simulation 4 unit

8/8/2019 Simulation 4 Unit

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UNIT I

INTRODUCTION TO MODELING ANDSIMULATION


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A System is defined as an aggregation or assemblage ofobjects joined in some regular interaction orinterdependence. While this definition is broad enough toinclude static systems, the principal interest will be indynamic systems where the interactions cause changes overtime.

Desire

Heading

Gyroscope Control

Surface

Airframe

Actual Heading

An aircraft under autopilot control

SYSTEM


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SIMULATION EXAMPLE

A FACTORY SYSTEM

PRODUCTION

CONTROL DEPT.

PURCHASING

DEPTFABRICATION

DEPT

ASSEMBLING

DEPT

SHIPPING

DEPT

CUSTOMER

ORDERRAW

MATERIALS

PRODUCT


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SYSTEM ENVIRONMENT

System is affected by changes occurring outside the system.

Such changes occurring Outside the systemare said to occur in system environment.

ENDOGENEOUS

Used to describe activitiesoccurring within the system

EXOGENEOUS

Used to describe activities

In the environment thataffect the system


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ACTIVITIES

DITERMINISTIC

Outcome of activity can bedescribe

completely in terms of input

STOCHASTIC

Effect of activity varyrandomly Over

various possible outcome.

SYSTEM SYSTEM


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CONTINUOUS ANDDISCRETE SYSTEM

In continuoussystem, changesare predominantly

smooth.

Example: Aircraft.

In discrete system,changes arepredominantly

discontinuous.

Example: Factory.


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SYSTEM MODELING

Model is defined as the body ofinformation about a system gathered forthe purpose of studying the system.


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TYPES OF MODEL

PHYSICAL

It is based on analogy

between such systemas mechanical andelectrical. In this,system attributes arerepresented by

measurements such asvoltage or position ofshaft

MATHEMATICAL

It uses symbolic

notation andmathematical equationto represent system.


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Physical Model Mathematical Model

Static Dynamic Static Dynamic

Numerical Analytical Numerical

SystemSimulation

MODEL


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STATIC PHYSICALMODEL

An example of a static physical model is a stick model of

a water molecule, with two small hydrogen "balls" stuck

with short sticks on either side of the oxygen "ball." This

model does not change with time. Another physical

model is that of a tank of water with sand, which shows

the effect of the wind and the movement of water.


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MATHEMATICAL MODEL

A mathematical model is a description of a system

using mathematical language. The process of

developing a mathematical model is termed

mathematical modeling (also written modeling).

Mathematical models are used not only in the natural

science (such as physics, biology, earth science,

meteorology) and engineering disciplines, but also in

the social science (such as economics, psychology,

sociology and political science); economists use

mathematical models most extensively.
http://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Biologyhttp://en.wikipedia.org/wiki/Earth_sciencehttp://en.wikipedia.org/wiki/Meteorologyhttp://en.wikipedia.org/wiki/Economicshttp://en.wikipedia.org/wiki/Psychologyhttp://en.wikipedia.org/wiki/Sociologyhttp://en.wikipedia.org/wiki/Political_sciencehttp://en.wikipedia.org/wiki/Economisthttp://en.wikipedia.org/wiki/Economisthttp://en.wikipedia.org/wiki/Political_sciencehttp://en.wikipedia.org/wiki/Sociologyhttp://en.wikipedia.org/wiki/Psychologyhttp://en.wikipedia.org/wiki/Economicshttp://en.wikipedia.org/wiki/Meteorologyhttp://en.wikipedia.org/wiki/Earth_sciencehttp://en.wikipedia.org/wiki/Biologyhttp://en.wikipedia.org/wiki/Physics


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EXAMPLES OFMATHEMATICAL MODEL

Population Growth.

Model of a particle in a potential-field.

Model of rational behavior for a

consumer.


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STATIC V/S DYNAMICMODEL

Static vs. dynamic: A static model does

not account for the element of time, while a

dynamic model does. Dynamic models

typically are represented with difference

equation or differential equations.


