and the dessert

routing rule 10% have dessert and others dont have.

Activity block(Server) Processes one or more items simultaneously. Processing

time is a constant or is based on a distribution or an

itemsattribute.

Purpose in cafeteria Model

Serves the customers as per the service distribution i.e. exponentially distributed

Random Number block Generates random numbers

Purpose in cafeteria Model

Outputs values to a Set block as per the look up table

Exit (Exit) Removes items from the simulation and counts them as

they leave.


Exits the customers from the model.


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Plotter, Discrete Event


Reports the length of the waiting line and how many jobs has been processed.

Starting at the right of the Executive block, place the blocks on the model worksheet

in a line from left to right, based on their order in the table.

Label the blocks as the system entities.

Enter the dialog parameters and settings for each block

Make the connections

Run the simulation

Verify the results

Animate the model

Results: (Program printout with output)Program: (Printed model developed in Extend Sim)

Output: (Printed results i.e. the plotter data)

Questions:

1. List down the benefits of using a special purpose package such as Extend Sim to

implement and run the simulation problem


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Outcomes:

Conclusion:



References References:




3. ExtendSim 8 Manual (Departmental Library)


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Experiment No.: 9

Title: Simulation of Manufacturing System

Page No:


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Batch: Roll No.: Experiment No.: 9

Aim: To Implement the model for Manufacturing System using Extend Sim and estimate

system performance

Resources needed: Extend Sim 7

Theory

Problem Statement:In a manufacturing unit three types of items (numbered as 1, 2, or 3 depending on the typeof item it will be) arrive on an assembly line exponentially with a mean 1.Out of these 25%

are type 1; 50% are type 2 and 25 % are type 3 respectively. At one step of the assembly

process, there are four machines. Two of the machines can work on all three types, but oneof the machines is old and can work on types 1 and 2, and the fourth machine can only

work on type 3. All the machines process one item at a time and take constant time of one

unit to process each item. Plot the utilization of each machine.

Model the problem as a queuing system.

Animate the model in 2D.

Run the simulation till 480 min I,e 8 hours.

Plot and verify the following results Length of the queue against the number of jobs exited from the system.

Concepts:

A basic discrete event modelThe most common discrete event model involves the handling of one or more waiting linesor queues, such as those found in supermarkets or factories.

Conceptual Model assumptions:

The Job Shop model represents a business operation where Jobs can be processed.

The assumptions for the model are:

Themodel runs for a simulated time of 8 hours (480 minutes)

Arrival of the items is exponentially distributed with a mean of 1.

Processing times for jobs are constant (I unit each)

Theblocks come from the Item, Value, and Plotter libraries


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Procedure:

Starting a model and setting simulation parameters

The following steps are typical when starting any discrete event model.

Open a new model worksheet

Give the command Run > Simulation Setup. In the Setup tab enter the simulationparameters:

o End time: 480o Global time units: minutes

If they arent already open, open the Item, Plotter, and Value libraries Place an Executive block (Item library) on the top left corner of the model

worksheet

The Executive block does event scheduling and manages discrete event simulations. It must

be present in every discrete event model.

Start small

In building any simulation model, it is easiest to start with a simple subset of the process

and add detail until you arrive at a completed model that approximates the system thats

being modelled. This allows you to test at various stages while making the model building

process more manageable.

The following table lists the blocks that will be added to the worksheet and their use in the

model.

Except for the Plotter block from the Plotter library and random number block from value

library, the blocks in the table are from the Item library.

Name (Label) Block Function

Create block (items) Generates items or values, either randomly or on

schedule. If used to generate items, it pushes them

into the simulation and should be followed by a

queue-type block.

Purpose in Manufacturing Model

Generates items that arrive exponentially with mean 1

Set (Item > Properties) Attaches user-assigned properties (attribute, priority,

and quantity) to items passing through.


Set property as Type (1, 2, 3)

Queue block (Buffer) Acts as a sorted queue or as a resource pool Queue.

As a sorted queue, holds items in FIFO or LIFO order,

or sorts items based on their attribute or priority.


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Holds the items based on the type when the machine is available, and releases them one by

one in first-in, first-out order.

Select I tem I n (Item > Routing) Selects an input and outputs its item.


For merging the inputs (items coming from the buffers) and output one (to a particular

machine).

Select I tem Out (Item > Routing) Sends each item it gets to a selected O/P


Apply the routing rule (items to be routed to the specified buffer)

Activity block (Machine) Processes one or more items simultaneously.

Processing time is a constant or is based on a

distribution or an itemsattribute.


Processes the items as per the service time i.e. constant

Random Number block Generates random numbers


Outputs values to a Set block as per the look up table

Exit (Exit) Removes items from the simulation and counts them

as they leave.


Exit the items after being processed from the system.

Starting at the right of the Executive block, place the blocks on the model worksheetin a line from left to right, based on their order in the table.

Label the blocks as the system entities.

Enter the dialog parameters and settings for each block

Make the connections

Run the simulation


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Verify the results

Animate the model

Results: (Program printout with output)Program: (Printed model developed in Extend Sim )Output: (Printed results i.e. the plotter data)

Questions:

1. Give the significance of Validation and verification in simulation?

Page No:


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Outcomes:

Conclusion:



References References:




3. ExtendSim 8 Manual (Departmental Library)


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Experiment No.: 10

Title: Case Study 2

Processor and Memory simulation using virtual lab.

Documents

ALl Lab Writeup SM 2015 2016