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PRINCIPLES USED INMODELING

Block Building

Description of system should be

organized in series of blocks.

Relevance

Model should only include those aspectsof the system that are relevant to thestudy objectives.


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UNIT II

SYSTEM SIMULATION AND CONTINUOUSSYSTEM SIMULATION


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TECHNIQUE OF SIMULATION

ANALYTICAL NUMERICAL

It produces directlythe general

solution

It produces solutionin

steps

Dynamic Problems Static Problems


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Monte Carlo Simulation

Select numbers randomly from aprobability distribution

Use these values to observe how amodel performs over time

Random numbers each have an equallikelihood of being selected at random


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MONTE-CARLOSIMULATION

.

RANDOM

NUMBER

DISTRIBUTION

RANDOM

VARIABLE

SIMULATION

OUTPUT

REAL

SYSTEM

MODEL


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TYPES OF SYSTEM

CONTINUOUSSYSTEM

In continuoussystem, changesare predominantlysmooth.

Example: Aircraft.

DISCRETE SYSTEM

In discrete system,changes arepredominantlydiscontinuous.

Example: Factory.


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DISTRIBUTED LAGMODEL

Model that have the properties of changingonly at fixed interval of time are calleddistributed lag model.

These are used in economic studies wherethe uniform steps corresponds to a timeinterval, such as month or a year.

These model consist of linear, algebraicequations.


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COBWEB MODEL

The cobweb model or cobweb theory is aneconomic model that explains why prices might

Be subject to periodic fluctuations in certain types

of markets. It describes cyclical supply anddemand in a market where the amount producedmust be chosen before prices are observed.Producers' expectations about prices are assumed

to be based on observations of previous prices.


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Two other possibilities are:

Fluctuations may also remain of constant magnitude,so a plot of the equilibria would produce a simplerectangle, if the supply and demand curves have

exactly the same slope. If the supply curve is less steep than the demand

curve near the point where the two curves cross, butmore steep when we move sufficiently far away, thenprices and quantities will spiral away from theequilibrium price but will not diverge indefinitely;instead, they may converge to a limit cycle.

COBWEB MODEL
http://en.wikipedia.org/wiki/Limit_cyclehttp://en.wikipedia.org/wiki/Limit_cycle


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Continuous SystemModels Continuous system models were the first widely

employed models and are traditionally described byordinary and partial differential equations.

Such models originated in such areas as physics andchemistry, electrical circuits, mechanics, andaeronautics.

They have been extended to many new areas such asbio-informatics, homeland security, and social

systems.

Continuous differential equation models remain anessential component in multi-formalism compositions.


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Analog computer

Analog computer measures andanswer the questions by the methodof HOW MUCH. The input data is not

a number infect a physical quantitylike tem, pressure, speed, velocity.

Signals are continuous of (0 to 10 V)

Accuracy 1% Approximately High speed

Output is continuous

Time is wasted in transmission time


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Digital Computers

Digital computer counts and answer thequestions by the method of HOW Many.The input data is represented by a number.

These are used for the logical andarithmetic operations.

Signals are two level of (0 V or 5 V)

Accuracy unlimited

low speed sequential as well as parallelprocessing

Output is continuous but obtain when

computation is completed.


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Hybrid Computer

The combination of features of analogand digital computer is called Digitalcomputer. The main example are

central national defense andpassenger flight radar system. Theyare also used to control robots.


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UNIT III

SYSTEM DYNAMICS & PROBABILITYCONCEPT IN SIMULATION


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EXPONENTIAL GROWTHMODEL

Exponential growth (including exponential decal occurswhen the growth rate of a mathematical function ispropotional to the function's current value.

Human Population, if the number of births and deaths perperson per year were to remain at current levels

Heat Transfer experiments yield results whose best fit line areexponential growth curves.

Compound Interest at a constant interest rate providesexponential growth of the capital.


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LIMITATIONS

Exponential growth models of physical phenomenaonly apply within limited regions, as unboundedgrowth is not physically realistic. Although growthmay initially be exponential, the modeledphenomena will eventually enter a region in whichpreviously ignored Negative feedback factors

become significant (leading to a Logistic growthmodel) or other underlying assumptions of theexponential growth model, such as continuity orinstantaneous feedback, break down.


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EXPONENTIAL DECAY

MODEL

A quantity is said to be subject to exponential

decay if it decreases at a rate proportional to itsvalue. Symbolically, this process can be modeledby the following differential equation, where Nisthe quantity and (lambda) is a positive number

called the decay constant:

EXPONENTIAL DECAY


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Exponential decay occurs in a wide variety ofsituations. Most of these fall into the domain of thenatural sciences. Any application of mathematics tothe

Social science or humanities is risky and uncertain,because of the extraordinary complexity of humanbehavior. However, a few roughly exponentialphenomena have been identified there as well.Many decay processes that are often treated as

exponential, are really only exponential so long asthe sample is large and the law of large numbersholds. For small samples, a more general analysis isnecessary, accounting for a Poission process.

EXPONENTIAL DECAYMODEL


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APPLICATIONS

Ecology

Neural Network

Statistics In medicine: modeling of growth of

tumors

S t d i


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System dynamicsDiagram

System dynamics is an approach tounderstanding the behaviour ofcomplexsystems over time. It deals with internalfeedback loops and time delays that affect

the behaviour of the entire system.[1]What makes using system dynamicsdifferent from other approaches tostudying complex systems is the use offeedback loops and stocks and flows.

These elements help describe how evenseemingly simple systems display bafflingnonlinearity.
http://en.wikipedia.org/wiki/Complex_systemhttp://en.wikipedia.org/wiki/Complex_systemhttp://en.wikipedia.org/wiki/Feedbackhttp://en.wikipedia.org/wiki/Stock_and_flowhttp://en.wikipedia.org/wiki/Nonlinearityhttp://en.wikipedia.org/wiki/Nonlinearityhttp://en.wikipedia.org/wiki/Stock_and_flowhttp://en.wikipedia.org/wiki/Feedbackhttp://en.wikipedia.org/wiki/Complex_systemhttp://en.wikipedia.org/wiki/Complex_system


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Causal loop diagrams


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Stock and flow diagrams


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MULTI SEGMENT MODEL A multi-segment model is used to investigate optimal

compliant-surface jumping strategies and is applied tospringboard standing jumps. The human model hasfour segments representing the feet, shanks, thighs,and trunkheadarms. A rigid bar with a rotationalspring on one end and a point mass on the other end(the tip) models the springboard. Board tip mass,length, and stiffness are functions of the fulcrumsetting. Body segments and board tip are connectedby frictionless hinge joints and are driven by joint

torque actuators at the ankle, knee, and hip.


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RANDOM NUMBER


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RANDOM NUMBERGENERATION

A random number generator (oftenabbreviated as RNG) is acomputational or physical device

designed to generate a sequence ofnumbers or symbols that lack anypattern, i.e. appear random.

P actical applications
http://en.wikipedia.org/wiki/Computerhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Randomhttp://en.wikipedia.org/wiki/Randomhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Computer


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Practical applicationsand uses

Gambling

Statistical sampling

Computer Simulation Cryptography

Completely randomized design

SIMULATION OF QUEUING SYSTEM


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UNIT IV

SIMULATION OF QUEUING SYSTEMAND DISCRETE SYSTEM SIMULATION


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El t f W iti


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Elements of WaitingLines

Queue is another name for a waitingline.

A waiting line system consists of two

components: The customer population (people or objects

to be processed)

The process or service system

Whenever demand exceeds availablecapacity, a waiting line or queue forms There is a tradeoff between cost and

service level.


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Customer PopulationCharacteristics

Finite versus Infinite populations: Is the number of potential new customers materially

affected by the number of customers already in queue?

Balking When an arriving customer chooses not to enter a queue

because its already too long.

Reneging When a customer already in queue gives up and exits

without being serviced. Jockeying

When a customer switches between alternate queues inan effort to reduce waiting time.


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Service System

The service system is defined by:

The number of waiting lines

The number of servers The arrangement of servers

The arrival and service patterns

The service priority rules


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Number of Lines

Waiting lines systems can havesingle or multiple queues.

Single queues avoid jockeying behaviorand perceived fairness is usually high.

Multiple queues are often used whenarriving customers have differingcharacteristics (e.g. paying with cash,less than 10 items, etc.) and can bereadily segmented.


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Servers

Single servers or multiple, parallelservers providing multiple channels

Arrangement of servers (phases) Multiple phase systems require customers

to visit more than one server

Example of a multi-phase, multi-server

system:

C C C CC DepartArrivals

1

2

3 6

5

4

Phase 1 Phase 2

Example Queuing


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Example QueuingSystems

A i l & S i


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Arrival & ServicePatterns

Arrival rate:

The average number of customers arriving

per time period Modeled using the Poisson distribution

Arrival rate usually denoted by lambda ()

Example: =50 customers/hour; 1/=0.02hours between customer arrivals (1.2 minutesbetween customers)


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Arrival & Service Patterns

Service rate: The average number of customers that can be

served during the period of time

Service times are usually modeled using theexponential distribution

Service rate usually denoted by mu ()

Example: =70 customers/hour; 1/=0.014

hours per customer (0.857 minutes percustomer).

Even if the service rate is larger than thearrival rate, waiting lines form!

Reason is the variation in specific customer


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Example Priority Rules First come, first served

Best customers first (reward loyalty)

Highest profit customers first

Quickest service requirements first

Largest service requirements first

Earliest reservation first Emergencies first

Etc.


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Waiting Line PerformanceMeasures

Lq = The average number of customerswaiting in queue

L = The average number of customersin the system

Wq = The average waiting time inqueue

W= The average time in the system

p = The system utilization rate (% oftime servers are busy)


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Single-Server Waiting Line Assumptions

Customers are patient (no balking, reneging, orjockeying)

Arrivals follow a Poisson distribution with a meanarrival rate of. This means that the timebetween successive customer arrivals follows anexponential distribution with an average of 1/

The service rate is described by a Poissondistribution with a mean service rate of . Thismeans that the service time for one customer

follows an exponential distribution with anaverage of 1/

The waiting line priority rule is first-come, first-served

Infinite population


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Formulas: Single-ServerCase

= lambda= mean arrival rate

=mu= mean service rate

p=

= average system utilization

Note:> for system stability. If this is not the case,

an infinitl lon line will eventuall form.


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Formulas: Single-ServerCase (continued)

L=

= average number of customers in system

Lq =pL=average number of customers in line

W=1

= average time in system including service

Wq =pW=average time spent waiting

Pn= 1 p pn= probability ofn customers in the system

at a given point in time


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Example

A help desk in the computer lab servesstudents on a first-come, first servedbasis. On average, 15 students need

help every hour. The help desk canserve an average of 20 students perhour.

Based on this description, we know: = 20 students/hour (average service time

is 3 minutes)

= 15 students/hour (average timebetween student arrivals is 4 minutes)


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Average Utilization

p=

=

15

20 = 0.75 or75


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h S


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Average Time in the System,and in Line

W=1

=

1

20

15= 0 .2 hours

or 12 minutes

Wq =pW=0 .75 0 .2 = 0 . 15 hours

or 9 minutes

P b bili f


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Probability ofnStudents in the Line

P0= 1 p p0= 1 0 . 75 1= 0.25

P1=

1

p p

1=

1

0. 75 0 . 75=

0.188P2= 1 p p

2= 1 0. 75 0 . 752= 0.141

P3= 1 p p3= 1 0 .75 0 . 75

3= 0.105

P 4= 1 p p4= 1 0 . 75 0 .754= 0.079


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Single Server: SpreadsheetApproach

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A B C

QueuingAnalysis: SingleServer

Inputs

Timeunit hour

Arrival Rate(lambda) 15 customers/hour

ServiceRate(mu) 20 customers/hour

IntermediateCalculations

Averagetimebetweenarrivals 0.066667 hour

Averageservicetime 0.05 hour

PerformanceMeasures

Rho(averageserver utilization) 0.75

P0(probabilitythesystemisempty) 0.25

L(averagenumberinthesystem) 3 customersLq(averagenumber waitinginthequeue) 2.25 customers

W(averagetimeinthesystem) 0.2 hourWq(averagetimeinthequeue) 0.15 hour

Probabilityof aspecificnumber of customersinthesystemNumber 2

Probability 0.140625

Key FormulasB9: =1/B5B10: =1/B6B13: =B5/B6B14: =1-B13B15: =B5/(B6-B5)B16: =B13*B15B17: =1/(B6-B5)B18: =B13*B17B22: =(1-B$13)*(B13^B21)

Use Data Table (trackingB22) to easily computethe probability ofncustomers in the

system.


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Multiple Server Case

Assumptions

Same as Single-Server, except here we

have multiple, parallel servers Single Line

When server finishes with customer, firstperson in line goes to the idle server

All servers are identical


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Multiple Server Formulas

= lambda= mean arrival rate

=mu= mean service rate for one server

s= number of parallel, identical servers

p=

s= average system utilization

Note:s> for system stability. If this is not the case,an infinitly long line will eventually form.


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l l l


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Multiple Server Formulas(continued)

Lq=P

0/

sp

s! 1 p 2=

average number of customers in line

Wq=L

q/=average time spent waiting in line

W=Wq

1

= average time in system including service

L=W= average number of customers in system


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Example: Multiple Server

Computer Lab Help Desk

Now 45 students/hour need help.

3 servers, each with service rate of18 students/hour

Based on this, we know: = 18 students/hour

s = 3 servers

= 45 students/hour

Flexible Spreadsheet Approach


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Flexible Spreadsheet Approach

Formulas are somewhat complex to set up initially, butyou only need to do it once!

For other multiple-server problems, can just change theinput values.

This approach also makes sensitivity analysis possible.

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A B C

QueuingAnalysis: MultipleServers

Inputs

Timeunit hour

Arrival Rate(lambda) 45 customers/hour

ServiceRateper Server (mu) 18 customers/hour

Number of Servers(s) 3 servers

IntermediateCalculations

Averagetimebetweenarrivals 0.022222 hour

Averageservicetimeper server 0.055556 hour

Combinedservicerate(s*mu) 54 customers/hour

PerformanceMeasures

Rho(averageserver utilization) 0.833333P0(probabilitythesystemisempty) 0.044944

L(averagenumberinthesystem) 6.011236 customers

Lq(averagenumber waitinginthequeue) 3.511236 customersW(averagetimeinthesystem) 0.133583 hour

Wq(averagetimeinthequeue) 0.078027 hour

Probabilityof aspecificnumber of customersinthesystemNumber 5

Probability 0.081279

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108

109

E F G H

WorkingCalculations, mainlyfor P0Calculation

lambda/mu 2.5

s! 6

n (/)n n! Sum0 1 1 1

1 2.5 1 3.5

2 6.25 2 6.625

3 15.625 6 9.229166667

4 39.0625 24 10.85677083

5 97.65625 120 11.67057292

6 244.14063 720 12.00965712

7 610.35156 5040 12.13075862

8 1525.8789 40320 12.16860284

9 3814.6973 362880 12.17911512

10 9536.7432 3628800 12.18174319

11 23841.858 39916800 12.18234048

12 59604.645 47900160012.18246492

13 149011.61 6.227E+0912.18248885

14 372529.03 8.718E+1012.18249312

15 931322.57 1.308E+1212.18249383

16 2328306.4 2.092E+1312.18249394

17 5820766.1 3.557E+1412.18249396

18 14551915 6.402E+1512.18249396

99 2.489E+39 9.33E+15512.18249396

100 6.223E+39 9.33E+15712.18249396


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Key Formulas for Spreadsheet

F10: =F$5^E10 (copied down)

G10: =E10*G9 (copied down)

H10: =H9+(F10/G10) (copied down)

F5: =B5/B6

F6: =INDEX(G9:G109,B7+1) B10: =1/B5

B11: =1/B6

B12: =B7*B6

B15: =B5/B12

B16: = (INDEX(H9:H109,B7)+ (((F5^B7)/F6)*((1)/(1-B15))))^(-1)

B17: =B5*B19 B18: =(B16*(F5^B7)*B15)/(INDEX(G9:G109,B7+1)*(1-B15)^2)

B19: =B20+(1/B6)

B20: =B18/B5

B24: =IF(B23


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Probability ofn students in thesystem

Probability of Number in System

0.0000

0.0200

0.0400

0.0600

0.0800

0.1000

0.1200

0.1400

0.1600

0 2 4 6 810

12

14

16

18

20

22

24

26

28

30

Number in System

Probability


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Changing System Performance

Customer Arrival Rates Try to smooth demand through non-peak discounts

or price promotions

Number and type of service facilities Increase or decrease number of servers, or dedicate

specific servers for certain tasks (e.g., express linefor under 10 items)

Change Number of Phases

Can use multi-phase system instead of single phase.This spreads the workload among more servers andmay result in better flow (e.g., fast food restaurantshaving an order phase, pay phase, and pick-upphase during busy hours)


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Server efficiency Add resources to each phase (e.g., bagger

helping a checker at the grocery store)

Use technology (e.g. price scanners) toimprove efficiency

Change priority rules Example: implement a reservation protocol

Change the number of lines Reduce multiple lines to single queue to

avoid jockeying Dedicate specific servers to specific

transactions


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Supplement D Highlights

The elements of a waiting line system include the customerpopulation source, the patience of the customer, the servicesystem, arrival and service distributions, waiting line priorityrules, and system performance measures. Understanding theseelements is critical when analyzing waiting line systems.

Waiting line models allow us to estimate system performance bypredicting average system utilization, average number ofcustomers in the service system, average number of customerswaiting in line, average time a customer waits in line, and theprobability ofn customers in the service system.

The benefit of calculating operational characteristics is to

provide management with information as to whether systemchanges are needed. Management can change the operationalperformance of the waiting line system by altering any or all ofthe following: the customer arrival rates, the number of servicefacilities, the number of phases, server efficiency, the priorityrule, and the number of lines in the system. Based on proposedchanges, management can then evaluate the expectedperformance of the system.


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Customer Arrival Rates Try to smooth demand through non-peak discounts

or price promotions

Number and type of service facilities Increase or decrease number of servers, or dedicate

specific servers for certain tasks (e.g., express linefor under 10 items)

Change Number of Phases

Can use multi-phase system instead of single phase.This spreads the workload among more servers andmay result in better flow (e.g., fast food restaurantshaving an order phase, pay phase, and pick-upphase during busy hours)


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Server efficiency Add resources to each phase (e.g., bagger

helping a checker at the grocery store)

Use technology (e.g. price scanners) toimprove efficiency

Change priority rules Example: implement a reservation protocol

Change the number of lines Reduce multiple lines to single queue to

avoid jockeying Dedicate specific servers to specific

transactions


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Supplement D Highlights

The elements of a waiting line system include the customerpopulation source, the patience of the customer, the servicesystem, arrival and service distributions, waiting line priorityrules, and system performance measures. Understanding theseelements is critical when analyzing waiting line systems.

Waiting line models allow us to estimate system performance bypredicting average system utilization, average number ofcustomers in the service system, average number of customerswaiting in line, average time a customer waits in line, and theprobability ofn customers in the service system.

The benefit of calculating operational characteristics is to

provide management with information as to whether systemchanges are needed. Management can change the operationalperformance of the waiting line system by altering any or all ofthe following: the customer arrival rates, the number of servicefacilities, the number of phases, server efficiency, the priorityrule and the number of lines in the system Based on proposed

simulation 4 unit

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